Error propagation on the Euclidean motion group arises in a number of areas such as in dead reckoning errors in mobile robot navigation and joint errors that accumulate from the base to the distal end of kinematic chains such as manipulators and biological macromolecules. We address error propagation in rigid-body poses in a coordinate-free way. In this paper we show how errors propagated by convolution on the Euclidean motion group, SE133, can be approximated to second order using the theory of Lie algebras and Lie groups. We then show how errors that are small (but not so small that linearization is valid) can be propagated by a recursive formula derived here. This formula takes into account errors to second order, whereas prior efforts only considered the first-order case. Our formulation is non-parametric in the sense that it will work for probability density functions of any form (not only Gaussians). Numerical tests demonstrate the accuracy of this second-order theory in the context of a manipulator arm and a flexible needle with bevel tip.

中文翻译：

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运动群上误差传播的非参数二阶理论

欧几里得运动群上的误差传播出现在许多领域，例如移动机器人导航中的航位推算误差和从运动链（如机械手和生物大分子）的底部到远端累积的关节误差。我们以无坐标的方式解决刚体姿势中的误差传播问题。在本文中，我们展示了如何使用李代数和李群理论将欧几里德运动群 SE133 上的卷积传播的误差近似为二阶。然后，我们展示了如何通过此处导出的递归公式来传播很小的错误（但不是小到线性化有效）。该公式考虑到二阶误差，而先前的努力仅考虑一阶情况。我们的公式是非参数的，因为它适用于任何形式的概率密度函数（不仅是高斯函数）。数值测试证明了这种二阶理论在机械臂和带有斜角尖端的柔性针的背景下的准确性。