Vehicle brake control method considering comfort

By establishing the vehicle braking state equation defined by the seven-degree polynomial, calculating the minimum value of the vehicle acceleration, and establishing a vehicle braking control equation that considers comfort, the shortcomings of the existing AEB system when combining emergency braking and comfort are solved, and the comfort improvement and maximum impact reduction during the vehicle braking process are achieved.

CN119659558APending Publication Date: 2025-03-21CHONGQING UNIV OF TECH
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Patent Information

Application Number
CN202411846069.6
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2024-12-16
Publication Date
2025-03-21

AI Technical Summary

Technical Problem

The existing AEB system has insufficient consideration of vehicle emergency braking and comfort, resulting in poor ride comfort.

Method used

By establishing the vehicle braking state equation defined by the seventh-order polynomial, the minimum value of the vehicle's acceleration (jerk) state equation is calculated, a vehicle braking control equation considering comfort is established, and its effectiveness is verified through simulation.

Benefits of technology

Improves the comfort during the vehicle braking process, reduces the maximum impact during the braking process by about 50%, and provides a fast and accurate verification method through simulation verification.

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Abstract

The invention discloses a vehicle braking control method considering comfort, and relates to the field of vehicle driving braking. The method comprises the following steps: establishing a septimal polynomial to define a vehicle braking state equation; a vehicle braking state equation is defined based on a septimal polynomial, and a vehicle braking control equation considering comfort is established; and verifying the vehicle brake control equation considering the comfort through simulation. According to the method, the safety and comfort of the vehicle can be well considered, and compared with a traditional automatic emergency braking control method Time-to-Collision (TTC) and a safety distance model, the method provided by the invention considers various constraint conditions at the starting moment and the ending moment of vehicle braking, and the braking comfort of the vehicle can be remarkably improved on the basis of ensuring collision avoidance.
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Description

Technical Field

[0001] The invention relates to the field of vehicle driving braking, and in particular to a vehicle braking control method taking comfort into consideration. Background Art

[0002] The automatic emergency braking (AEB) system is an important part of modern automobile safety technology. The early research on the control method of the AEB system mainly focused on the automatic emergency braking control method (Time-to-Collision, TTC) model. It is an indicator used to evaluate the collision risk between a vehicle and other objects (such as other vehicles, pedestrians or obstacles). By calculating the relative distance and relative speed, it predicts the time required for a collision under current conditions. The smaller the TTC value, the higher the collision risk; conversely, a larger TTC value indicates higher safety. Since the TTC model error is large in close distance situations, the inverse of TTC can be used to make risk judgments.

[0003] However, the existing technologies of AEB control system based on safety distance model and TTC model mainly consider the situation of vehicle lane change, but do not consider the emergency braking and comfort of actual vehicles. The existing technology combines the two, which makes the ride comfort poor. Therefore, in order to improve people's acceptance of AEB system, it is necessary to simulate the comfort of vehicle braking control strategy to avoid economic losses caused by using real vehicles for testing. Summary of the invention

[0004] In order to solve the above-mentioned problem of considering the comfort of vehicle braking, the present invention provides a vehicle braking control method taking comfort into consideration.

[0005] To implement the above technical solution, the details are as follows:

[0006] S1. Establish a seventh-order polynomial to define the vehicle braking state equation; the expression is as follows:

[0007] x(t)=b0+b1t+b2t 2 +b3t 3 +b4t 4 +b5t 5 +b6t 6 +b7t 7

[0008] Wherein, t represents time; b0 represents the zeroth coefficient; b1 represents the first coefficient; b2 represents the second coefficient; b3 represents the third coefficient; b4 represents the fourth coefficient; b5 represents the fifth coefficient; b6 represents the sixth coefficient; b7 represents the seventh coefficient;

[0009] S2, defining the vehicle braking state equation based on a seventh-order polynomial and establishing a vehicle braking control equation taking comfort into consideration;

[0010] Here are the steps:

[0011] S2.1. Take the first derivative of the vehicle braking state equation defined by the seventh-order polynomial to obtain the vehicle speed state equation; the expression is as follows:

[0012] v(t)=b1+2b2t+3b3t 2 +4b4t 3 +5b5t 4 +6b6t 5 +7b7t 6 ;

