In this paper, we investigate the problem of estimating the 4-DOF (three-dimensional position and orientation) robot-robot relative frame transformation using odometers and distance measurements between robots. Firstly, we apply a two-step estimation method based on maximum likelihood estimation. Specifically, a good initial value is obtained through unconstrained least squares and projection, followed by a more accurate estimate achieved through one-step Gauss-Newton iteration. Additionally, the optimal installation positions of Ultra-Wideband (UWB) are provided, and the minimum operating time under different quantities of UWB devices is determined. Simulation demonstrates that the two-step approach offers faster computation with guaranteed accuracy while effectively addressing the relative transformation estimation problem within limited space constraints. Furthermore, this method can be applied to real-time relative transformation estimation when a specific number of UWB devices are installed.

翻译：本文研究了利用机器人里程计及机器人间距离测量值估计机器人间4自由度（三维位置与方向）相对框架变换的问题。首先，我们采用基于最大似然估计的两步估计方法。具体而言，通过无约束最小二乘与投影获得良好的初始值，再通过一步高斯-牛顿迭代实现更精确的估计。此外，本文给出了超宽带（UWB）的最优安装位置，并确定了不同UWB设备数量下的最小运行时间。仿真表明，该两步法在保证精度的同时具有更快的计算速度，并能有效解决有限空间约束下的相对变换估计问题。进一步地，当安装特定数量的UWB设备时，该方法可应用于实时相对变换估计。