In this paper, we consider the closed-loop control problem of nonlinear robotic systems in the presence of probabilistic uncertainties and disturbances. More precisely, we design a state feedback controller that minimizes deviations of the states of the system from the nominal state trajectories due to uncertainties and disturbances. Existing approaches to address the control problem of probabilistic systems are limited to particular classes of uncertainties and systems such as Gaussian uncertainties and processes and linearized systems. We present an approach that deals with nonlinear dynamics models and arbitrary known probabilistic uncertainties. We formulate the controller design problem as an optimization problem in terms of statistics of the probability distributions including moments and characteristic functions. In particular, in the provided optimization problem, we use moments and characteristic functions to propagate uncertainties throughout the nonlinear motion model of robotic systems. In order to reduce the tracking deviations, we minimize the uncertainty of the probabilistic states around the nominal trajectory by minimizing the trace and the determinant of the covariance matrix of the probabilistic states. To obtain the state feedback gains, we solve deterministic optimization problems in terms of moments, characteristic functions, and state feedback gains using off-the-shelf interior-point optimization solvers. To illustrate the performance of the proposed method, we compare our method with existing probabilistic control methods.

翻译：本文研究了存在概率不确定性与干扰的非线性机器人系统的闭环控制问题。具体而言，我们设计了一种状态反馈控制器，旨在最小化因不确定性和干扰导致的系统状态相对于标称状态轨迹的偏差。现有解决概率系统控制问题的方法通常局限于特定类别的不确定性与系统，例如高斯不确定性、高斯过程及线性化系统。本文提出了一种能够处理非线性动力学模型与任意已知概率不确定性的方法。我们将控制器设计问题表述为一个基于概率分布统计量（包括矩与特征函数）的优化问题。特别地，在所提出的优化问题中，我们利用矩与特征函数将不确定性传播至机器人系统的非线性运动模型。为降低轨迹跟踪偏差，我们通过最小化概率状态协方差矩阵的迹与行列式，减少概率状态围绕标称轨迹的不确定性。为获取状态反馈增益，我们使用现成的内点优化求解器，求解基于矩、特征函数及状态反馈增益的确定性优化问题。最后，通过与传统概率控制方法的对比实验，验证了所提方法的性能。