In this paper, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present CPL-SLAM, an efficient and certifiably correct algorithm to solve planar graph-based SLAM using the complex number representation. We formulate and simplify planar graph-based SLAM as the maximum likelihood estimation (MLE) on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to convex complex semidefinite programming (SDP). Furthermore, we simplify the corresponding complex semidefinite programming to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region (RTR) method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through applications of CPL-SLAM and comparisons with existing state-of-the-art methods on planar graph-based SLAM, which indicates that our proposed algorithm is capable of solving planar graph-based SLAM certifiably, and is more efficient in numerical computation and more robust to measurement noise than existing state-of-the-art methods. The C++ code for CPL-SLAM is available at https://github.com/MurpheyLab/CPL-SLAM.

中文翻译：

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CPL-SLAM：使用复数表示的高效且可证明正确的基于平面图的 SLAM

在本文中，我们考虑了基于平面图的同时定位和映射 (SLAM) 问题，该问题涉及自主代理的姿势和观察到的地标的位置。我们提出了 CPL-SLAM，这是一种使用复数表示解决基于平面图的 SLAM 的有效且可证明正确的算法。我们将基于平面图的 SLAM 公式化并简化为单位复数乘积的最大似然估计 (MLE)，并将这个非凸二次复优化问题放松到凸复半定规划 (SDP)。此外，我们在复杂斜流形上将相应的复杂半定规划简化为黎曼阶梯优化（RSO），可以用黎曼信任域（RTR）方法求解。此外，我们证明，只要噪声幅度低于某个阈值，SDP 松弛和 RSO 简化就会很严格。通过 CPL-SLAM 的应用以及与现有的基于平面图的 SLAM 的最先进方法的比较，验证了这项工作的有效性，这表明我们提出的算法能够可靠地解决基于平面图的 SLAM，并且与现有的最先进方法相比，它在数值计算方面更有效，对测量噪声更鲁棒。CPL-SLAM 的 C++ 代码可从 https://github.com/MurpheyLab/CPL-SLAM 获得。这表明我们提出的算法能够可靠地解决基于平面图的 SLAM，并且比现有的最先进方法在数值计算方面更有效，对测量噪声更鲁棒。CPL-SLAM 的 C++ 代码可从 https://github.com/MurpheyLab/CPL-SLAM 获得。这表明我们提出的算法能够可靠地解决基于平面图的 SLAM，并且比现有的最先进方法在数值计算方面更有效，对测量噪声更鲁棒。CPL-SLAM 的 C++ 代码可从 https://github.com/MurpheyLab/CPL-SLAM 获得。