[0013] S2.2. Deriving the vehicle velocity state equation, we obtain the vehicle acceleration state equation; the expression is as follows:

[0014] a(t)=2b2+6b3t+12b4t 2 +20b5t 3 +30b6t 4 +42b7t 5 ;

[0015] S2.3. Derivative the vehicle acceleration state equation to obtain the vehicle jerk state equation; the expression is as follows:

[0016] j(t)=6b3+24b4t+60b5t 2 +120b6t 3 +210b7t 4 ;

[0017] Furthermore, the impact degree of the vehicle can be reflected by the vehicle jerk state equation;

[0018] S2.4. By calculating the minimum value of the vehicle jerk state equation, establish the vehicle braking control equation considering comfort; the expression is as follows:

[0019]

[0020] In the formula, t f Indicates the moment when vehicle braking ends, calculated as follows:

[0021] When the vehicle starts braking at t=0, it can be obtained that: x(0)=0; v(0)=v0; a(0)=0; j(0)=0;

[0022] The vehicle ends braking at t = t f We can get: x(t f ) = t f ; v(t f )=0;a(tf )=0;j(t f )=0;

[0023] Taking the conditions of the vehicle at the start of braking and the end of braking as constraints, substituting into the seventh-order polynomial to define the vehicle braking state equation, vehicle speed state equation, vehicle acceleration state equation and vehicle jerk state equation, we can get:

[0024] b0=0

[0025] b1=v0

[0026] b2=0

[0027] b3=0

[0028]

[0029]

[0030] Furthermore, the vehicle stopping time t can be obtained by calculation. f for:

[0031]

[0032] In the formula, x f Indicates stopping distance;

[0033] S3, verifying the vehicle braking control equation considering comfort through simulation;

[0034] include:

[0035] S3.1, point mass model simulation;

[0036] S3.1.1, set the initial vehicle speed to 60km / h and the road adhesion coefficient to 0.9 for simulation;

[0037] S3.1.2, set the road adhesion coefficient to 0.9, and simulate different initial vehicle speeds;

[0038] S3.1.3. Set the simulation test under the condition that the front vehicle is stationary, the initial distance between the rear vehicle and the front vehicle is 150m, the initial speed of the rear vehicle is 80km / h, and the road adhesion coefficient is 0.9;

[0039] S3.2, Carsim model CCR simulation.

[0040] Beneficial effects of the present invention

[0041] The control algorithm of the present invention takes into account the constraints of the vehicle braking start time and end time, so that the deceleration of the vehicle during the braking process changes evenly, greatly improving the braking comfort of the vehicle.

[0042] Compared with the prior art, the maximum impact degree during braking is reduced by about 50%.

[0043] The present invention verifies its effectiveness by adopting simulation, and provides a faster and more accurate verification method compared with the prior art. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a schematic diagram of the process of the present invention;

[0045] Figure 2 Schematic diagram of simulation results of the present invention when the initial vehicle speed is 60 km / h and the road adhesion coefficient is 0.9; wherein, part (a) is a comparison diagram of braking distances of the fifth-order polynomial and the seventh-order polynomial; part (b) is a comparison diagram of vehicle speeds of the fifth-order polynomial and the seventh-order polynomial; part (c) is a comparison diagram of decelerations of the fifth-order polynomial and the seventh-order polynomial; part (d) is a comparison diagram of jerks of the fifth-order polynomial and the seventh-order polynomial;

[0046] Figure 3 Schematic diagram of simulation results of different initial vehicle speeds when the road adhesion coefficient is 0.9; wherein, part (a) is a braking distance diagram of a fifth-order polynomial; part (b) is a vehicle speed diagram of a fifth-order polynomial; part (c) is a deceleration diagram of a fifth-order polynomial; part (d) is a jerk diagram of a fifth-order polynomial; part (e) is a braking distance diagram of a seventh-order polynomial; part (f) is a vehicle speed diagram of a seventh-order polynomial; part (g) is a deceleration diagram of a seventh-order polynomial; part (h) is a jerk diagram of a seventh-order polynomial;

