International Journal of Crashworthiness Vol. 16, No. 2, April 2011, 181–194
The first collision point position identification method in vehicle–pedestrian impact accident Zhang Xiaoyuna∗ , Jin Xianlonga , Chai Xianghaia and Hou Xinyib a
Department of Mechanical Engineering, Shanghai Jiaotong University, Shanghai, China; b Traffic and Police Office of Shanghai, Shanghai, China (Received 15 September 2010; final version received 13 December 2010) Vehicle–pedestrian accidents share high frequency of occurrence in fatal traffic accidents in China. According to the Chinese Traffic Safety Regulations, the first collision point is the key factor for responsibility cognisance in related traffic accidents. Usually, the police or other accident investigators determine the first collision point position between vehicle and pedestrian only through experience. The position error of the collision point will lead to inaccurate accident analysis and responsibility judgement. This paper applies computer-simulated reconstruction in vehicle–pedestrian accident investigation and uses optimisation methods to analyse the simulation result. The first collision point position coordinates are set as variables of the optimisation objective function. Through optimisation analysis, first collision point position coordinates can be obtained. And, the reliability of simulation result can be evaluated through the reliability analysis. By reconstructing a real-world vehicle–pedestrian impact accident, the performance of the method is evaluated. Keywords: vehicle–pedestrian impact accident; first collision point position; optimisation; accident reconstruction
Introduction As one of the major types of traffic accidents, pedestrian fatalities in developed countries accounted for 11–15% of road accident fatalities in recent years [10]. In developing countries, this proportion is even higher. In India, the proportion is 41%. In People’s Republic of China, according to the data collected by Shanghai Jiao Tong University from January 2007 to July 2009, 31% of fatal traffic accidents in the Shanghai urban area are vehicle–pedestrian accidents. Accident analysis can provide useful information to policy-makers or car manufacturers for the improvement of infrastructure and vehicles, thus helping to avoid accidents. Therefore, finding the root cause of accidents and determining the responsibility for the accident is of great importance. So, it is very important to analyse the cause of accidents and determine correctly the responsibility for the accident. Now, the police and other accident investigators are more inclined to utilise some specialised tools to reconstruct and analyse the vehicle–pedestrian accidents than before. As one of the powerful tools for accident investigation, numerical simulation for accident reconstruction has recently attracted increasing attention in the field of traffic safety. Through traffic accident reconstruction, usually the location of the collision point, the vehicle speed at the impact moment and the pedestrian behaviour can be figured out according to vehicle braking distance and the pedestrian thrown location [2, 3, 6]. In a traditional
∗
Corresponding author. Email: general
[email protected]
ISSN: 1358-8265 print / ISSN: 1754-2111 online C 2011 Taylor & Francis DOI: 10.1080/13588265.2010.548135 http://www.informaworld.com
numerical simulation of traffic accident, some simulation parameters, such as the first collision point position between vehicle and pedestrian, need to be determined by practical experience or subjective judgement [15]. The simulation result tends to differ from the real accident process dramatically. This study proposes a novel process that integrates the traditional simulation method and optimisation method for the reconstruction of traffic accidents. The first collision point position coordinates are set as a variable of the optimisation objective function. Through optimisation analysis, the first collision point position coordinates can be obtained. And, the reliability can be evaluated through the reliability analysis. Through the application of traffic police to real-world vehicle–pedestrian impact accident, the results show that the method has a positive social significance in the area of accident prevention and pedestrian safety protection. This study presents vehicle–pedestrian impact reconstruction method based on the numerical simulation and optimisation theory, especially the first collision point position identification method. The structure of the paper is as follows: the optimisation methods and applications are introduced in Section 2; optimisation analysis for accident simulation is presented in Section 3; a real-world vehicle–pedestrian accident is demonstrated with the proposed method in Section 4; and the conclusions and discussion are given in Section 5.
