CN105946863A - Stable vehicle driving zone determining method - Google Patents

Abstract

The invention discloses a stable vehicle driving zone determining method. The stable vehicle driving zone determining method mainly comprises the following steps that step one, a complete vehicle side dynamical model is established; step one, mass center side slip angle speed-mass center side slip angle phase trajectory curve family of a vehicle is drawn; step three, a stable boundary phase trajectory is determined; step four, a stable boundary is fit in the form of a straight line; step five, a three-dimensional map of stable boundary slope and intercept about the vehicle speed and a road adhesion coefficient is made. The stable vehicle driving zone determining method quantitatively represent a stable vehicle driving zone, is simple modeling and has a certain real-timeliness and effectiveness.

Description

A kind of determination method in vehicle run stability region

Technical field

The present invention relates to a kind of determination method in vehicle run stability region, particularly relate to one and pass through side slip angle The process of phase plane stability boundaris three-dimensional map carries out the method that vehicle run stability region determines.

Background technology

Phase plane is a kind of figure solution for nonlinear system movement locus, it is not necessary to carry out nonlinear equation specifically Solve, by the movement locus of drawing system in phase plane, the relation between correlated variables can be observed the most accurately, So that it is determined that the forms of motion that system solves under different initial condition.The system that be in nonlinearity state this for vehicle For, phase plane analysis method is a kind of comprehensively, the intuitively and effectively method characterizing its characteristic.Currently for intact stability Phase plane mainly comprise two kinds, a kind of phase plane trajectory being based on directly on vehicle-state, another kind is to be entered by vehicle-state The deformation of row definite meaning, thus the energy phase plane trajectory formed.The wherein side slip angle speed-side slip angle of vehicle Phase plane directly reflects vehicle and follows driveability for desired trajectory, has important for the stability of vehicle Research Significance.But, the research for this phase plane is extremely limited at present, stablizing of the most systematic description phase plane Property region and instability region, versatility is very poor, therefore can not be well used for analyzing, control and evaluate the steady of vehicle Qualitative characteristics.Under this research background, for the stability region of side slip angle speed-side slip angle phase plane Describe and research becomes particularly important.

Summary of the invention

It is an object of the invention to, for side slip angle speed-side slip angle phasor, propose a kind of novel vehicle row Sail the processing method of stability region, and determine the three-dimensional map of the lateral boundary of stability of vehicle.

It is an object of the invention to be achieved through the following technical solutions,

A kind of determination method in vehicle run stability region, comprises the following steps:

Step one, combine Vehicle Lateral Motion Based mechanism and Unitire UniTire model, set up complete vehicle and laterally move Mechanical model；

Step 2, the vehicle lateral dynamic model set up according to step one, at the beginning of parametric form transmission vehicle-state Value, including side slip angle and side slip angle speed, by parameter assignment input driver intention and surface conditions, i.e. speed And coefficient of road adhesion, under different original states, draw the side slip angle speed-barycenter lateral deviation under vehicle difference operating mode Angle phase path family of curves；

Step 3, according to the side slip angle speed-side slip angle phase under the vehicle difference operating mode obtained in step 2 Figure, utilizes interpolation method to determine the phase path of stability boundaris under every kind of operating mode；

Step 4, the phase path of the stability boundaris obtained in step 3 is carried out linear fit, confront by the form of straight line The stability boundaris of heart yaw angle speed-side slip angle phase plane is described, and obtains the stability boundaris straight line under different operating mode；

Step 5, slope and intercept to the stability boundaris straight line under operating modes different in step 4 collect, and make Stability boundaris slope and intercept are about speed and the three-dimensional map of coefficient of road adhesion, so that it is determined that go out the stability region of vehicle.

Due to the fact that and have employed above-mentioned technical scheme, the present invention has the positive effect that:

1. the present invention carries out the lateral Dynamic Modeling of vehicle based on data, and modeling process is simple and practical, it is possible to complete is anti- Answer the dynamics of vehicle；

2. the present invention uses the method for fitting a straight line to be fitted boundary locus, decreases operand, highly versatile；

3. utilize stability boundaris three-dimensional map that the present invention draws, it is possible to quickly cook up the stability region of vehicle, right The real-time being applied on real vehicle and accuracy have guarantee reliably；

4. inventive algorithm is simple and direct, highly reliable, has saved substantial amounts of resource and cost, for intact stability control and Evaluation lays a solid foundation.