[0047] Figure 4 The present invention sets a simulation test under the condition that the front vehicle is stationary, the initial distance between the rear vehicle and the front vehicle is 150m, and the initial speed of the rear vehicle is 80km / h; the schematic diagram of the simulation results when the road adhesion coefficient is 0.9; wherein, part (a) is the distance diagram between the two vehicles; part (b) is the speed diagram of the rear vehicle; part (c) is the acceleration diagram; part (d) is the jerk diagram; and part (e) is the response diagram of TTC;

[0048] Figure 5 It is a simulation diagram of Carsim model CCR of the present invention;

[0049] Figure 6 The figure is a schematic diagram of the simulation results in the CCRs scenario of the present invention, wherein (a) is a diagram of the distance between two vehicles; (b) is a diagram of the speed of the rear vehicle; (c) is a diagram of the relative speed of the two vehicles; (d) is a diagram of the deceleration of the rear vehicle; and (e) is a diagram of the response of TTC.

[0050] Figure 7Schematic diagram of simulation results in the CCRm scenario of the present invention, wherein (a) is the distance between two vehicles; (b) is the speed of the rear vehicle; (c) is the relative speed of the two vehicles; (d) is the deceleration of the rear vehicle; (e) is the response of TTC;

[0051] Figure 8 It is a schematic diagram of the simulation results in the CCRb scenario of the present invention, wherein part (a) is the distance between the two vehicles; part (b) is the speed diagram of the rear vehicle; part (c) is the relative speed diagram of the two vehicles; part (d) is the deceleration diagram of the rear vehicle; and part (e) is the response diagram of TTC. DETAILED DESCRIPTION

[0052] The present invention is further described in detail below in conjunction with specific embodiments.

[0053] like Figure 1 As shown, a vehicle braking control method considering comfort includes the following steps:

[0054] S1. Establish a seventh-order polynomial to define the vehicle braking state equation; the expression is as follows:

[0055] x(t)=b0+b1t+b2t 2 +b3t 3 +b4t 4 +b5t 5 +b6t 6 +b7t 7

[0056] Wherein, t represents time; b0 represents the zeroth coefficient; b1 represents the first coefficient; b2 represents the second coefficient; b3 represents the third coefficient; b4 represents the fourth coefficient; b5 represents the fifth coefficient; b6 represents the sixth coefficient; b7 represents the seventh coefficient;

[0057] S2. Define the vehicle braking state equation based on the seventh-order polynomial to establish a vehicle braking control equation that takes comfort into consideration; the steps are as follows:

[0058] S2.1. Take the first derivative of the vehicle braking state equation defined by the seventh-order polynomial to obtain the vehicle speed state equation; the expression is as follows:

[0059] v(t)=b1+2b2t+3b3t 2 +4b4t 3 +5b5t 4 +6b6t 5 +7b7t 6 ;

[0060] S2.2. Deriving the vehicle velocity state equation, we obtain the vehicle acceleration state equation; the expression is as follows:

[0061] a(t)=2b2+6b3t+12b4t 2 +20b5t 3 +30b6t 4 +42b7t 5 ;

[0062] S2.3. Derivative the vehicle acceleration state equation to obtain the vehicle jerk state equation; the expression is as follows:

[0063] j(t)=6b3+24b4t+60b5t 2 +120b6t 3 +210b7t 4 ;

[0064] Furthermore, the impact degree of the vehicle can be reflected by the vehicle jerk state equation;

[0065] S2.4. By calculating the minimum value of the vehicle jerk state equation, establish the vehicle braking control equation considering comfort; the expression is as follows:

[0066]

[0067] In the formula, t f Indicates the moment when vehicle braking ends, calculated as follows:

[0068] When the vehicle starts braking at t=0, it can be obtained that: x(0)=0; v(0)=v0; a(0)=0; j(0)=0;

[0069] The vehicle ends braking at t = t f We can get: x(t f ) = t f ; v(t f )=0;a(t f )=0;j(t f )=0;

[0070] Taking the conditions of the vehicle at the start of braking and the end of braking as constraints, substituting into the seventh-order polynomial to define the vehicle braking state equation, vehicle speed state equation, vehicle acceleration state equation and vehicle jerk state equation, we can get:

[0071] b0=0

[0072] b1=v0

[0073] b2=0

[0074] b3=0

[0075]

[0076]

[0077] Furthermore, the vehicle stopping time t can be obtained by calculation. f for:

[0078]

