As robots are being increasingly used in close proximity to humans and objects, it is imperative that robots operate safely and efficiently under real-world conditions. Yet, the environment is seldom known perfectly. Noisy sensors and actuation errors compound to the errors introduced while estimating features of the environment. We present a novel approach (1) to incorporate these uncertainties for robot state estimation and (2) to compute the probability of collision pertaining to the estimated robot configurations. The expression for collision probability is obtained as an infinite series, and we prove its convergence. An upper bound for the truncation error is also derived, and the number of terms required is demonstrated by analyzing the convergence for different robot and obstacle configurations. We evaluate our approach using two simulation domains which use a roadmap-based strategy to synthesize trajectories that satisfy collision probability bounds.

随着机器人越来越接近人类和物体使用，当务之急是在现实条件下安全有效地操作机器人。然而，很少有人完全了解环境。噪声传感器和致动误差会增加估算环境特征时引入的误差。我们提出了一种新颖的方法（1）并入这些不确定性以进行机器人状态估计，以及（2）计算与估计的机器人配置有关的碰撞概率。碰撞概率的表达式是一个无穷级数，我们证明了它的收敛性。还得出了截断误差的上限，并且通过分析不同机器人和障碍物配置的收敛性来证明所需项的数量。