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The optimisation methods and applications The traffic accident simulation is an inverse dynamics problem. It can also be treated as a continuous feedback optimisation problem [12]. Firstly, according to the information left at the accident scene and the human body injuries, which are necessary for the preliminary reverse inference, a hypothetical situation at crash time can be acquired. Then, by comparing the simulation result with the accident scene information, the initial assumption is amended. The steps described above are repeated until the simulation result coincides with the result of accident scene investigation. The management and analysis of data in simulation will take much time, and the various constraints may be in contradiction with each other. Because of these limitations, the approximate results with real accidents’ process can usually be obtained, though far from the true course of accident [16]. At the same time, the simulation method cannot take into consideration and evaluate the inevitable uncertainty and variability caused by changes in the input variables [8]. In the study, the optimisation methods are adopted to replace classical manual optimisation, which can automatically change input parameters to minimise the error between simulation and real data. Through the coupling calculation between simulation and optimisation, it can gain the best coincidence result with scene investigation if optimised for all parameters. Design of experiments Design of experiments (DOE) is used to find out the parameters that have relatively large impact on the simulation result. The establishment of response surface model (RSM) can help visualise the problem so that it can be easier to understand. The objective of DOE is usually to get information as much as possible out of a limited number of experiments. The goal of accident simulation analysis is usually to show the statistical significance of an effect that a particular factor exerts on the dependent variable of interest. Many DOE methods have been developed and are available for different kinds of applications. The methods can be classified into two categories: orthogonal designs and random designs [5]. In this paper, we adopt one of the random methods, the Latin hypercube design (LHD) method [11], which is a multidimensional stratified sampling method. Compared with other sampling methods, the biggest advantage of LHDs is that any number of samples can be produced easily. In this way, each level is present in the design and the number of levels is maximised. A design is space-filling if the points are spread out and do not cluster in one portion of the experimental region. As a measure for the space-filling of a design, we take the minimal distance between two of its design points. The larger this minimal distance, the better the design will be. These features can meet the requirements of parameters analysis in traffic accident simulation.
The response surface is a hyper-surface, which describes the relationship between the experimental factors (input parameters) and the values of one or more measurable responses (output parameters). Once a DOE is performed, usually an RSM will be generated to fit the experimental results so that the researchers can get a better insight into the specific problem. Generally, there are two types of RSMs implemented, least squares fitting and interpolation. Optimisation analysis for traffic accident simulation To ensure the effectiveness of the optimisation, selection of a suitable optimisation model and strategy is of great importance. A traditional optimisation method includes the simplex method based on linear programming and the various iterations gradient algorithms based on non-linear programming such as the steepest descent method and conjugate gradient method [7]. The traditional optimisation method has the following limitations: (1) the single calculation method decreases the efficiency greatly; (2) moving towards the optimal direction restricts the capacity beyond the local optimisation; (3) the stop conditions are only the local optimum conditions; and (4) the requirements for the objective function and the restrict function limit the scope of the application of the algorithm. The optimisation algorithm is a heuristic algorithm that arose in the 1980s, including taboo search, simulated annealing, genetic algorithm, neural networks and ant colony algorithm. These methods are mainly used to solve the large number of practical problems [1]. At present, these algorithms have been developed in the theory and applied in practical problems. All of these algorithms have a common goal – finding the global optimum known as NP-hard combinatorial optimisation problems. On the grounds that these algorithms are independent of gradient information, they are especially suitable for solving the large-scale complex problem that cannot be solved by conventional methods. In vehicle–pedestrian accidents, injury of the pedestrian is not only caused by crashing with the vehicle but also by contact with the ground. Therefore, the vehicle–pedestrian traffic accident reconstruction calculation is a highly nonlinear optimisation problem, with objective function and constraints being implicit. As mentioned above, the sequential quadratic programming (SQP) method, known as a local optimisation method, and self-adaptive evolution (SAE) method, known as a global optimisation method, are used as the optimisation strategy in this study. In traffic accident optimisation, the parameters can be divided into two groups: pre-impact and post-impact parameters. Among them, vehicle speed at the impact moment, the first collision point position between vehicle and pedestrian, relative vehicle–pedestrian position and vehicle–pedestrian rest position are the primary factors
