Identification for Speed of Vehicle and Its Uncertainty Analysis of Road Traffic Accident by Momentum Method

2017 3rd International Conference on Computer Science and Mechanical Automation (CSMA 2017) ISBN: 978-1-60595-506-3

Identification for Speed of Vehicle and Its Uncertainty Analysis of Road

Traffic Accident by Momentum Method

Hao-Cun DONG

1

and Zhong-Guo NIE

2

1_{Automobile and Traffic College, Shenyang Ligong University, Shenyang 110159, China} Email: Donghaocun@163.com

2

Shenyang Jiashi Judicial identification Office, Shenyang 110025, China Email: lnwsfdckf @163.com

Keywords: Road traffic accident, Identification for speed of vehicle, Momentum conservation, Uncertainty.

Abstract. For vehicle collision on road traffic accident, this paper establishes a mathematical model

for calculating the driving speed of the vehicle with the momentum method, applies the uncertainty theory, analyzes the influence of the uncertainty of the accident parameters on the speed calculation result, and gives the evaluation method of the speed uncertainty of the accident vehicle, and then determines the reasonable range of the speed of vehicle. The analysis of a practical case proves that this method is feasible.

Introduction

Vehicle collision is a typical form of road traffic accident, the core content of forensic expertise is often the driving speed of the collision accident vehicle, which is the important basis to determine the nature of the accident and the responsibility of the accident, therefore, the method of identifying the speed of the accident vehicle shall adopt a more scientific, accurate and objective method

The vehicle involved in the collision form on road traffic accident all has the certain exterior di-mension, which is the complex rigid body, in the collision, this part of vehicle simultaneously has the distortion, the mass loss and so on, but because the vehicle collision process is extremely short, generally in 0.1~0.2s around [1], at the same time, the impact force is very large, momentum ex-change can be regarded to be completed approximately in a moment, so the vehicle can be simpli-fied, regarding it as the quality of the particle or a single rigid body, considering the beginning and end of the collision process, the system of the collision vehicle has the momentum conservation, which can be used to calculate the instantaneous speed of vehicle collision. The momentum conser-vation method is one of the most commonly used methods for traffic accident speed identification.

However, in the analysis and calculation of the speed of the accident vehicle, it is bound to use some of the physical significance of the accident parameters, because of the measurement condi-tions, measurement technology and other reasons, resulting in the accuracy of the data is different, some accident parameters are often inaccurate, that is, the accident parameters are uncertain, vehicle collision traffic accidents happen and end in an instant, some parameters cannot be obtained after the event, only to estimate the range of its value, which makes the part of the parameters of vehicle collision uncertain, This will inevitably lead to the uncertainty of the calculation result of the speed of traffic accident. The uncertainty analysis technique is applied to the analysis and calculation of vehicle speed of traffic accident, and the influence degree of accident parameter error to the calcula-tion result can be obtained, which can be used to support for analysis on the accuracy and confi-dence of the calculation result of the vehicle traveling speed [2].

Uncertainty Theory

a range of values, and is a parameter in the measurement that represents the dispersion of the meas-ured value, indicating that the measurement result is not a definite value but an interval in which the infinite number of possible values are dispersed. The uncertainty of measurement generally contains several components, and all the uncertainty components are characterized by standard deviation. The types of uncertainty can be divided into standard uncertainty, synthetic standard uncertainty and extension uncertainty (also known as extended uncertainty).The uncertainty of standard devia-tion representadevia-tion is the standard uncertainty. The uncertainty of synthesizing the standard uncer-tainty of each component is the synthetic unceruncer-tainty. The extension unceruncer-tainty gives an interval range of the measurement results, where the measured value is located in with high confidence probability [3].

The mathematical model of the measured parameter y is the multivariate function

(

)

n x x x x f

y= ₁, ₂, ₃, _{, and} y_{is affected by the number of}n_inputx

(

_{i n}

)

i =1,2,3, with uncertainty, if

the true value of each directly measured xi_{is recorded as}x_{z, the uncertainty of}xi _{is recorded as} uxi

which can also be represented as absolute error (measured value and difference of truth)∆x

, then the

uncertainty of the inputxi_is

z i

xi

x

x

u

>

∆

=

−

x (1)

The uncertainty of the measured parameter

y

depends on the uncertainty of the input

x

_i, which can be calculated as follows.

Standard Uncertaintyu_x

The evaluation of standard uncertainty is classified into class A and class B, the evaluation of class A adopts statistical analysis method, whose standard uncertainty is just the standard deviation ob-tained from series observations, and the class B evaluation is based on the theory of probability, and the standard deviation is evaluated by analyzing the distribution characteristics of the measured col-umns. In the practical application, the method of class B evaluation is more adopted, and for the uncertainty evaluation of the speed of road traffic accident, if it is not used statistical analysis, then the class B is usually used to evaluate [4].

The standard uncertainty u_xi of

x

_i is obtained by dividing half width of the range

a

i of the input

i

x

with the inclusion factor k_i.

i i xi

k a

u = ₍₂₎

The inclusion factor

k

_i is also called a range factor. If the input

x

_i obeys the normal distribution,

the inclusion factor is obtained by querying the normal distribution integral table; If the input

x

_i is

uniformly distributed, then the inclusion factor is 3[3].

