This paper describes continuous-space methodologies to estimate the collision
probability, Euclidean distance and gradient between an ellipsoidal robot model
and an environment surface modeled as a set of Gaussian distributions.
Continuous-space collision probability estimation is critical for
uncertainty-aware motion planning. Most collision detection and avoidance
approaches assume the robot is modeled as a sphere, but ellipsoidal
representations provide tighter approximations and enable navigation in
cluttered and narrow spaces. State-of-the-art methods derive the Euclidean
distance and gradient by processing raw point clouds, which is computationally
expensive for large workspaces. Recent advances in Gaussian surface modeling
(e.g. mixture models, splatting) enable compressed and high-fidelity surface
representations. Few methods exist to estimate continuous-space occupancy from
such models. They require Gaussians to model free space and are unable to
estimate the collision probability, Euclidean distance and gradient for an
ellipsoidal robot. The proposed methods bridge this gap by extending prior work
in ellipsoid-to-ellipsoid Euclidean distance and collision probability
estimation to Gaussian surface models. A geometric blending approach is also
proposed to improve collision probability estimation. The approaches are
evaluated with numerical 2D and 3D experiments using real-world point cloud
data.

该论文描述了使用连续空间方法估计椭球形机器人模型与将环境表面建模为一组高斯分布之间的碰撞概率、欧几里得距离和梯度。

高斯表面模型下的距离和碰撞概率估计

Distance and Collision Probability Estimation from Gaussian Surface  Models

We present an approach for safe motion planning under robot state and
environment (obstacle and landmark location) uncertainties. To this end, we
first develop an approach that accounts for the landmark uncertainties during
robot localization. Existing planning approaches assume that the landmark
locations are well known or are known with little uncertainty. However, this
might not be true in practice. Noisy sensors and imperfect motions compound to
the errors originating from the estimate of environment features. Moreover,
possible occlusions and dynamic objects in the environment render imperfect
landmark estimation. Consequently, not considering this uncertainty can wrongly
localize the robot, leading to inefficient plans. Our approach thus
incorporates the landmark uncertainty within the Bayes filter estimation
framework. We also analyze the effect of considering this uncertainty and
delineate the conditions under which it can be ignored. Second, we extend the
state-of-the-art by computing an exact expression for the collision probability
under Gaussian distributed robot motion, perception and obstacle location
uncertainties. We formulate the collision probability process as a quadratic
form in random variables. Under Gaussian distribution assumptions, an exact
expression for collision probability is thus obtained which is computable in
real-time. In contrast, existing approaches approximate the collision
probability using upper-bounds that can lead to overly conservative estimate
and thereby suboptimal plans. We demonstrate and evaluate our approach using a
theoretical example and simulations. We also present a comparison of our
approach to different state-of-the-art methods.

本文提出了一种考虑机器人状态和环境不确定性的安全运动规划方法，旨在解决机器人定位中地标不确定性的问题，并给出了在高斯分布机器人运动、感知和障碍位置不确定性下的碰撞概率的精确表达式。

考虑环境不确定性的安全运动规划

Safe motion planning with environment uncertainty

Classical map-based navigation methods are commonly used for robot
navigation, but they often struggle in crowded environments due to the Frozen
Robot Problem (FRP). Deep reinforcement learning-based methods address the FRP
problem, however, suffer from the issues of generalization and scalability. To
overcome these challenges, we propose a method that uses Collision Probability
(CP) to help the robot navigate safely through crowds. The inclusion of CP in
the observation space gives the robot a sense of the level of danger of the
moving crowd. The robot will navigate through the crowd when it appears safe
but will take a detour when the crowd is moving aggressively. By focusing on
the most dangerous obstacle, the robot will not be confused when the crowd
density is high, ensuring scalability of the model. Our approach was developed
using deep reinforcement learning (DRL) and trained using the Gazebo simulator
in a non cooperative crowd environment with obstacles moving at randomized
speeds and directions. We then evaluated our model on four different
crowd-behavior scenarios with varying densities of crowds. The results shown
that our method achieved a 100% success rate in all test settings. We compared
our approach with a current state-of-the-art DRLbased approach, and our
approach has performed significantly better. Importantly, our method is highly
generalizable and requires no fine-tuning after being trained once. We further
demonstrated the crowd navigation capability of our model in real-world tests.

提出了一种基于深度强化学习和碰撞概率（Collision Probability，CP）的机器人人群导航方法，以解决机器人导航不稳定性问题。在不同密度的人群行为情景中进行了测试，并与当前最先进的 DRL 方法进行了比较，结果表明该方法表现出色，而且高度泛化。

基于深度强化学习的无地图人群导航移动机器人感知移动风险

Deep Reinforcement Learning-Based Mapless Crowd Navigation with  Perceived Risk of the Moving Crowd for Mobile Robots

The method of 1-bit ("sign-sign") random projections has been a popular tool
for efficient search and machine learning on large datasets. Given two $D$-dim
data vectors $u$, $v\in\mathbb{R}^D$, one can generate $x = \sum_{i=1}^D u_i
r_i$, and $y = \sum_{i=1}^D v_i r_i$, where $r_i\sim N(0,1)$ iid. The
"collision probability" is ${Pr}\left(sgn(x)=sgn(y)\right) =
1-\frac{\cos^{-1}\rho}{\pi}$, where $\rho = \rho(u,v)$ is the cosine
similarity.
We develop "sign-full" random projections by estimating $\rho$ from (e.g.,)
the expectation $E(sgn(x)y)=\sqrt{\frac{2}{\pi}} \rho$, which can be further
substantially improved by normalizing $y$. For nonnegative data, we recommend
an interesting estimator based on $E\left(y_- 1_{x\geq 0} + y_+ 1_{x<0}\right)$
and its normalized version. The recommended estimator almost matches the
accuracy of the (computationally expensive) maximum likelihood estimator. At
high similarity ($\rho\rightarrow1$), the asymptotic variance of recommended
estimator is only $\frac{4}{3\pi} \approx 0.4$ of the estimator for sign-sign
projections. At small $k$ and high similarity, the improvement would be even
much more substantial.

本文提出了一种新的随机投影估计器，其使用期望和正则化方法从数据中估计余弦相似度，并表明在高相似度下，该估计器比标准 Method of 1-bit random projections 更准确。