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Table 1. The primary parameters in accident reconstruction. Type Pre-impact
Post-impact
Parameter Vehicle impact speed Braking force Vehicle steering angle Pedestrian position Vehicle position Pedestrian posture Pedestrian contact characteristic Pedestrian walking speed Pedestrian rest position Pedestrian rest direction Pedestrian rest posture Vehicle rest position Pedestrian injury
Reference
Scene records Scene records Vehicle inspection report, autopsy report, hospital medical records
Role in optimisation Optimisation variables Optimisation variables Optimisation variables Optimisation variables
Optimisation variables Witness testimony Scene records Witness testimony Scene records Autopsy report, hospital medical records
for the identification of the kinematics in vehicle–pedestrian accidents. The vehicle–pedestrian rest position can be obtained from the accident scene information. Theoretically, if any three of the four factors are known, the remaining one can be figured out [4]. Thus, the aim of the optimisation analysis is to find the best pre-impact parameters to minimise the error of the rest position between simulation and real-world data. For vehicle–pedestrian accident reconstruction, the parameters involved in the optimisation model are listed in Table 1. Apart from vehicle speed at the impact moment, the first collision point position between vehicle and pedestrian, vehicle–pedestrian relative position and other unknown pre-impact parameters that have little correlativity with optimisation object, such as pedestrian walking speed and initial pedestrian posture, are dependent on estimation of engineering experience in this study. And, if they are regarded as design variables, the noise level of objective and constraint functions will increase. Some post-impact parameters, such as the pedestrian rest direction, posture and injuries, are used as the constraints to narrow down the feasible region. The relationship between input parameters and optimisation object posed by post-impact parameters is the implicit function of the design variables and often has highly non-linear behaviour. Reliability analysis for simulation result Simulation is typically run in a deterministic way. For a given set of input variables, the corresponding output will always be the same. However, in the real world, the output is usually not a fixed value but appears to be a distribution around a mean value due to the variability of the input variables. Such behaviour tends to result in an uncertain simulation result. To avoid this problem, we have to apply reliability methods as well as take the variability of the design variables into account so as to ensure the reliability of the simulation result.
Object function Constraint function Constraint function Object function Constraint function
Before making a reliability analysis, the input distribution, which is generally described by the probability distribution function, needs to be considered. In OPTIMUS, there are 11 types of built-in distributions that can be assigned to the input variable, including normal distribution, lognormal distribution, uniform distribution and so on. In vehicle–pedestrian accident reconstruction, the distribution of input parameters is a normal distribution. The output that exceeds the range of allowable error is considered as a failed result. The study adopts the MonteCarlo method to calculate the failure probability of an output by dividing the number of failures with the number of simulations, which yields an approximation of the actual failure probability. The confidence interval analysis of input parameters, which is an expression mean to the reliability analysis result, can acquire the influence of value range of the different input parameters on the distribution of output parameters value. By setting the value range of different input parameters, different reliability analysis calculations can be carried out. As a result, the change in simulation result that is caused by the change in value range of input parameters can be analysed. By calculating the ratio of the feasible result of output parameter, the reliability of simulation result under the value range of different input parameters can be verified [13]. Usually, the parameters have allowable error in a traffic accident’s responsibility cognizance, which is the permission range of output parameters in simulation. By comparing this range with the distribution range of output parameters obtained by reliability analysis, the influence of change of input parameters on simulation result can be evaluated. Optimisation analysis for accident simulation The first collision point position between vehicle and pedestrian can be obtained by optimisation analysis for accident simulation. In this case, vehicle braking distance and pedestrian throw distance are conditions of establishing
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the optimisation objective function. And, changes are observed with respect to distances with the first collision point position between vehicle and pedestrian. As one of the input variables of optimisation analysis, the first collision point position is a major factor affecting convergence of optimisation objective. So, the optimisation result of the first collision point position can be considered that is consistent with the actual accident data. At the same time in the optimisation objective convergence, it is necessary to satisfy the constraints of pedestrian rest posture and human body injury.
trian. When xk increases, the real vehicle brake distance or pedestrian throw distance decreases due to vehicle stop location, but the pedestrian thrown location does not change. Figure 1 shows that yobj is replaced by yobj − xk .