[0079] In the formula, x f Indicates stopping distance;

[0080] Furthermore, the braking acceleration a max Maximum corresponding time t a and parking distance x f The expression is as follows:

[0081]

[0082] In the formula, x s =v0 2 / (2a max ); x f5 The braking distance required is a fifth-order polynomial;

[0083] It can be seen that the minimum distance required for braking with the 7th-order polynomial is approximately 1.78 times the braking distance with the maximum deceleration and 1.25 times the braking distance with the 5th-order polynomial;

[0084] S3, verifying the vehicle braking control equation considering comfort through simulation;

[0085] S3.1. Point mass model simulation; details are as follows:

[0086] S3.1.1, set the initial vehicle speed to 60km / h, the road adhesion coefficient to 0.9, the results are as follows Figure 2 As shown;

[0087] Figure 2 This is a simulation comparison of 5th-order polynomial braking and 7th-order polynomial braking. The blue solid line is the 5th-order polynomial result, and the red dotted line is the 7th-order polynomial result. Figure 2Parts (a), (b), (c) and (d) are the braking distance, vehicle speed, deceleration and jerk responses, respectively. Under the same initial conditions, the minimum braking distance required for the 5th-order polynomial braking is 21.9m, and that for the 7th-order polynomial is 27.4m; the braking time is 3.29s and 3.84s, respectively; the time to reach the peak deceleration is 1.11s and 1.54s, respectively; the maximum Jerk values ​​are 1.85g / s and 0.92g / s, corresponding to t=0 and t=0.6, respectively. Therefore, it can be concluded that, compared with the 5th-order polynomial, although the braking distance required for the 7th-order polynomial increases by 25%, the maximum braking jerk decreases by about 50%, and the comfort of the vehicle during braking is significantly improved.

[0088] S3.1.2, set the road adhesion coefficient to 0.9, the simulation results of different initial vehicle speeds are as follows Figure 3 As shown;

[0089] Figure 3 Parts (a), (b), (c), (d), (e), (f), (g), and (h) are the responses of the vehicle's braking distance, speed, acceleration, and Jerk, respectively. It can be seen that with the increase of the initial vehicle speed, the braking distance required for different polynomial braking increases accordingly, the braking time increases proportionally with the increase of the initial vehicle speed, and the maximum Jerk value of the vehicle decreases with the increase of the initial vehicle speed. The specific evaluation indicators are shown in Table 1. Under the same initial conditions, the braking distance of the 7th-order polynomial is 25% larger than that of the 5th-order polynomial, and the maximum jerk value is reduced by about 50%.

[0090] Table 1 Comparison of braking results at different initial vehicle speeds

[0091]

[0092] S3.1.3, set the simulation test under the condition of the front vehicle being stationary, the initial distance between the rear vehicle and the front vehicle is 150m, the initial speed of the rear vehicle is 80km / h, the road adhesion coefficient is 0.9; the results are as follows Figure 5 As shown;

[0093] Figure 4Parts (a), (b), (c), (d) and (e) are the responses of the distance between the two vehicles, the speed of the rear vehicle, acceleration, Jerk and TTC respectively. It can be seen that the vehicle equipped with the 7th-order polynomial automatic braking control system can effectively prevent the rear vehicle from rear-ending. The distance between the two vehicles when the rear vehicle is stationary is about 2m. The vehicle braking deceleration increases first and then decreases, reaching the peak deceleration at t = 6.5s. At about t = 9.5, the vehicle speed drops to zero. The vehicle comfort is good during the entire braking process, and the vehicle Jerk changes smoothly, with troughs and peaks appearing successively, reaching -0.69g / s and 0.51g / s at t = 5.3s and t = 7.8s respectively. TTC reaches the minimum value of 0.84s at t = 7.2s.

[0094] S3.2, such as Figure 5 As shown, Carsim model CCR simulation; the details are as follows:

[0095] To further verify the effectiveness of the control algorithm, the vehicle dynamics model of CarSim software was used for simulation testing. CarSim software is a mature commercial software for vehicle dynamics simulation developed by Mechanical Simulation Corporation. It has been widely used in the automotive academic and industrial circles. The test conditions adopt the European New Car Assessment Programme (Euro NCAP) conditions, which mainly include the following three test scenarios: CCRs, CCRm and CCRb. See Table 2 for details. In the CCRs test condition, the front vehicle remains stationary. In the CCRm test condition, the front vehicle travels at a speed of 20km / h. Figure 7 The following figure shows the AEB test scenario. The main parameters of the rear vehicle model are shown in Table 3.

