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Let's Make a Splan: Risk-Aware Trajectory Optimization in a Normalized Gaussian Splat

Jonathan Michaux, Seth Isaacson, Challen Enninful Adu, Adam Li, Rahul Kashyap Swayampakula, Parker Ewen, Sean Rice, Katherine A. Skinner, Ram Vasudevan

TL;DR

This paper addresses safe real-time planning in scenes represented by radiance fields, specifically Gaussian Splats. It introduces SPLANNING, a risk-aware trajectory optimizer that operates directly in a Normalized 3D Gaussian Splat, deriving a tractable bound on collision probability from the rendering equation and embedding a differentiable chance constraint into a receding-horizon planner. Key contributions include a rigorous definition of rigid-body collision within a radiance field, a computationally efficient upper bound on collision probability via H(S) and a tunable parameter α that connects to a risk budget β, and a normalized 3DGS formulation that preserves probabilistic validity for planning. Experimental results on reconstruction quality, simulation benchmarks, and real-world hardware demonstrations show SPLANNING outperforms baselines in collision avoidance while maintaining real-time performance, highlighting its practical potential for safe robotic manipulation in cluttered environments.

Abstract

Neural Radiance Fields and Gaussian Splatting have recently transformed computer vision by enabling photo-realistic representations of complex scenes. However, they have seen limited application in real-world robotics tasks such as trajectory optimization. This is due to the difficulty in reasoning about collisions in radiance models and the computational complexity associated with operating in dense models. This paper addresses these challenges by proposing SPLANNING, a risk-aware trajectory optimizer operating in a Gaussian Splatting model. This paper first derives a method to rigorously upper-bound the probability of collision between a robot and a radiance field. Then, this paper introduces a normalized reformulation of Gaussian Splatting that enables efficient computation of this collision bound. Finally, this paper presents a method to optimize trajectories that avoid collisions in a Gaussian Splat. Experiments show that SPLANNING outperforms state-of-the-art methods in generating collision-free trajectories in cluttered environments. The proposed system is also tested on a real-world robot manipulator. A project page is available at https://roahmlab.github.io/splanning.

Let's Make a Splan: Risk-Aware Trajectory Optimization in a Normalized Gaussian Splat

TL;DR

This paper addresses safe real-time planning in scenes represented by radiance fields, specifically Gaussian Splats. It introduces SPLANNING, a risk-aware trajectory optimizer that operates directly in a Normalized 3D Gaussian Splat, deriving a tractable bound on collision probability from the rendering equation and embedding a differentiable chance constraint into a receding-horizon planner. Key contributions include a rigorous definition of rigid-body collision within a radiance field, a computationally efficient upper bound on collision probability via H(S) and a tunable parameter α that connects to a risk budget β, and a normalized 3DGS formulation that preserves probabilistic validity for planning. Experimental results on reconstruction quality, simulation benchmarks, and real-world hardware demonstrations show SPLANNING outperforms baselines in collision avoidance while maintaining real-time performance, highlighting its practical potential for safe robotic manipulation in cluttered environments.

Abstract

Neural Radiance Fields and Gaussian Splatting have recently transformed computer vision by enabling photo-realistic representations of complex scenes. However, they have seen limited application in real-world robotics tasks such as trajectory optimization. This is due to the difficulty in reasoning about collisions in radiance models and the computational complexity associated with operating in dense models. This paper addresses these challenges by proposing SPLANNING, a risk-aware trajectory optimizer operating in a Gaussian Splatting model. This paper first derives a method to rigorously upper-bound the probability of collision between a robot and a radiance field. Then, this paper introduces a normalized reformulation of Gaussian Splatting that enables efficient computation of this collision bound. Finally, this paper presents a method to optimize trajectories that avoid collisions in a Gaussian Splat. Experiments show that SPLANNING outperforms state-of-the-art methods in generating collision-free trajectories in cluttered environments. The proposed system is also tested on a real-world robot manipulator. A project page is available at https://roahmlab.github.io/splanning.
Paper Structure (55 sections, 7 theorems, 89 equations, 5 figures, 7 tables)

