SUMMARYA nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker–Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space,SO(3), and the Euclidean motion group of the plane,SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.

中文翻译：

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使用指数映射的随机非完整系统的运动学状态估计和运动规划

总结 一个非完整系统受到来自环境的外部噪声或在其自身的执行器中的内部噪声的影响，将以由一组轨迹描述的随机方式演化。这种轨迹集合等价于通常在李群上演化的 Fokker-Planck 方程的解。如果要估计这样一个系统的最可能状态，并制定从当前状态开始的后续运动计划，以便将系统以高概率移动到期望的状态，那么建模系统的概率密度如何进化很关键。求解在李群上演化的 Fokker-Planck 方程的方法变得很重要。可以使用组傅立叶变换的操作特性来求解此类方程，其中不可约酉表示 (IUR) 矩阵起着关键作用。因此，我们开发了一种简单的方法，用于对机器人技术中最感兴趣的两个组的所有 IUR 矩阵进行数值逼近：三维空间中的旋转组，所以(3)，以及平面的欧几里得运动群，东南(2)。这种方法使用来自这些组的李代数的指数映射，并利用李代数表示矩阵的稀疏性质。还探索了其他用于组密度估计的技术。计算的密度应用于平面运动推车的概率路径规划和三维空间中的灵活针转向。在这些示例中，将人工噪声注入计算模型（而不是实际物理系统中的噪声）用作搜索配置空间和规划路径的工具。最后，我们说明了密度估计问题是如何在诸如陀螺仪等定向传感器中的物理噪声表征中出现的。