Robot Motion Planning With Radiance Fields and Collision Probability Bounds
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Solution Overview
Problem
Existing motion planning methods for robots using radiance fields are non-differentiable and struggle to reason rigorously about collisions in real-time, leading to overly conservative or inefficient collision avoidance.
Innovation Solution
A risk-aware trajectory planner using Gaussian Splatting models rigorously computes collision probabilities within radiance fields, integrating polynomial zonotopes to over-approximate robot positions and velocities, ensuring real-time collision-free motion planning.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If reachability-based trajectory design uses polynomial zonotopes to construct reachable sets, then collision-free motion planning is achieved, but the reachable sets become overly conservative
Solution Approach 1:
The patent changes the mathematical representation parameters from polynomial zonotopes to Gaussian Splatting radiance fields. This allows the system to maintain rigorous collision guarantees while reducing conservatism by using continuous probability density functions that better represent actual obstacle occupancy, thereby improving motion planning efficiency.
Solution Approach 2:
The patent replaces the traditional geometric set-based reachability analysis with a radiance field-based probability density approach. This substitution enables more accurate collision probability computation while maintaining real-time performance, resolving the contradiction between reliability and productivity.
2Productivity
If existing motion planning methods use simple geometric primitives for collision checking, then real-time computation is achieved, but collision avoidance becomes a soft penalty rather than hard constraint
Solution Approach 1:
The patent introduces collision probability computation as an intermediary step between trajectory generation and execution. By computing the actual collision probability using radiance fields and using this probability as a hard constraint in the optimization problem, the system achieves both real-time computation and provable collision avoidance safety.
Solution Approach 2:
The patent implements feedback by computing collision probabilities based on the planned trajectory and using this information to adjust the trajectory through constrained optimization. This closed-loop approach ensures that collision avoidance is enforced as a hard constraint while maintaining real-time performance.
3Measurement precision
If radiance field methods are used for collision detection, then detailed environmental modeling is achieved, but the methods become non-differentiable and difficult to reason about rigorously
Solution Approach 1:
The patent changes the radiance field representation to use normalized Gaussian functions with explicit probability density properties. This parameter change enables rigorous collision probability computation through integration of the Gaussian functions, making the method both detailed and mathematically tractable for optimization.
Solution Approach 2:
The patent segments the collision reasoning process into discrete steps: (1) representing the environment as normalized Gaussian Splats, (2) computing collision probability through integration, and (3) using the probability as a constraint in optimization. This segmentation makes the complex radiance field approach rigorous and differentiable.
4Productivity
If discrete time instances are used for collision checking, then computational efficiency is improved, but continuous-time collision avoidance cannot be guaranteed
Solution Approach 1:
The patent maintains continuity by using continuous probability density functions (Gaussian Splatting) to represent obstacle occupancy throughout the entire time horizon. The collision probability is computed continuously along the trajectory, and the optimization enforces this continuous constraint, ensuring continuous-time collision avoidance while maintaining computational efficiency through the properties of Gaussian functions.
Data Source
AI summary
A system and method for estimating a collision probability between a robot and a radiance field within a three-dimensional environment includes: determining forward occupancy information indicating potential over-approximating volumes that are able to be occupied by a robot disposed at a starting position; modeling an environment of the robot using a radiance field; and computing a probability of collision between the robot and an obstacle within the environment based on the forward occupancy information and the radiance field. The estimation method is useful for robotic trajectory determination that involves discretizing a trajectory of the robot into a sequence of trajectory segments over time subintervals; computing an upper-bound for a probability of collision between the robot and a radiance field at each of the time subintervals using a Gaussian Splatting model that normalizes 3D Gaussians within the radiance field; and performing real-time trajectory adjustments based on the computed collision probability upper-bounds.