Synthetic Standard Uncertainty

u

y

The influence of the standard uncertainty uxi _{of each direct measured value} xi _{on the uncertainty}

transfer coefficient obtained from estimated value of the measured parameter y is

i x

y

∂

∂ _{, and the}

xi i

yi u

x y u

∂ ∂

= ₍₃₎

According to the theory of error synthesis, when the input is irrelevant, each input will introduce an uncertain factor, the uncertainty of the measured parameter y should be the synthesis of all the

uncertainty components, so the synthetic standard uncertainty uy of the function y is

∑

∑

= =

       

∂ ∂ =

=

n

i

xi i n

i yi

y u

x y u

u

1

2 2

1

2 ₍₄₎

Expansion Uncertainty

U

Extension uncertainty is also known as extended uncertainty, which multiplies the synthetic

stand-ard uncertainty uy _{of a function} y _{with the inclusion factor} k _{to obtain an extension uncertainty of} U_.

y

ku

U

=

₍₅₎

Using the extension uncertainty as the uncertainty of the measured parameter y, the function y can be expressed as

U

y

Y

=

±

₍₆₎

This means that the optimal value of the measured parameter Y is y and most of the values that contain Y in the range between y-U and the y+U, thereby determining the range of values of the function.

Small Uncertainty and Uncertainty Factors

By the synthetic standard uncertainty u_y of the formula (4) function y, the uncertainty of the

syn-thetic standard of the measured parameter y is the form of the square superposition of the

uncer-tainty caused by the input xi_{. Compared with the total effect of all uncertainties, if the difference}

between the effect of an uncertainty and the original effect is less than 1/10 of the original effect, the uncertainty can be regarded as a tiny uncertainty, and may be omitted from the total uncertainty.

If the value of a certain input xi _{is most uncertain, the uncertainty of the parameter} uxi _is

signifi-cantly greater than that of the other parameters. The uncertainty of the calculation results is the most important, occupying the dominant position, then such a decisive character of the input parameter can be regarded as the uncertainty factor [5], according to the uncertainty of the factor (value range) to calculate the measured parameter y corresponding range and uncertainty.

The Momentum Method for Calculating Speed of Vehicle and Evaluation of Uncertainty

The Momentum Method for Calculating Speed of Vehicle

A system of colliding vehicles maintains a constant momentum of the system in the event of a colli-sion, and obeys law of conservation of momentum. Assuming that the mass of the vehicle 1 and

is α₁ and α₂ respectively, then the momentum conservation equation of the vehicle 1 and vehicle 2 is projected as x on the rectangular coordinate.







+

=

+

+

=

+

2 2 2 1 1 1 20 20 2 10 10 1 2 2 2 1 1 1 20 20 2 10 10 1

sin

sin

sin

sin

cos

cos

cos

cos

α

α

α

α

α

α

α

α

v

m

v

m

v

m

v

m

v

m

v

m

v

m

v

m

(7)

By the equation group (7), the driving speed before the collision between two vehicles is

(

)

(

)

(

)

(

)

(

)

(

)

       − − + − = − − + − = 20 10 2 2 10 2 2 1 10 1 1 20 10 20 1 2 20 2 2 1 20 1 1 10 sin sin sin sin sin sin α α α α α α α α α α α α m v m v m v m v m v m v (8)

Analysis of Small Uncertainty and Uncertain Factors

The formula (8) shows that the instantaneous driving speed v₁₀ and v₂₀ of vehicle 1 and vehicle 2 are all non-linear multivariate functions with 8 input quantities. In general, the uncertainty of the vehicle quality parameters [6,7] and the geometric parameters of the accident evidence on the scene such as braking (or taxiing) distance, speed direction angle, etc., are relatively small [8,9], which can be regarded as small uncertainty to be ignored, however, the uncertainty of the parameters, such as the coefficient of adhesion, which needs to be measured and selected are large enough to be set as the uncertainty factor, and according to its value range, the corresponding range and uncertainty of the accident speed can be calculated. This characteristic of uncertainty evaluation can simplify the speed identification of vehicles involved in the complicated road traffic accident.

Evaluation of Uncertainty of Speed of Vehicle

In the practice of road traffic accident speed identification, it is often based on the law of conserva-tion of energy, that is, the kinetic energy of vehicle after collision is converted into fricconserva-tional power, and the instantaneous speed

i

v is calculated first according to the vehicle braking distance or glide

distance.

i i g s

v = ₂ ϕ ₍₉₎

In the formula, gis the gravitational acceleration, ϕ_{is the attachment coefficient, and}

s

_{iis the}

moving distance of the vehicle after the collision. Then, according to the formula (8) which ignores the influence of the small uncertainty, the uncertainty of the driving speed before the collision can be evaluated.