The optimisation objective function based on first collision point position In traffic accident reconstruction, the vehicle speed at the impact moment, the first collision point position between vehicle and pedestrian, the relative location of the vehicle and pedestrian, and so on are usually set as the input variables. And, the objective function for optimisation is usually the vehicle brake distance and pedestrian throw distance. Usually, the optimisation problems take the mathematical form as below: F (x) =
ωk
k
yk − yobj yobj
2 (1)
k = 1, 2, . . . , m gi (x) ≤ ci , l(m) si
≤ xi ≤
i = 1, 2, . . . n u(m) si ,
(2)
i = 1, 2, . . . p
(3)
where Equation (1) is the objective function that can be formulated as the quality function based on the weighted sum of square of the error, Inequality (2) is the constraints and Inequality (3) is the design domain for input parameter. Among them, yobj is the target value of the vehicle brake distance or pedestrian throw distance that comes from accident scene investigation reports. yk is the calculated result in every optimisation cycle, ωk is the weighting coefficient l(m) u(m) and si and si are the lower and upper bounds of the input parameter, respectively, that is defined according to the engineer experience. When the first collision point position between vehicle and pedestrian is set as an input parameter, yobj is changed with the first collision point position between vehicle and pedestrian. So, in this study, the objective function for the optimisation needs to be modified to Equation (4). F (x) =
k
ωk
yk − (yobj − xk ) (yobj − xk )
Figure 1. The change from A to B for first collision point position coordinates.
2
Pedestrian rest posture constraints In some simulation and optimisation analyses of traffic accidents, the reconstruction is based on the rest position of vehicle and pedestrian [14]. However, it is not enough to deduce the whole impact process. This study adds the pedestrian orientation and poses on the ground as constraints that provide a more precise reconstruction result. The study provides two types of methods to determine the pedestrian orientation, that is the pedestrian head and pelvis localisation method and the pedestrian head and twofoot localisation method. The pedestrian head and pelvis localisation method is shown in Figure 2(a). It defines the pedestrian position at impact moment as datum mark. Then, it sets the head and pelvis horizontal distance dx1 (dx2 ) and vertical distance dy1 (dy2 ) as the output of optimisation. The pedestrian’s horizontal orientation is confirmed by dx2 − dx1 . The pedestrian’s vertical orientation is confirmed by dy2 − dy1 . In 50-percentile standard pedestrian model as an example, the distance of the centroid between head and pelvis is 0.73 m. If the angle between rest direction of pedestrian and vehicle traffic direction is α, the upper limit of dy2 − dy1 can be calculated by 0.73 × cos α m. The pedestrian head and two-foot localisation method is shown in Figure 2(b). It is similar to the head and pelvis localisation method, which determines pedestrian’s orientation through the displacements of head and two-foot. The pedestrian rest posture includes supine and prostration. The supine and prostration constraint definition method is shown in Figure 3. In the pedestrian dummy model, it sets the back centroid coordinates zba and the sternum centroid coordinates zbo as the output of optimisation. If zba is smaller than zbo , the rest posture is supine, otherwise it is prostration.
(4)
k = 1, 2, . . . , m Among them, xk is the first collision point position coordinate in motion direction between vehicle and pedes-
Pedestrian body injury constraints The injury criterion of pedestrian body can be reflected through some physical parameters or function definition, which is used to reflect a certain degree of damage caused by
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Figure 2. The method defining pedestrian rest orientation. (a) Pedestrian head and pelvis localisation method. (b) Pedestrian head and two-foot localisation method.
the injury. These physical parameters include acceleration, velocity, force, displacement, deformation and so on. These can usually be tested by dynamic response of the human body. Various parts of the human body have their own injury indicators, and different loading methods need to study the different damage indexes. The tolerance level refers to the payload size leading to an injury or making an injury criterion reach the threshold value. The tolerance level is always aimed at the formulation of a damage criterion, specific human body parts and loading type. Pedestrian body damage constraints include tolerance level of head, chest and lower limb injuries based on TNO (The Netherlands Organization) pedestrian injury criterion [9]. Similar to the pedestrian orientation and pose constraints, pedestrian body injury is also set to constraints of convergence for optimisation objective. Head injury criterion In response to a comparison of the WSTC (Wayne State Tolerance Curve) and the GSI (Gadd Severity Index), the US government defined an injury criterion, the Head Injury Criterion (HIC), for the head: HIC =
⎡ max ⎣ 1 T 0 ≤ t1 ≤ t2 ≤ T E t2 −t1
t2 t1
⎤2.5 R(t)dt ⎦ (t2 −t1 ) (5)
Figure 3. The method defining pedestrian supine/prostration.