Accompanying drawing explanation

Fig. 1 is double track auto model schematic diagram；

Fig. 2 is vehiclePhase path curve synoptic diagram；

Fig. 3 (a) be coefficient of friction be 0.8, speed is the Phase plane curve figure of 10m/s

Fig. 3 (b) be coefficient of friction be 0.8, speed is the Phase plane curve figure of 20m/s

Fig. 3 (c) be coefficient of friction be 0.8, speed is the Phase plane curve figure of 25m/s

Fig. 3 (d) be coefficient of friction be 0.8, speed is the Phase plane curve figure of 35m/s

Fig. 4 (a) be speed be 30m/s, coefficient of friction is the Phase plane curve figure of 0.1

Fig. 4 (b) be speed be 30m/s, coefficient of friction is the Phase plane curve figure of 0.3

Fig. 4 (c) be speed be 30m/s, coefficient of friction is the Phase plane curve figure of 0.5

Fig. 4 (d) be speed be 30m/s, coefficient of friction is the Phase plane curve figure of 0.7

Fig. 5 is stability boundaris slope k three-dimensional map

Fig. 6 is stability boundaris intercept b three-dimensional map

Detailed description of the invention

Below in conjunction with the accompanying drawings, the technical scheme proposed invention is further elaborated and illustrates.

The present invention provides a kind of determination method in intact stability region, and the method comprises the following steps:

1. combine Vehicle Lateral Motion Based mechanism and Unitire UniTire model, set up complete vehicle lateral dynamics Model

In order to obtain the side slip angle speed-side slip angle phasor of vehicle, the present invention is directed to the weaving of vehicle Set up complete vehicle dynamic model with lateral movement, first make hypothesis below:

Do not consider the impact of wheel steering system, front wheel angle δ_fInput as system；

Ignore the effect of suspension, it is believed that automobile is along the displacement of z-axis, the angle of heel around x-axis with the angle of pitch around y-axis Zero；

Do not consider change and the impact on tire cornering characteristics of the ground tangential force of tyre load；

Ignoring aerodynamic effect, driving force is little；

Automobile is constant along the longitudinal velocity of x-axis.

The present invention uses iso standard vehicle axis system, and zero is located at the barycenter of vehicle, vehicle along headstock to moving ahead The direction sailed is set to x-axis positive direction, and level is the positive direction of y-axis to the left, and z-axis positive direction is perpendicular to x-axis, the plane of y-axis composition, Direction is determined by right-hand screw rule.According to reaching bright BELL'S THEOREM, the power that moment of inertia suffered by vehicle and inertia force cause Square and equal with all moment of face sums suffered by vehicle, thus obtain the moving equilibrium equation of vehicle.

Vehicle is along the resulting side force ∑ F of y-axis_y:

${\mathrm{\ΣF}}_y=m(r\·v_x+{\stackrel{\·}{v}}_y)---\left(1\right)$

The yaw moment sum ∑ M that complete vehicle quality produces around z-axis_z:

${\mathrm{\ΣM}}_z=I_z\·\stackrel{\·}{r}---\left(2\right)$

The most only consider the lateral deviation motion of vehicle, ignore the straight skidding of vehicle, owing to setting longitudinal speed v_xIt is constant, Then side slip angle β and rate of change thereofCan approximate is expressed as:

$\begin{array}{cc}\mathrm{\β}=\frac{v_y}{v_x}& \stackrel{\·}{\mathrm{\β}}=\frac{{\stackrel{\·}{v}}_y}{v_x}\end{array}---\left(3\right)$

Arrange formula (1)-(3), finally give vehicle as follows about the expression formula of yaw velocity r and side slip angle β Shown in formula (4):

$\left\{\begin{array}{c}\stackrel{\·}{r}=\frac{{\mathrm{\ΣM}}_z}{I_z}\\ \stackrel{\·}{\mathrm{\β}}=\frac{{\mathrm{\ΣF}}_y}{m\·v_x}-r\end{array}\right.---\left(4\right)$

According to the simplification double track auto model shown in Fig. 1, the force analysis of comprehensive four tires, resulting side force ∑ F_yWith conjunction Moment ∑ M_zCan be represented by formula (5):

