【泡泡一分钟】最小二维位姿图SLAM问题的测地线与弦线成本的最小值分析

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标题：Analysis of minima for geodesic and chordal cost for a minimal 2D pose-graph SLAM problem

作者：Felix H. Kong,Jiaheng Zhao

来源：2020 IEEE International Conference on Robotics and Automation (ICRA)

编译：张宁

审核：柴毅，王靖淇

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摘要

在本文中，我们表明，对于最小的2D姿态图SLAM问题，即使在理想测量和球面协方差的理想情况下，也可以使用测地距离（在2D中为“包裹函数”）比较角度，从而得出多个次优局部极小值。对于某些示例，我们通过数值估算吸引到这些局部极小值的区域，并提供证据表明它们具有非零度量，并且这些区域的大小会随着添加噪声而增大。相比之下，在相同的假设下，我们表明角度误差的弦距离表示在周期性上具有唯一的最小值。对于弦成本，我们发现无法收敛到全局最小值的初始条件要少得多，由于数值问题而失败，并且在我们的示例中似乎不随噪声而增长。

图1.（a）此热图说明了为什么在使用测地距离时可能会有局部最小值。

图2.使用测地成本从第五节中无噪声的三位示例问题中绘制的图。

图3.使用弦成本从第IV节中无噪声的3个姿势示例问题中得出的图。

Abstract

In this paper, we show that for a minimal 2D pose-graph SLAM problem, even in the ideal case of perfect measurements and spherical covariance, using geodesic distance (in 2D, the “wrap function”) to compare angles results in multiple suboptimal local minima. We numerically estimate regions of attraction to these local minima for some examples, give evidence to show that they are of nonzero measure, and that these regions grow in size as noise is added. In contrast, under the same assumptions, we show that the chordal distance representation of angle error has a unique minimum up to periodicity. For chordal cost, we find that initial conditions failing to converge to the global minimum are far fewer, fail because of numerical issues, and do not seem to grow with noise in our examples.