where T 0 is the starting time of the simulation, TE is the ending time of the simulation, R(t) is the resultant head acceleration in g (measured at the head’s centre of gravity) over the time interval T0 ≤ t ≤ TE, and t1 and t2 are the initial and final times of the interval, respectively, during which the HIC attains a maximum value. As for the GSI, a value of 1000 is specified for the HIC as a concussion tolerance level in frontal (contact) impact. For practical reasons, the maximum time interval (t2 − t1), which is considered to give appropriate HIC values, was set to 36 ms. The length of the time interval greatly affects the HIC calculation. In order to restrict the use of the HIC to hard head contact impacts, this time interval has been proposed to be further reduced to 16 ms.
Thoracic trauma index The thorax contains the next most critical organs to be protected from injuries. The bony cage structure of the thorax consists of 12 thoracic vertebrae, the sternum and 12 pairs of ribs that form a relatively rigid and also a flexible shell. A commonly stated human tolerance level for severe chest injury (AIS ≥ 4) is a maximum linear acceleration in the centre of gravity of the upper thorax of 60 g, sustained for 3 ms or longer. Thus, the criterion is not based on a single maximum value but on a sustainable level of linear acceleration. To predict the probability of serious injury to the ‘hard’ or bony thorax as a result of blunt lateral impact, the Thoracic Trauma Index (TTI) was proposed in
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1984. The TTI, as defined by Morgan, is calculated by the following equation: TTI = 1.4 × AGE + 0.5 × (RIBg + T 12g ) × MASS/MSTD,
(6)
where AGE is the age of the test subject in years, RIBg is the maximum absolute value of acceleration in g’s of the fourth and eighth ribs on struck side, in lateral direction, T 12g is the maximum absolute acceleration value in g’s of the 12th thoracic vertebra, in lateral direction, MASS is the test subject mass in kilograms and MSTD is the standard reference mass of 75 kg. Real-world impact reconstruction In this section, a real-world case is presented to illustrate and test the above-mentioned method. Through the method, the first collision point position between vehicle and pedesTable 2. The accident information in the case.
trian can be determined. This case is from well-documented fatal pedestrian accidents, which were collected during 2009 in Shanghai, People’s Republic of China. The police provided us with the detailed case reports, including scene investigation reports, autopsy reports, photographs and witnesses. All reconstructions are implemented through a dynamic simulation package in MADYMO. Optimisation procedures coupling with simulations are performed with the optimisation tool in OPTIMUS.
Accident reconstruction Case brief introduction A male pedestrian was struck by a bus heading southbound when he was walking across the street. According to the vehicle inspection report and forensic report, the bus front hit the pedestrian on his right side and is one of the reasons for pedestrian’s death. The accident data are shown in Table 2. The braking mark of the left front wheel and the right
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Figure 4. The initial impact modelling of bus and pedestrian. (a) the bus picture in accident. (b) The initial impact modelling.
front wheel are 11.54 and 11.74 m in length, respectively. The pedestrian bumped his head on the left windshield and knocked his leg and body against the bumper. The bumper’s distortion was notable, and the windshield was partially shattered in radial shape. The speed limit here is 50 km/h. Thus, the impact speed is one of the unknown parameters of interest. The accident investigation shows that the bus braking marks pass crosswalk lines. Therefore, whether the impact point locates in the crosswalk framework or not is the key for responsibility judgement. The pedestrian rest direction and posture was recorded from the picture of the traffic accident. The victim’s lying direction was found with his head towards the west and feet towards the east. The distance between his head and the road traffic line was 76 cm, and the distance between his breech and the road traffic line was 38 cm.