This paper contains 55 sections, 7 theorems, 89 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Trajectory Optimization
  4. Reachability Analysis
  5. Radiance Fields
  6. Proposed Method Overview
  7. Notation
  8. Sets and Functions
  9. Probabilities
  10. Robot Descriptions
  11. Overview of Optimization
  12. Robot Representation
  13. Arm Kinematics
  14. Arm Occupancy
  15. Detailed Description of SPLANNING
  16. ...and 40 more sections

Key Result

Theorem 3

Given a serial manipulator with $n_q \in \mathbb{N}$ revolute joints and a time partition $T$ of a finite set of intervals $T_i$ (i.e., $T = \cup_{i=1}^{n_t} T_i$), the swept volume corresponding to the robot's motion over $T$ is overapproximated by a collection of $L_2$ balls in $\mathbb{R}^3$, wh where each $S_{j,i,m}(q(T_i; k))$ is an $L_2$ ball in $\mathbb{R}^3$, $n_S \in \mathbb{N}$ is a pa

Figures (5)

  • Figure 1: SPLANNING constructs risk-aware trajectories in a Gaussian Splatting map in real-time. The top image shows a real-world scene that a 7DOF serial manipulator must plan through, starting at the blue configuration (left) and ending at the green configuration (right). The scene is represented as a normalized 3D Gaussian Splat. Then, in real-time, SPLANNING solves an optimization problem that constrains the probability that the robot's forward occupancy (bottom center, purple) collides with the scene.
  • Figure 2: SPLANNING optimizes trajectories in a Normalized 3D Gaussian Splat given a start configuration (blue) and goal configuration (green). Offline, a Normalized 3D Gaussian Splat is constructed to represent the scene geometry (Sec. \ref{['subsec:approach_splatting']}, bottom right). Online, a family of parameterized trajectories (App. \ref{['app:modeling_trajectory']}) is partitioned into a finite set of intervals (App. \ref{['app:reachability']}, top left). Then, for each time interval, the Spherical Forward Occupancy ($\altmathcal{SFO}$) (purple) is computed as an overapproximation of the robot's swept volume (Sec. \ref{['subsec:modeling_arm_occupancy']}, bottom left). Finally, during online trajectory optimization (Sec. \ref{['subsec:approach_motion_planning']}, bottom middle), a novel constraint (Sec. \ref{['subsec:approach_collision']}--\ref{['subsec:approach_integral_evaluation']}) bounds the probability that the $\altmathcal{SFO}$ intersects with the scene, as represented by a Normalized 3D Gaussian Splat .
  • Figure 3: SPLANNING generates safe trajectories in densely cluttered environments in simulation. The left panel shows discrete time steps of sequential trajectories that brings the arm from the start configuration (blue) safely to the goal configuration (green). The right panel shows an intermediate planning step where the Spherical Forward Occupancy (purple) avoids the obstacles represented by a Normalized 3D Gaussian Splat.
  • Figure 4: The collision constraint representation of CATNIPS, Splat-Nav, and SPLANNING are compared. CATNIPS transforms a NeRF into a 3D occupancy grid by relating it to a Poisson Point Process; after convolving this grid with a robot kernel, the center of the robot is checked for collision with the grid. Splat-Nav deterministically evaluates whether the confidence ellipsoids of each Gaussian intersect with a spherical robot. Finally, SPLANNING integrates a Normalized 3D Gaussian Splat over an overapproximation of the robot arm to form a risk constraint.
  • Figure 5: Two Precision-Recall curves are presented. Treating the collision constraints from SPLANNING, CATNIPS, Splat-Nav as classifiers, where a positive indicates collision and a negative indicates no collision, the top plot shows the Precision-Recall over individual spheres while the bottom plot show Precision-Recall over configurations. Highlighted markers indicate nominal parameters $\alpha = \beta = 0.025$, $\sigma=0.99$, and $\sigma=1$ for SPLANNING, CATNIPS, and Splat-Nav respectively.

Theorems & Definitions (11)

  • Theorem 3
  • Theorem 5
  • Theorem 6
  • Definition 7: Trajectory Parameters
  • Lemma 8: Parameterized Trajectory PZs
  • Theorem 9: Spherical Forward Occupancy
  • Remark 10
  • Theorem 11
  • proof
  • Theorem 12
  • ...and 1 more