$\begin{array}{c}{\mathrm{\ΣF}}_y=(F_{yfl}+F_{yfr}){\mathrm{cos\δ}}_f+F_{yrl}+F_{yrr}\\ {\mathrm{\ΣM}}_z=F_{yfl}(L_f\·{\mathrm{cos\δ}}_f+\frac{d_1}{2}{\mathrm{sin\δ}}_f)-F_{yrl}{L}_r+F_{yfr}(L_f\·{\mathrm{cos\δ}}_f-\frac{d_1}{2}{\mathrm{sin\δ}}_f)-F_{yrr}{L}_r\end{array}---\left(5\right)$

Wherein: m is complete vehicle quality, I_zIt is the vehicle rotary inertia around z-axis, L_fAnd L_rIt is that vehicle centroid arrives antero posterior axis respectively Wheelbase, d₁Represent the front tread of vehicle, v_xAnd v_yRepresenting the longitudinally and laterally speed of vehicle respectively, β represents side slip angle, r Represent yaw velocity, δ_fRepresent front wheel angle, F_yfl、F_yfr、F_yrl、F_yrrRepresent vehicle the near front wheel, off-front wheel, left rear wheel respectively And the side force of tire of off hind wheel.

In order to react the nonlinear characteristic of vehicle the most accurately, to vehicle about side slip anglePhase plane is entered Row research, here introduces non-linear tire model and is described the side force of tire.Due to speed, coefficient of friction and direction The extraneous factors such as dish corner all can result in tire and leave entrance zone of saturation, the range of linearity, so carrying out phase plane stable region During domain analysis, the whether suitable key factor being to determine vehicle phase plane accuracy that tire model is chosen.At each warp In the tire model of allusion quotation, ' unified index tire model ' that Guo Konghui academician proposes be based on to substantial amounts of tire test result and Mechanics of tire Analysis on Mechanism is set up, it is possible to fully demonstrate the nonlinear characteristic of vehicle tyre under the influence of different factor, be into The ideal tire model of a driving stability analysis, therefore in the research of phase plane, ' unification refers in present invention selection Number ' Unitire UniTire model.

Because only considering the lateral deviation motion of vehicle in the analysis to vehicle phase plane, ignore the straight skidding of vehicle, because of This carries out a series of simplification to Unitire tire model, finally gives side force F of four wheels_yiExpression formula such as following formula (6) Shown in:

$F_{yi}=\{1-\exp \[-|{\mathrm{\Φ}}_{yi}|-E\·{{\mathrm{\Φ}}^2}_{yi}-(E^2+\frac{1}{12})\·|{\mathrm{\Φ}}_{yi}{|}^3\]\}\·\mathrm{\μ}\·F_{zi}\·\mathrm{sgn}\left({\mathrm{\Φ}}_{yi}\right)---\left(6\right)$

Wherein,

In above formula, i={fl, fr, rl, rr}, represent the near front wheel, off-front wheel, left rear wheel and off hind wheel respectively, lower same.μ For surface friction coefficient, E is the structural parameters of tire, K_yiCornering stiffness for tire.

In longitudinal speed v_xConstant, longitudinal acceleration a_xOn the premise of being zero, it is considered to vehicle is at body quality m, laterally accelerate Degree a_yEffect under, vertical load F of each tire_ziShown in expression formula such as following formula (7):

$\begin{array}{c}{F}_{zfl}=m\·g\·\frac{L_r}{2(L_f+L_r)}-m\·a_y\·\frac{h}{d}\·\frac{L_r}{(L_f+L_r)}\\ F_{zfr}=m\·g\·\frac{L_r}{2(L_f+L_r)}+m\·a_y\·\frac{h}{d}\·\frac{L_r}{(L_f+L_r)}\\ F_{zrl}=m\·g\·\frac{L_f}{2(L_f+L_r)}-m\·a_y\·\frac{h}{d}\·\frac{L_f}{(L_f+L_r)}\\ F_{zrr}=m\·g\·\frac{L_f}{2(L_f+L_r)}+m\·a_y\·\frac{h}{d}\·\frac{L_f}{(L_f+L_r)}\end{array}---\left(7\right)$

In above formula, h is automobile height of center of mass, and d is the average wheelspan of front and back wheel, and g is acceleration of gravity.