Accident simulation modelling and calculation The bus model adopts finite element model of shell element and rigid material. The bus body’s movement and rotation are described through a free hinge, and the spin motion of four wheels is presented through four rotary hinges. Correspondingly, the bus model has seven degrees of freedom (DOFs), which includes the four vertical DOFs of wheels and the three translational DOFs of vehicle centroid. The bus model includes the windscreen, bumper, the front lights, and the bus body and wheels. The parameters of the bus mass are set according to the accident identification report. According to the accident vehicle identification report of the automobile brake test, the friction coefficient between the road and wheels is 0.75. In MADYMO, the front collision contact characteristics can be defined by the force-deformation curves. The study defines the characteristic curve of the bus model based on database of DSD2006 (Dr. Steffan, Datentechnik). The pedestrian model adopts mid-size male dummy of TNO pedestrian model. Then, by multi-rigid-body dummy
model parameter settings, the dummy’s height, weight and shape characteristic parameters are adjusted on the basis of body characteristics of the victim. According to the information furnished by witnesses, the pedestrian’s initial state is set in a walking stance with the impacted leg in the rearmost position and the arms in a natural position. The pedestrian is given an initial velocity of 2 m/s. The initial impact modelling of bus and pedestrian is shown in Figure 4. To improve the efficiency of simulation and optimisation calculation, the contact definition only includes the front of the bus and pedestrian. It uses coupling contact algorithm between multi-rigid-body and finite element. The debris at accident scene, the traces of the crash and rest position of bus, the site of injuries, the walking direction of the pedestrian and other comprehensive information determine the start time of simulation. With manual optimisation calculations, the result of the simulation is able to meet the basic accuracy requirements of post-impact parameters, which cover the stop location of bus and pedestrian and so on. The simulation result is shown in Figure 5.
Figure 5. The simulation calculation result.
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Table 3. Optimisation variables and value range. Initial value
Xi
Optimisation variable
Cvel Posx
The bus speed (m/s) 8.33 The first collision point 0.0 position coordinate (m) The relative position −1.5 coordinate of pedestrian and bus (m) 0.75 Friction coefficient of bus ground Friction coefficient of 0.35 pedestrian ground
Posy
Cu Mu
Lower boundary
Table 4. Constraint conditions. Upper boundary
8.06 −0.78
9.72 2.58
−1.52
−1.48
Str1 Str2 ArmRDT ArmRDF LegRDT LegRDF
0.7
0.8
0.3
0.4
The bus speed is estimated to be 11.84 m/s at the impact moment.
Optimisation modelling Using the simulation method described above, it is difficult to accurately calculate the bus speed at the impact moment. And, the first collision point position between vehicle and pedestrian cannot be determined. In this study, the results of simulation are corrected by an optimisation method.
Figure 6. The optimisation process in the case.
VC HIC
Constraint function
Boundary
Pedestrian rest direction Pedestrian rest posture The minimum impact torque of right arm (Nm) The minimum impact force of right arm (N) The minimum impact torque of thigh (Nm) The minimum impact force of thigh (N) The thorax injury HIC (m/s) The head injury HIC
>0.4 >0 >100 >1960 >430 >6300 1000
As shown in Table 3, the study sets the bus speed at impact moment Cvel, the first collision point position coordinate between vehicle and pedestrian Posx, the relative position of bus and pedestrian Posy, the friction coefficient between vehicle and ground Cu, and the friction coefficient between pedestrian and ground Mu as the input variables. The possible range of optimisation variables can be estimated by the braking mark, bus deformation and human body damage. As shown in Table 4, the constraints of rest posture are treated as values of head displacement and pelvis displacement in simulation, which must meet the range of error
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Figure 7. The correlation between input and output parameters.
Figure 10. The local optimisation convergence curve.
Figure 8. Response surface models.
allowed. The constraints of human body damage are listed, including the arm contact force, the arm torque, the leg contact force, the leg torque, VC of thorax and the HIC of head. The optimum problem for this case is to find the bus speed at impact moment and the first collision point posi-
Figure 9. The global optimisation convergence curve.
tion, which can minimise the error of the displacement of pedestrian and vehicle between simulation and real-world data. The optimisation analysis process is shown in Figure 6; the output parameters are set as the pedestrian throw distance Dman and the vehicle braking distance Dcar. The optimisation objective function F (x) is expressed by the variable Opt. The Analysis1 defines MADYMO as a dynamic simulation tool. The case.xml and case.lds, and case.peak are input and output files, respectively.
