On The Evaluation of Collision Probability along a Path

TL;DR

This article proposes an index it calls “Risk Density,” which offers a theoretical link between conceptually distant assumptions about the interplay of single collision events along a continuous path and shows how this index can be used to approximate the collision probability in the case where the robot evolves along a nominal continuous curve from random initial conditions.

Abstract

Characterizing the risk of operations is a fundamental requirement in robotics, and a crucial ingredient of safe planning. The problem is multifaceted, with multiple definitions arising in the vast recent literature fitting different application scenarios and leading to different computational approaches. A basic element shared by most frameworks is the definition and evaluation of the probability of collision for a mobile object in an environment with obstacles. We observe that, even in basic cases, different interpretations are possible. This paper proposes an index we call "Risk Density", which offers a theoretical link between conceptually distant assumptions about the interplay of single collision events along a continuous path. We show how this index can be used to approximate the collision probability in the case where the robot evolves along a nominal continuous curve from random initial conditions. Indeed, under this hypothesis the proposed approximation outperforms some well-established methods either in accuracy or computational cost.

Figures (19)

• Figure 1: Depiction of different scenarios where collisions have to be modeled in different ways. (a) shows an environment with areas where falling rocky debree could hit the robot at any point in time. (b) shows a robot boat navigating in a randomly mined body of water. (c) shows a warehouse robot navigating an uncertain environment.
• Figure 2: Graphical depiction of the decomposition of the robot and obstacle sets. The result of the Minkowski sum is also illustrated. Each object is represented by its shape, the vector pointing to its frame of reference, and a Random Variable describing the uncertainty in its position.
• Figure 3: Visual representation of how considered assumptions work on two subsequent events. The events considered are $\{\omega | N_{RO}(s_0,\omega)\in A\}$ and $\{\omega | N_{RO}(s_1,\omega) \in B\}$, where the sets $A$ and $B$ are shown as red and blue disks. $P_0(\cdot)$ and $P_1(\cdot)$ stand respectively for the probability measure of the first and the second random variable to be inside a set.
• Figure 4: Interpretation of \ref{['eq:totalprobconfigurationmoving']}. The problem is moved from the initial domain $\mathbb{W}$ to the domain of the difference between the robot and the obstacle shape. This results from the transformation shown in \ref{['fig:decompositionuncertainset']}, which moves all the uncertainty on a single distribution centered in zero, while the integration set is $D_T$.
• Figure 5: Example of discretization choice effect on the computed probability by grid-based methods. The path crossed by the robot (green) lives in $\mathbb{R}^2$, and $3$ distinct tessellations of the environment are given (red, blue, black). The robot sensors measure different probability values \ref{['eq:bayesbelief']} for the cells depending on the image's color. The probability computed using \ref{['eq:gridprobbelief']} assumes different values depending on the discretization, although the underlying environment is the same. Indeed, the probability changes as a function of the cardinality of the set $\mathbf{m_c}$.

Theorems & Definitions (10)

• Lemma 1: H1
• proof
• Lemma 2: H2
• proof
• Lemma 3: H3
• proof
• Lemma 4
• proof
• Remark
• Remark