Lateral acceleration a of vehicle_yExpression formula be:

$a_y=v_x\·(r+\stackrel{\·}{\mathrm{\β}})---\left(8\right)$

Correspondingly, the side drift angle α of each tire of vehicle_iCan be specifically described as follows by formula (9):

$\begin{array}{c}{\mathrm{\α}}_{fl}={\tan }^{-1}\left(\frac{v_x\·\sin \mathrm{\β}+L_f\·r}{v_x\·\cos \mathrm{\β}-\frac{d_1}{2}\·r}\right)-{\mathrm{\δ}}_f\\ {\mathrm{\α}}_{fr}={\tan }^{-1}\left(\frac{v_x\·\sin \mathrm{\β}+L_f\·r}{v_x\·\cos \mathrm{\β}+\frac{d_1}{2}\·r}\right)-{\mathrm{\δ}}_f\\ {\mathrm{\α}}_{rl}={\tan }^{-1}\left(\frac{v_x\·\sin \mathrm{\β}-L_f\·r}{v_x\·\cos \mathrm{\β}-\frac{d_2}{2}\·r}\right)\\ {\mathrm{\α}}_{rr}={\tan }^{-1}\left(\frac{v_x\·\sin \mathrm{\β}-L_r\·r}{v_x\·\cos \mathrm{\β}+\frac{d_2}{2}\·r}\right)\end{array}---\left(9\right)$

Aggregative formula (4)-(9), mate Vehicular system structural parameters, and utilize matlab/simulink module to build instead Answer the complete vehicle dynamic model of vehicle lateral stability.

2. drawPhase path curve

Owing to side slip angle phasor can react the stability status of vehicle really, its stability region is along with extraneous bar The change of part and change, the additional horizontal stroke applied including speed, surface friction coefficient, the steering operation of driver and control system Pendulum moment etc..Owing to, during the control of vehicle stability systems, the change of steering wheel angle is wanted frequently in speed and road surface The change of coefficient of friction, and the yaw moment reasonably applying to add by theory analysis can increase stability region, therefore this Bright considers speed and the surface friction coefficient impact on side slip angle phase plane stability region.

According to the complete vehicle dynamic model set up in step one, first with m function sets operating mode, including speed and Surface friction coefficient, by input vehicle-state initial value i.e. side slip angle and side slip angle speedThrough running Phantom, i.e. can get a phase path curve of vehicle, changes state initial value, under i.e. available a kind of operating mode vehicle about Side slip anglePhase path family of curves.The present invention organizes emulation experiment by running more, finally gives surface friction coefficient For 0.1-1 (being spaced apart 0.1), speed is 60 groups under the straight line operating mode of 10m/s-35m/s (being spaced apart 5m/s)Phase plane Figure.

3. determine stability boundaris phase path

Above-mentioned steps 2 depicts many group vehiclesPhase path curve, this step is with v_x=20m/s, μ=0.1 work The vehicle drawn under condition(as shown in Figure 2) to determining as a example by phase path curveStability boundaris phase path and differentiation The method of stability region and unstable region illustrates.

From figure 2 it can be seen that zero point is the focus of system stability, if phase path finally converges to stable focus, say Bright vehicle finally returns to steady statue, and this phase path is in the stability region of vehicle, if phase path can not return to stable Focus, illustrates that this phase path is in the unstable region of vehicle.When vehicle centroid side drift angle state is positioned at one or three quadrant, its Phase path majority is in divergent state, and is positioned at two four-quadrant phase paths due to the rate of change of side slip angle and barycenter lateral deviation Angle in opposite direction, makes side slip angle have the trend of reduction, so majority belongs to stability region.The method utilizing interpolation determines Shown in figure two phase path curve l₁And l₂It is the theoretical stability boundaris phase path of this phase plane.These two curves are by phase Plane is divided into two regions, i.e. stability region and unstable region.Every original state is positioned at stability region, vehicle equal Surface curve finally can converge to stable focus by own dynamics characteristic, and the movement locus converted in time can court always The Long-term change trend making the absolute value of side slip angle and side slip angle rate of change constantly diminish, the vehicle in region is in Steady statue.And the phase path curve movement that original state is positioned at unstable region can be dissipated into infinity, it is impossible to converge to Metastable focus, phase path has the trend making side slip angle and side slip angle rate of change absolute value increase, or needs A very long time to be experienced just can return to stable state, now defines vehicle and has been in instability status.

