Wang Yunfeng, Chirikjian Gregory S
Department of Mechanical Engineering, The College of New Jersey, Ewing, NJ 08628.
Int J Rob Res. 2008;27(11-12):1258-1273. doi: 10.1177/0278364908097583.
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, SE(3), 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 nonparametric 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.
欧几里得运动群上的误差传播出现在多个领域，例如移动机器人导航中的航位推算误差，以及从运动链（如机械手和生物大分子）的基座到远端累积的关节误差。我们以无坐标的方式处理刚体姿态中的误差传播。在本文中，我们展示了如何利用李代数和李群理论，将在欧几里得运动群SE(3)上通过卷积传播的误差近似到二阶。然后我们展示了如何通过这里推导的递归公式传播小误差（但又不至于小到线性化有效的程度）。这个公式考虑了二阶误差，而之前的研究仅考虑了一阶情况。我们的公式在非参数意义上适用于任何形式的概率密度函数（不仅是高斯函数）。数值测试证明了该二阶理论在机械手和带斜角尖端的柔性针的情况下的准确性。