[0096] Table 2 CCR test scenarios

[0097]

[0098]

[0099] Table 3 Main parameters of the vehicle

[0100] parameter symbol unit value Sprung mass m kg 1370 Wheelbase L m 2.866 Center of mass height <![CDATA[h g ]]> m 0.52 Rolling moment of inertia <![CDATA[I xx ]]> <![CDATA[kg·m 2 ]]> 671.3 Pitch moment of inertia <![CDATA[I yy ]]> <![CDATA[kg·m 2 ]]> 1972.8 Yaw moment of inertia <![CDATA[I zz ]]> <![CDATA[kg·m 2 ]]> 2315.3 Distance from center of mass to front axle <![CDATA[L1]]> m 1.11 Distance from center of mass to rear axle <![CDATA[L2]]> m 1.756 Front suspension stiffness <![CDATA[K sf ]]> N / m 153000 Rear suspension stiffness <![CDATA[K sr ]]> N / m 82000 Tire model - - 215 / 55R17 Braking system - - Hydraulic Road adhesion coefficient μ - 0.9

[0101] S3.2.1, CCRs scenario;

[0102] like Figure 6 As shown in the figure, in the CCRs test scenario, the front vehicle remains stationary, the rear vehicle has an initial speed of 80 km / h, and the initial distance between the two vehicles is 60 m.

[0103] Figure 6Parts (a), (b), (c), (d) and (e) are the distance between the two vehicles, the speed of the rear vehicle, the relative speed of the two vehicles, the deceleration of the rear vehicle and the response of TTC. It can be seen that the vehicle equipped with the 7th-order polynomial automatic braking system can effectively prevent rear-end collisions. The distance between the rear vehicle and the front vehicle is about 2.1m when the rear vehicle stops. During the braking process, the vehicle deceleration changes smoothly, and there is no step change similar to the TTC control, which ensures good comfort. However, since the vehicle model takes into account the characteristics of the suspension system, during the braking process, the pitch of the vehicle body will cause the vertical load of the tire to fluctuate, thereby causing the wheel braking force to fluctuate, and ultimately causing a certain degree of fluctuation in the vehicle deceleration. During the entire braking process, the minimum TTC is 0.94s, which occurs at t = 2.9s.

[0104] S3.2.2, CCRm scenario;

[0105] like Figure 7 As shown in the figure, in the CCRm test scenario, the front vehicle is traveling at a constant speed of 20 km / h, and the initial speed of the rear vehicle is 60 km / h. The initial distance between the two vehicles is 60m.

[0106] Figure 7 Parts (a), (b), (c), (d) and (e) are the distance between the two vehicles, the speed of the rear vehicle, the relative speed of the two vehicles, the deceleration of the rear vehicle and the response of TTC. It can be seen that the vehicle equipped with a 7th-order polynomial automatic braking system can also effectively prevent rear-end collisions in this scenario. The distance between the rear vehicle and the front vehicle when it stops is about 2.1m. The rear vehicle reduces its speed to the same as the front vehicle at around t=6.5s. The vehicle deceleration increases first and then decreases, reaching the peak deceleration at around t=5.4s. During the entire braking process, the minimum TTC is 0.75s, which occurs at t=5.0s.

[0107] S3.2.3, CCRb scenario;

[0108] like Figure 8 As shown in the figure, in the CCRb test scenario, both vehicles are traveling at a constant speed of 50 km / h. The initial distance between the two vehicles is 12 m. At t = 0, the leading vehicle brakes at a deceleration of 6 m / s2.

[0109] Figure 8 Parts (a), (b), (c), (d) and (e) are the distance between the two vehicles, the speed of the rear vehicle, the relative speed of the two vehicles, the deceleration of the rear vehicle and the response of TTC. It can be seen that the rear vehicle can also effectively avoid collision in this scenario. The distance between the rear vehicle and the front vehicle when the rear vehicle stops is about 2.1m. The speed of the rear vehicle drops to 0 at t = 3.5s. The vehicle deceleration also increases first and then decreases, reaching a peak at t = 1.9s. During the entire braking process, the minimum TTC is 1.06s, which occurs at t = 2.32s.