Owing to phase plane trajectory has symmetric property, border phase path curve l₁And l₂About origin symmetry.Therefore by really FixedStability boundaris phase path l₁, i.e. may utilize symmetric property and determineStability boundaris phase path l₂.Utilize the method, this Invention is for 60 groups drawn in step 2Often group phase plane stability boundaris phase path l determined by phase-plane diagram₁。

4. fitting a straight line stability boundaris

Step 3 utilizes the method for interpolation to determinePhase plane stability boundaris phase path, in order to the simplest and accurate The stability boundaris to side slip angle phase plane be described, in this step, determine method according to conservative, with straight line Form is to theoretical stability boundaris track l₁It is fitted.Choose the such as A (x of the characteristic point in the phase path of border₁,y₁),B(x₂,y₂), and With form such asStraight line L₁To stability boundaris track l₁It is described, wherein slopeInterceptL the most at last₁As theoretical stability boundaris lower limit.Owing to phase path has the character of symmetry, the most rightSymmetrical stability boundaris phase path l of phase plane₂Carry out stability boundaris upper limit L that fitting a straight line obtains₂ForTwo Region between bar straight line is stability region, can be described asArticle two, the region beyond straight line is range of instability Territory, is described as

Utilize the method stability boundaris phase path l to the phase plane under speeds different in step 3 and surface friction coefficient₁ It is fitted, and determines the value of corresponding k and b.

Fig. 3 and Fig. 4 sets forth the vehicle under the conditions of different speed and surface friction coefficientPhase path figure, and refer to Go out stability boundaris phase path l₁With the stability boundaris L after matching₁.It can be seen that in the identical situation of other conditions Under,Stability region scope can reduce, stability boundaris phase rail along with the increase of speed and the reduction of surface friction coefficient Mark can move right with the intersection point of transverse axis, and stability region also can reduce therewith.

5. make stability boundaris slope and intercept three-dimensional map

Combining step 2,3,4, is 0.1-1 (being spaced apart 0.1) for surface friction coefficient, speed be 10m/s-35m/s ( Be divided into 5m/s) straight line operating mode under 60 groupsPhase-plane diagram, obtains 60 groups of differences v_x, stability boundaris L under μ₁Slope k and The value of intercept b, as shown in table 1, table 2.

Table 1 slope k and v_x, the mapping table of μ

Table 2 intercept b and v_x, the mapping table of μ

The method utilizing linear interpolation makes x-axis respectively for speed v_x, y-axis is surface friction coefficient μ, and z-axis is that straight line is oblique Rate k and x-axis are speed v_x, y-axis is surface friction coefficient μ, and z-axis is the three-dimensional map of Linear intercept b, as shown in Figure 5 and Figure 6.This Sample can find vehicle about side slip angle by searching mapThe slope k of phase plane stability boundaris lower limit and intercept B, so that it is determined that intact stability region is

Claims (2)

1. the determination method in a vehicle run stability region, it is characterised in that comprise the following steps:

Step one, combine Vehicle Lateral Motion Based mechanism and Unitire UniTire model, set up complete vehicle lateral dynamics Model；

Step 2, the vehicle lateral dynamic model set up according to step one, transmit vehicle-state initial value, bag by parametric form Include side slip angle and side slip angle speed, by parameter assignment input driver intention and surface conditions, i.e. speed and road Face attachment coefficient, under different original states, draws the side slip angle speed-side slip angle phase under vehicle difference operating mode Geometric locus race；

Step 3, according to the side slip angle speed-side slip angle phasor under the vehicle difference operating mode obtained in step 2, profit The phase path of stability boundaris under every kind of operating mode is determined with interpolation method；

Step 4, the phase path of the stability boundaris obtained in step 3 is carried out linear fit, by the form of straight line to barycenter side The stability boundaris of drift angle speed-side slip angle phase plane is described, and obtains the stability boundaris straight line under different operating mode；

Step 5, slope and intercept to the stability boundaris straight line under operating modes different in step 4 collect, and make stable Border slope and intercept are about speed and the three-dimensional map of coefficient of road adhesion, so that it is determined that go out the stability region of vehicle.

The determination method in a kind of vehicle run stability region the most as claimed in claim 1, it is characterised in that described step one The process setting up complete vehicle lateral dynamic model includes:

Vehicle about the expression formula of yaw velocity r and side slip angle β is:

$\left\{\begin{array}{c}\stackrel{\·}{r}=\frac{{\mathrm{\ΣM}}_z}{I_z}\\ \stackrel{\·}{\mathrm{\β}}=\frac{{\mathrm{\ΣF}}_y}{m\·v_x}-r\end{array}\right.$

The force analysis of comprehensive four tires, resulting side force ∑ F_yWith resultant moment ∑ M_zIt is expressed as:

∑F_y=(F_yfl+F_yfr)cosδ_f+F_yrl+F_yrr

${\mathrm{\ΣM}}_z=F_{yfl}(L_f\·{\mathrm{cos\δ}}_f+\frac{d_1}{2}{\mathrm{sin\δ}}_f)-F_{yrl}{L}_r+F_{yfr}(L_f\·{\mathrm{cos\δ}}_f-\frac{d_1}{2}{\mathrm{sin\δ}}_f)-F_{yrr}{L}_r$

Wherein: m is complete vehicle quality, I_zIt is the vehicle rotary inertia around z-axis, L_fAnd L_rIt is the vehicle centroid axle that arrives antero posterior axis respectively Away from, d₁Represent the front tread of vehicle, v_xAnd v_yRepresenting the longitudinally and laterally speed of vehicle respectively, β represents side slip angle, and r represents Yaw velocity, δ_fRepresent front wheel angle, F_yfl、F_yfr、F_yrl、F_yrrRespectively represent vehicle the near front wheel, off-front wheel, left rear wheel and The side force of tire of off hind wheel；

To Unitire UniTire model, finally give side force F of four wheels_yiIt is expressed as:

$F_{yi}=\{1-\exp \[-|{\mathrm{\Φ}}_{yi}|-E\·{{\mathrm{\Φ}}^2}_{yi}-(E^2+\frac{1}{12})\·|{\mathrm{\Φ}}_{yi}{|}^3\]\}\·\mathrm{\μ}\·F_{zi}\·\mathrm{sgn}\left({\mathrm{\Φ}}_{yi}\right)$

Wherein,

${\mathrm{\Φ}}_{yi}=\frac{-K_{yi}\·{\mathrm{tan\α}}_i}{\mathrm{\μ}\·F_{zi}}$

In above formula, i={fl, fr, rl, rr}, represent the near front wheel, off-front wheel, left rear wheel and off hind wheel respectively；μ is to rub in road surface Wipe coefficient；E is the structural parameters of tire；K_yiCornering stiffness for tire；

In longitudinal speed v_xConstant, longitudinal acceleration a_xOn the premise of being zero, it is considered to vehicle is in body quality m, lateral acceleration a_y Effect under, vertical load F of each tire_ziExpression formula is shown below:

$F_{zfl}=m\·g\·\frac{L_r}{2(L_f+L_r)}-m\·a_y\·\frac{h}{d}\·\frac{L_r}{(L_f+L_r)}$

$F_{zfr}=m\·g\·\frac{L_r}{2(L_f+L_r)}+m\·a_y\·\frac{h}{d}\·\frac{L_r}{(L_f+L_r)}$

$F_{zrl}=m\·g\·\frac{L_f}{2(L_f+L_r)}-m\·a_y\·\frac{h}{d}\·\frac{L_f}{(L_f+L_r)}$

$F_{zrr}=m\·g\·\frac{L_f}{2(L_f+L_r)}+m\·a_y\·\frac{h}{d}\·\frac{L_f}{(L_f+L_r)}$

In formula, h is automobile height of center of mass, and d is the average wheelspan of front and back wheel, and g is acceleration of gravity；

Lateral acceleration a of vehicle_yExpression formula be:

$a_y=v_x\·(r+\stackrel{\·}{\mathrm{\β}})$

Correspondingly, the side drift angle α of each tire of vehicle_iCan specifically be expressed by following formula:

${\mathrm{\α}}_{fl}={\tan }^{-1}\left(\frac{v_x\·sin\mathrm{\β}+L_f\·r}{v_x\·cos\mathrm{\β}-\frac{d_1}{2}\·r}\right)-{\mathrm{\δ}}_f$

${\mathrm{\α}}_{fr}={\tan }^{-1}\left(\frac{v_x\·sin\mathrm{\β}+L_f\·r}{v_x\·cos\mathrm{\β}+\frac{d_1}{2}\·r}\right)-{\mathrm{\δ}}_f$

${\mathrm{\α}}_{rl}={\tan }^{-1}\left(\frac{v_x\·sin\mathrm{\β}-L_r\·r}{v_x\·\cos \mathrm{\β}-\frac{d_2}{2}\·r}\right)$

${\mathrm{\α}}_{rr}={\tan }^{-1}\left(\frac{v_x\·sin\mathrm{\β}-L_r\·r}{v_x\·\cos \mathrm{\β}+\frac{d_2}{2}\·r}\right)$

Comprehensive above each formula, mates Vehicular system structural parameters, and the complete vehicle building reaction vehicle lateral stability is lateral Kinetic model.