Probabilistic Inverse Simulation and Its Application in Vehicle Accident Reconstruction

Transcription

1 ASME 2013 IDETC/CIE 2013 Paper number: DETC Probabilistic Inverse Simulation and Its Application in Vehicle Accident Reconstruction Xiaoyun Zhang Shanghai Jiaotong University, Shanghai, China Zhen Hu, Xiaoping Du Missouri S &T, Rolla, MO, USA

2 Outline Inverse simulation example: traffic accident reconstruction Generalized probabilistic inverse simulation Inverse simulation with the highest probability density Examples Conclusions Future work 2

3 An Example for Inverse Simulation Traffic Accident Reconstruction Direct simulation: Cause => Consequence Inverse simulation: Consequence => Cause Vehicle accident reconstruction involves inverse simulation Given: accident consequences Find: pre-accident events 3

4 More Examples Identify pre-impact velocity Determine vehicle trajectory Identify cause of injury 4

5 Challenges in Traffic Accident Reconstruction Accident reconstruction simulation is computationally expensive Many uncertainties Input information is limited Traditional reconstruction may generate multiple solutions 5

6 gx ( ) Inverse Simulation Under Uncertainty Direct simulation: Given x find y x gx ( ) y Inverse simulation Given: y Find: x Vehicle Accident Reconstruction x: vehicle velocity just before collision y: accident consequences from the scene Random variables exist, such as coefficient of friction 6

7 Probabilistic Inverse Simulation Classify inputs into three groups x ( x, x, x ) unkn rand kn - x unkn, unknown deterministic variables - x rand, unknown variables with known distributions x kn -, variables that are known deterministically - Simulation equations y gx ( ) gx (, x ) unkn rand T 7

8 The New Method Maximize the joint probability density Function (PDF) Max: Joint PDF Subject to: Simulation Equations 8

9 Detailed Model Reliability analysis and optimization are employed Maximize the joint probability density A unique solution is identified The solution of x is x, where the joint rand * rand probability density is maximum. y y y ( x, x ) 1 1 unkn rand ( x, x ) 2 2 unkn rand m g g g m ( x, x ) unkn rand max f ( xrand) ( xunkn, xrand) subject to y gx ( unkn, xrand) 9

10 When Model Uncertainty Included max f ( xunc, xrand) ( xunkn, xrand) subject to (1 1 ) y1 g1( xunc, xrand) (1 1 ) y1 (1 2 ) y2 g2( xunc, xrand) (1 2 ) y2 (1 ) y g ( x, x ) (1 ) y m m m unc rand m m Model uncertainty is treated in an interval The bound is the percentage error 10

11 Implementation Transform random variables into standard normal random variables U nrand min u2 i ( xunkn, u) i1 subject to 1 T (1 1 ) y1 g1( xunkn, F ( ( u)) ) (1 1 ) y1 1 T (1 2 ) y2 g2( xunkn, F ( ( u)) ) (1 2 ) y2 1 T (1 ) y g ( xunkn, F ( ( u)) ) (1 ) y m m m m m Control variables are xunkn and u 11

12 Advantages All information available is used. Highest confidence is obtained. A unique solution is identified. 12

13 Examples A Mathematical Example y y g ( x, x ) 1 1 unkn rand x x x g unkn unkn rand,1 ( x, x ) 2 2 unkn rand rand,1 rand,2 x 2x 3x rand,2 13

14 Examples A Mathematical Example y y g ( x, x ) 1 1 unkn rand x x x g unkn unkn rand,1 ( x, x ) 2 2 unkn rand rand,1 rand,2 x 2x 3x rand,2 x 0.5( u unkn 1 u2) 1 x u unkn u2 0 * * u1 0.4 u u 1 2 2u 2 14

15 Application Traffic Accident Reconstruction The vehicle speed at the moment of accident needs to be determined. Post-accident data were collected at the accident scene, such as the rest position of the victim. 15

16 Task Given: y, rest position of the victim according to blood marks Find: xunkn, vehicle speed at the moment of accident xrand, coefficient of friction and relative distance between vehicle and victim 16

17 Direct Crash Simulation 17

18 Inverse Simulation Model Probabilistic inverse simulation 2 2 Min u i v, u1, u2 i1 Subject to ξ1 u1, ξ2 u2 ξ3 (2 v vl vu) ( vu vl) 3 3i 3 i j *(1 ) x sx x (, i j, k) Li( ξ1) i0 j0 k0 ( ξ ξ * Hj 2) Hk( 3) sx(1 x) 3 3i 3 i j * y sy( 1 y) (, i j, k) Li( ξ1) i0 j0 k0 H ( * j ξ2) Hk( ξ3) sy(1 y) Polynomial Chaos Expansion (PCE) Surrogate models are constructed to replace the direct simulation model 18

19 Results Velocity estimated from video record = [67, 69] km/h Velocity from inverse simulation = km/h 19

20 Conclusions Uncertainties in inverse simulation should be considered. Probabilistic analysis methods can be used to accommodate uncertainties. The proposed method involves reliability analysis and optimization. It maximizes the joint PDF. Examples demonstrate the effectiveness of the method. 20

21 Future work Consider conditional probabilities Incorporate probabilistic model uncertainty Acknowledgments China Scholarship Council The Intelligent Systems Center at Missouri S&T 21