Inconsistency issue is one crucial challenge for the performance of extended Kalman filter (EKF) based methods for state estimation problems, which is mainly affected by the discrepancy of observability between the EKF model and the underlying dynamic system. In this work, some sufficient and necessary conditions for observability maintenance are first proved. We find that under certain conditions, an EKF can naturally maintain correct observability if the corresponding linearization makes unobservable subspace independent of the state values. Based on this theoretical finding, a novel affine EKF (Aff-EKF) framework is proposed to overcome the inconsistency of standard EKF (Std-EKF) by affine transformations, which not only naturally satisfies the observability constraint but also has a clear design procedure. The advantages of our Aff-EKF framework over some commonly used methods are demonstrated through mathematical analyses. The effectiveness of our proposed method is demonstrated on three simultaneous localization and mapping (SLAM) applications with different types of features, typical point features, point features on a horizontal plane and plane features. Specifically, following the proposed procedure, the naturally consistent Aff-EKFs can be explicitly derived for these problems. The consistency improvement of these Aff-EKFs are validated by Monte Carlo simulations.

翻译：不一致性问题是基于扩展卡尔曼滤波（EKF）的状态估计方法性能面临的关键挑战之一，其主要受EKF模型与底层动态系统之间可观测性差异的影响。本文首先证明了一些可观测性保持的充分必要条件。我们发现，在一定条件下，若对应的线性化过程能使不可观测子空间独立于状态值，则EKF可自然地保持正确的可观测性。基于这一理论发现，本文提出了一种新颖的仿射EKF（Aff-EKF）框架，通过仿射变换克服标准EKF（Std-EKF）的不一致性问题；该框架不仅能自然地满足可观测性约束，还具有清晰的设计流程。通过数学分析，我们证明了所提Aff-EKF框架相较于一些常用方法的优势。本文方法在三种具有不同类型特征（典型点特征、水平面上的点特征以及平面特征）的同步定位与建图（SLAM）应用中验证了其有效性。具体而言，遵循所提出的流程，可针对这些问题显式推导出天然一致的Aff-EKF。蒙特卡洛仿真验证了这些Aff-EKF在一致性方面的提升。