# LIF-M: A Manifold-Based Approach for 3D Robot Localization in Unstructured Environments
## 1 Introduction  
With the advance of technology, mobile robots have become increasingly familiar, and the pursuit of enhanced localization accuracy through multi-sensor fusion has gained substantial attention across various fields [1]. In the domain of multi-sensor fusion, achieving high-quality real-time Simultaneous Localization and Mapping (SLAM) methods has emerged as a prominent research avenue. Methods centered around laser rangefinders excel in capturing intricate environmental features over long distances; however, they face challenges in scenarios lacking distinctive features, such as extended corridors or vast open spaces [2]. Common strategies involve amalgamating data from Inertial Measurement Units (IMU), satellite localization modules (e.g., Real-Time Kinematic, RTK), and laser rangefinders to bolster their individual robustness and precision. In contrast to SLAM, standalone robot localization predominantly involves establishing prior knowledge about pose within a known environment [3, 4], followed by iterative refinement within the mapped context. In recent years, localization has evolved more toward data processing. [PERSON] et al. [5] used AprilTags to construct a relative spatial coordinate system to provide accurate coordinate distribution for the algorithm, then proposed a time series prediction model [6] to correct unexpected errors. The data can be updated by replacing similar sensors [7]or establishing active robot exploration criteria [8], allowing for better localization accuracy. Fusion methods can be broadly categorized into two major classes: methods based on filtering [9; 10; 11], and methods based on optimization [12; 13; 14]. Filter-based approaches typically employ the Extended Kalman Filter (EKF) to propagate system states using observations from laser radar and IMU. This is accomplished via linearization and the application of the core of the Kalman filter to the remaining nonlinear processes and measurement models of the system. Optimization-based approaches often utilize techniques such as Factor Graph Optimization [15] or [PERSON]'s method [16; 17], treating measurements as constraints or factors to achieve optimal pose estimation. LINS [18], which relies on iterative Kalman filtering, has demonstrated superior performance across various scenarios. However, the computation of Jacobian matrices for individual sensors and the recording of landmark points present significant challenges for the EKF series of filters, particularly for larger scenes and situations involving weaker computational resources. The Unscented Kalman Filter (UKF), introduced by [PERSON] et al. [19], reduces the computational demands of handling nonlinear functions by transforming the problem into a probability density distribution of the system state variables. Additionally, there is substantial evidence that the use of the Lie group structure in the manifold space SE(3) plays a significant role in probabilistic robotics [20]. In the literature [21], the concept of an Invariant Extended Kalman Filter (INEKF) based on Lie groups has been proposed to address the inconsistency issues in the EKF series. Compared to the probability density defined on groups, as mentioned in [22], the exponential mapping method employed by INEKF is more efficient and stable. Lie group-based extended Kalman filters have been applied in the context of vision [23]. Research has shown that applying a Lie group to the filter requires a consistency improvement method similar to multi-group multiplication to obtain state errors in order to maintain system consistency by estimating the Jacobian matrix or observability constraints. Despite the improved consistency, there may be inaccurate state estimates or subelegant comparable matrices. It requires considerable time and computing power to manually solve the scanned images of RGB cameras, and it is challenging even with the experience and prior information of practitioners [24]. For such localization tasks, researchers have explored the use of deep learning techniques. [PERSON] et al.