Optimisation analysis Using the LHD methods, DOE analysis can choose the most relevant variables from the input parameters. The correlation values between input variables and optimisation
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objectives are shown in Figure 7. The analysis results show that the bus speed at impact moment Cvel and the first collision point position coordinate between vehicle and pedestrian Posx have the highest correlation with the simulation result. So, the optimisation analysis sets Cvel and Posx as input variables. Using the least squares method to build RSMs, the RSM will be generated to fit the experimental results to get a better insight in the design problem. y(x) = y(x; a1 , . . . , am ) =
m
aj Xj (x),
(7)
j =1
Figure 11. The distribution curve of the Posx.
where y is the objective function Obj. There are j independent variables, being fit to a model that has m adjustable parameters aj , j = 1, . . . , m. The general model of a linear least squares problem can be obtained, which is linear with respect to the parameters a1 , . . . , am . The functions X1 (x), . . . , Xm (x) are arbitrary fixed functions of x = (x1 , . . . , xm )t ∈ R k . For these linear models, the general model can be generalised by defining the optimisation problem, namely
min
a1 ,···am
n i=1
⎡ ⎣y i −
m
⎤2 aj Xj (x i )⎦
(8)
j =1
The RSM between objective function Obj and Cvel, Posx, is shown in Figure 8. The range of Cvel and Posx can be redefined as (8.5, 9.5) and (0.0, 2.0), which can narrow the search scope of optimisation and speed up the convergence rate. Then, on the basis of RSM, the model can perform global optimisation using the SAE method. The optimisation result is shown in Figure 9. The estimated value of input parameters that meet the optimisation goal can be obtained. On the basis of the result of global optimisation, Cvel and Posx adopt central values of 9.0 m/s and 1.0 m, respectively, which are set as initial value of the local optimisation. The SQP method is used to perform the local optimisation. As shown in Figure 10, the optimisation converges to 9.26 m/s, 1.25 m after six cycles. The bus speed at impact moment is 9.26 m/s. The distance of pedestrian to the crosswalk line in the first collision point position is 1.25 m.
Analysis of reconstruction result Applying reliability analysis methods, the simulation result reliability can be evaluated. The input variable value of the optimisation results is used as the source for reliability analysis. The normal distribution curve of the input variables value can be defined. The distribution curve of the Posx is
shown in Figure 11. The value of centre and σ (standard deviation) are calculated to be 1.25 m and 0.01, respectively. The Monte-Carlo method is applied to the reliability analysis. Through the histogram of distribution, we figure out the ratio of the output variables falling within the constraint range. The distribution histogram of optimisation object is shown in Figure 12. In the 120 groups of reliability analysis tests, σ is estimated to be 0.00181. Moreover, there are four groups beyond the scope of the output error tolerance. All the facts meet the error request. The vehicle and pedestrian rest position is shown in Figure 13. The process of the accident is shown in Figure 14. The results showed that, after the pedestrian is hit by the right side of the bus, his head and chest severely impact the left side of the windshield and the front hoarding. Then, the left shoulder contacts ground first. The head left impacts the ground. Comparing with the real situation, we can see that the contact position of people and vehicles and pedestrian injury from simulation is accurate enough.
Figure 12. The optimisation object distribution histogram.
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Figure 13. The result of optimisation. Table 5. Pedestrian injury simulation results. Autopsy result
Damage value
Tolerance limit
Head Thorax Upper limb
Fatal Did not test the obviously injury Fracture
1000, 50% prob. of AIS 3+ 1 m/s, 50% prob. of AIS 3+ 100 Nm 1960 N, 50% prob. of fracture
Lower limb
No fracture
HIC36 = 2016 VC = 0.22652 m/s Peak force of the thigh 3745 N Peak torque of the thigh 187 Nm Peak force of the thigh 4340 N Peak torque of the thigh 283 Nm
430 Nm 6300 N, 50% prob. of fracture
Figure 14. The accident process by simulation. (a) 50 ms. (b) 100 ms. (c) 150 ms. (d) 300 ms. (e) 400 ms. (f) 4000 ms.
