Simultaneous Localisation and Mapping (SLAM) is the archetypal chicken and egg problem: Localisation of a robot with respect to a map requires an estimate of the map, while mapping an environment from data acquired by a robot requires an estimate of the robot localisation. The nonlinearity and co-dependence of the SLAM problem has made it an ongoing research problem for more than thirty years. The present paper details recent advances in understanding the SLAM problem, specifically the existence of an underlying geometry and symmetry structure that provides significant insight into the difficulties that have plagued many SLAM algorithms. To demonstrate the power of the geometric insight we derive a constant gain observer for the SLAM problem that; that does not depend on linearisation, has globally asymptotically stable error dynamics, is very robust, and operates in dynamic environments (estimating the landmark velocities as states in the observer).

同时定位和映射 (SLAM) 是典型的鸡和蛋问题：机器人相对于地图的定位需要对地图进行估计，而根据机器人获取的数据映射环境需要对机器人定位进行估计。SLAM 问题的非线性和相互依存性使其成为一个持续研究三十多年的问题。本论文详细介绍了在理解 SLAM 问题方面的最新进展，特别是潜在几何和对称结构的存在，它提供了对困扰许多 SLAM 算法的困难的重要洞察。为了证明几何洞察力的力量，我们为 SLAM 问题导出了一个恒定增益观察器：不依赖于线性化，具有全局渐近稳定的误差动态，