典型汽车碰撞模型病态问题的成因分析与处理

Cause Analysis and Solution of Ill-posed Problems in Typical Vehicle Collision Model

为解决典型汽车碰撞模型病态问题,根据其几何特征,应用线性空间与内积空间理论计算了模型中任意两个方程之间的夹角。结果表明,病态问题的成因在于模型中某些方程之间存在很强的线性相关性。通过对这些方程进行数学变换,在病态范围内建立了改进模型。应用摄动理论,提出了对改进模型可靠性的检验方法。实例检验结果表明,在病态范围内当模型系数存在1%误差时改进模型能将碰撞前车速误差控制在合理的范围内,说明在病态范围内建立改进模型能解决模型病态问题。

For solving the ill-posed problems in typical vehicle collision model,the theories of linear space and inner product space is applied to calculate the angle between any two equations in the model according to the geometric feature of the problems. The results show that ill-posed problems are caused by the strong linear correlation among some equations in the model. Improved models can be developed in ill-posed ranges by mathematical transform on equations with strong linear correlation. The perturbation theory is also applied to establish a method for examining the reliability of improved model. The results of an example examination show that when the errors of model parameters are within 1% in ill-posed ranges,the error of pre-impact speed in the improved models can be controlled within a reasonable range,demonstrating that ill-posed problems can be solved by creating improved models in ill-posed ranges.