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Figure 15. The pedestrian rest position. (a) Pedestrian vertical rest position. (b) Pedestrian horizontal rest position.
The rest position of the pedestrian is shown in Figure 15, which includes the pedestrian vertical rest position and horizontal rest position. As shown in Table 3, the real-world distance between the pedestrian positions is described by accident investigation. The rest position of simulation result corresponds well with the accident data. As shown in Figure 16, the impact point between head and windshield from the simulation is also compared. The autopsy report indicates that the impact location is the right front side of the head. The lower left side of the vehicle’s front windshield is broken radially. The pedestrian head injury and the windshield breakage correspond well with the accident data. As shown in Table 5, the optimisation result values of ArmRDT, ArmRDF, LegRDT, LegRDF, VC and HIC are calculated to be 187 N/m, 3745 N, 283 N/m, 4340 N, 0.22652 m/s and 2016, respectively. The rest position and the pedestrian injuries correspond well with the accident data. The acceleration curve of pedestrian head is shown in Figure 17. There are two peaks in the curve that express the impact acceleration of head and bus at 180 ms and the impact acceleration of head and ground at 1070 ms. It shows that the impact acceleration of head and ground is higher than the impact acceleration of head and bus. The impact of head and ground is a more fatal cause than the impact of head and bus. The HIC of head is 2016. On the basis of the HIC, the head injury may cause the death of the pedestrian. The simulation results show that the bus speed at the impact moment is 9.26 m/s (33.34 km/h), which does not exceed the local speed limit. In addition, it is proved that the impact most probably happened at the point outside the crosswalk. In conclusion, according to the reconstruction findings, the pedestrian, without crossroad inside crosswalk, should take the main responsibility for the accident.
Figure 16. The comparison between bus deformation and simulation results. (a) Bus deformation picture. (B) Contact of pedestrian with bus in simulation.
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Figure 17. The acceleration curve of pedestrian head.
Conclusion In vehicle–pedestrian impact accidents, the first collision point position between vehicle and pedestrian is the key for responsibility judgement. This paper applies computersimulated reconstruction in vehicle–pedestrian accident investigation and uses optimisation methods to analyse the simulation result. The application of this method shows that it is effective in accident prevention and pedestrian safety protection. The optimisation method is adopted to replace classical manual optimisation, which automatically adjusts input parameters to minimise the error between simulation results and the real data. The first collision point position coordinates are set as variables of optimisation objective function. The pedestrian rest posture and pedestrian body injury is set as the constraint conditions. Because the displacement and posture parameters have higher non-linearity, the automatic optimisation method is more suitable for the convergence of optimisation objective. The systematic methods in the research are designed to perform optimisation, deviation and reliability analysis on the basis of parameters such as tyre marks, rest positions, position of vehicle–pedestrian contact point and human body injuries, which can be directly obtained from the accident scene. Because of the fact that the first collision point does not belong to those kinds of parameters that
can be determined directly at the accident scene, this paper adopts an alternative way to evaluate the accuracy and reliability of the reconstruction method, that is, to consider those direct parameters as the key factors for optimisation, carry out the error and reliability analysis, and finally recalculate the first collision point and perform statistical analysis on the collision point position after identifying the correctness of reconstruction findings by making comparison between the simulation results and the real investigation data. Therefore, if the derivation of the first collision point goes beyond the range that the method has found, those directly obtained parameters will suffer from greater calculation errors, which can lead to significant calculation errors. The results show that the method is very effective in finding an optimum solution and receiving a sufficiently accurate value of the first collision point position. The simulation results can meet the requirements for traffic accident responsibility and judgement. Although the proposed method can provide an effective solution for obtaining the first collision point position between vehicle and pedestrian, it should also be noticed that some pre-impact parameters still need to be estimated through engineering experience because of the limitations of computational capability. For example, pedestrian initial posture is estimated by pedestrian injuries and the relationship between vehicle and pedestrian is not modelled
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mathematically. Also, some accurate finite element models for local body are needed to investigate the mechanisms of pedestrian injuries for our future study. Acknowledgments The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Nos. 60970049, 50875166, 50705058).
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