INS/GPS组合导航中的病态问题及其处理方法

Solution to Ill-posed Problems in INS/GPS Integrated Navigation

在地球表面附近的组合导航中,一般以经、纬度表示水平位置,单位为弧度(rad),以高程表示竖直位置,单位为米(m)。1m的定位误差仅相当于约10-7rad,从而造成定位协方差矩阵的条件数达1013量级。Kalman滤波中要对协方差矩阵求逆,过高的条件数将引起严重的病态问题,从而造成很大的数值误差,影响滤波精度,甚至造成滤波发散。提出一种直接的解决方法,即构建一种避免病态问题的组合导航滤波模型。具体过程为:引入一种尺度因子(即平均地球半径)对组合导航系统的状态量、观测量、状态方程以及观测方程进行线性变换,从而对经、纬度误差进行适当的尺度化,明显降低协方差矩阵的条件数,有效避免了滤波过程中病态问题的出现。新方法在不明显增加计算量的前提下有效解决了病态问题,并保证了Kalman滤波的最优性质。数值仿真验证了该方法的有效性。

In the integrated navigation near the earth surface,horizontal positions are represented by latitude and longitude with radians as their units while vertical positions by heights with meters as units.However the position error of one meter is equivalent to about 10^-7 radians, resulting in a positioning error covariance with conditioning number being about 10^13.As the inverse of the covariance matrix is necessary in the Kalman filter, a large conditioning number may lead to serious ill-posed problems, which further cause a bad impact on the accuracy of the system, and sometimes make the filter diverge.In this contribution, a more direct method is proposed, i.e.that a kind of filtering model without potential ill-posed problems is constructed. A scaling factor, i. e. the average radius of the earth, is introduced to linearly transform the state process equation and the measurement equation, hence the latitude and longitude error is enlarged in the number and the covariance matrix is properly scaled.As a result, the conditioning number is reduced significantly to avoid the potential ill-posed problem effectively. The proposed method can retain the optimality of Kaman filter without obviously increasing the computational load. Simulation results validate the effectiveness of the proposed method.