First of all, to evaluate the standard uncertainty u_vi of the driving speed

v

_i of each input after the

vehicle collision, the distribution interval half width

a

_vi of the input v_i is divided by the inclusion factor k_vi, which is obtained:

vi vi vi

k

a

u

=

₍₁₀₎

Second, determine the standard uncertainty component of each input

u

vi0.vi

Evaluate the synthetic standard uncertainty

u

vi0 of driving speed

v

i0 before collision of vehicle

∑

∑

=

=













∂

∂

=

=

n

i

vi i i n

i

vi vi

vi

u

v

v

u

u

1

2 2

0

1 2 . 0

0 (12)

The extension uncertainty Uvi0of the function

v

i0can be obtained by multiplying the synthetic

standard uncertainty

u

_vi0of the function

v

_i0with the inclusion factor

k

.

0

0 vi

vi

ku

U

=

₍₁₃₎

Use the extension uncertainty as the uncertainty of the measured parameter

v

i0, and the

meas-ured driving speed

v

i0 before the vehicle collision can be expressed as

0 0

0 i vi

i

v

U

V

=

±

₍₁₄₎

This means that the measured parameter, that is, the best value of the driving speed is

v

_i0 and

most of the values that contain V_i0 in the range between

v

i0

-

U

vi0and

v

i0

+

U

vi0, thereby determin-ing the range of the drivdetermin-ing speed before the collision.

Actual Case Analysis

The bus moves from west to east, in the early overtaking across the yellow line in the middle of the road, the truck from east to west, in the late overtaking is returning to its own lane, the left front wheel is on yellow line in the middle of the road, two opposite driving vehicles across the midline overtaking produce the collision on the left front corner, the key content of the accident identifica-tion is the driving speed of vehicle before the collision. Now the bus is regarded as vehicle 1, and the truck is regarded as vehicle 2, composing of a vehicle collision system with momentum conser-vation.

Equivalent Driving Speed

v

₁and

v

₂ of Two Vehicles After Collision

After the collision, because of the glide distance _s1=34.1m_{, the turning angle} o 5 .

53 , non-braking and irregular rotation sideslip of the bus on the road, a comprehensive estimate [10,11,12] can ob-tain its adhesion coefficient

ϕ

₁ =0.26±0.02 _{as an uncertain factor, according to the law of}

conser-vation of energy, the equivalent speed v1 after the bus crash is

km/h 83 . 1 42 . 47

2 _{1 1}

1 = g s = ±

v ϕ ₍₁₅₎

After the collision, because of the glide distance s2 =33.17m, the turning angle

o

7 .

48 , non-braking and irregular rotation sideslip of the truck on the road, a comprehensive estimate can obtain its adhesion coefficient ϕ1=0.24±0.02 as an uncertain factor, according to the law of conservation

of energy, the equivalent speed v₂ after the bus crash is

km/h 98 . 1 74 . 43

2 2 2

2 = g s = ±

Speed

v

₁₀of Bus before Collision

According to the law of conservation of momentum, record the speed direction of the bus (vehicle 1) and the truck (vehicle 2), the momentum conservation equation of the collision system is projected on the rectangular coordinate.

   + = + = 2 2 2 1 1 1 20 20 2 10 10 1 2 2 2 1 1 1 20 20 2 10 10 1 sin sin sin sin cos -cos cos -cos

α

α

α

α

α

α

α

α

v m v m v m v m v m v m v m v m (17)

In the formula: the quality of the bus is _m1=13035kg_{, the speed direction before the collision is}

o

5

10 =

α _{, and the angle between the speed direction and the longitudinal axis of the road is} o

2

1=

α ;

the quality of the truck is _m2=17250kg, the speed direction before the collision is o

10

20=

α , and the

angle between the speed direction and the longitudinal axis of the road is o

18

2=

α .

Solve equation group (17), obtain: driving speed

v

10 of the bus (vehicle 1) before the collision is

(

)

(

)

(

)

69.22 2.88km/h

sin sin sin 10 20 1 20 2 2 2 1 20 1 1

10 == ±

+ − + + =

α

α

α

α

α

α

m v m v m v (18)

Evaluation of Uncertainty of Driving Speed

v

10of Bus Before Collision

The uncertainty of driving speed of vehicle is not calculated by statistical analysis, so it should be evaluated by class B. According to the uncertainty theory, the confidence probability of 0.95 corre-sponds to the inclusion factor _k = ₂, and calculate according to formula (10), the standard

uncer-tainty of 1

v

and v2

is ~ _{0.9150 km/h}

1 = v

u and u v2 = 0 .9900 km/h , respectively.

According to the formula (11) (14), the standard uncertainty component of the driving speed

v

10 of the bus before collision is _10.1 = 0.7350 km/h

v v

u and uv10.v2 =0.7045km/h , the synthetic standard

uncertainty is uv10 =1.0181 km/h , and the extension uncertainty is Uv10=2.0362km/h. After ignor-ing the small uncertainty introduced in some input parameters, the drivignor-ing speed range of the bus before collision can be

71.26km/h -67.18 km/h 04 . 2 22 . 69

10 = ± =

v (19)

The uncertainty calculation method of driving speed

v

20 of the truck before collision is the same. The judicial appraisal of this case concludes that: before the collision, the driving speed of the bus is 9km/h6 .

Conclusion

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