[0110] The control algorithm proposed in this paper has a certain role in promoting the popularization of intelligent connected vehicles and autonomous driving control technology. At present, this paper only uses simulation methods to verify the control algorithm, and will verify it through real vehicle tests in the future.

Claims

1. A vehicle braking control method considering comfort, characterized in that: The following steps are involved: S1. Establish a seventh-order polynomial to define the vehicle braking state equation; S2, defining the vehicle braking state equation based on a seventh-order polynomial and establishing a vehicle braking control equation taking comfort into consideration; S3, verifying the vehicle braking control equation considering comfort through simulation; The simulation includes: S3.1, point mass model simulation; S3.1.1, set the initial vehicle speed to 60km / h and the road adhesion coefficient to 0.9 for simulation; S3.1.2, set the road adhesion coefficient to 0.9, and simulate different initial vehicle speeds; S3.1.

3. Set the simulation test under the condition that the front vehicle is stationary, the initial distance between the rear vehicle and the front vehicle is 150m, the initial speed of the rear vehicle is 80km / h, and the road adhesion coefficient is 0.9; S3.2, Carsim model CCR simulation.

2. The vehicle braking control method considering comfort according to claim 1, characterized in that: The seventh-order polynomial defines the vehicle braking state equation as follows: <h2 style=";text-align:left;direction:ltr">x(t) = b0 + b1t + b2t<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> +b3t<h2 style=";text-align:left;direction:ltr"> 3 <h2 style=";text-align:left;direction:ltr"> +b4t<h2 style=";text-align:left;direction:ltr"> 4 <h2 style=";text-align:left;direction:ltr"> +b5t<h2 style=";text-align:left;direction:ltr"> 5 <h2 style=";text-align:left;direction:ltr"> +b6t<h2 style=";text-align:left;direction:ltr"> 6 <h2 style=";text-align:left;direction:ltr"> +b7t<h2 style=";text-align:left;direction:ltr"> 7 In the formula, t represents time; b0 represents the zeroth coefficient; b1 represents the first coefficient; b2 represents the second coefficient; b3 represents the third coefficient; b4 represents the fourth coefficient; b5 represents the fifth coefficient; b6 represents the sixth coefficient; and b7 represents the seventh coefficient.

3. The vehicle braking control method considering comfort according to claim 1, characterized in that: The steps of the vehicle braking control equation considering comfort are as follows: S2.

1. Take the first derivative of the vehicle braking state equation defined by the seventh-order polynomial to obtain the vehicle speed state equation; the expression is as follows: v(t)=b1+2b2t+3b3t 2 +4b4t 3 +5b5t 4 +6b6t 5 +7b7t 6 ; S2.

2. Deriving the vehicle velocity state equation, we obtain the vehicle acceleration state equation; the expression is as follows: a(t)=2b2+6b3t+12b4t 2 +20b5t 3 +30b6t 4 +42b7t 5 ; S2.

3. Deriving the vehicle acceleration state equation, we obtain the vehicle acceleration state equation; the expression is as follows: j(t)=6b3+24b4t+60b5t 2 +120b6t 3 +210b7t 4 ; Furthermore, the impact degree of the vehicle can be reflected through the vehicle acceleration state equation; S2.

4. By calculating the minimum value of the vehicle acceleration state equation, establish the vehicle braking control equation considering comfort; the expression is as follows: Where, t f Indicates the moment when vehicle braking ends, calculated as follows: When the vehicle starts braking at t=0, it can be obtained that: x(0)=0; v(0)=v0; a(0)=0; j(0)=0; The vehicle ends braking at t = t f We can get: x(t f ) = t f ; v(t f )=0;a(t f )=0;j(t f )=0; Taking the conditions of the vehicle at the start of braking and the end of braking as constraints, substituting into the seventh-order polynomial to define the vehicle braking state equation, vehicle speed state equation, vehicle acceleration state equation and vehicle jerk state equation, we can get: b0=0 b1=v0 b2=0 b3=0 Furthermore, the vehicle stopping time t can be obtained by calculation. f for: In the formula, x f Indicates stopping distance.