Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.14063

Semidefinite Relaxations for Collision-Free Motion Planning

Bernhard Paus Graesdal·Alexandre Amice·Pablo A. Parrilo·Russ Tedrake
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study semidefinite relaxations for collision-free motion planning. We focus on a point robot moving from start to goal through spherical obstacles in $\mathbb{R}^n$, subject to path continuity constraints and squared derivative costs; a setting that is conceptually simple yet captures the hardness of collision-free motion planning. We formulate this problem exactly as a nonconvex problem over polynomial curves, and present a natural semidefinite relaxation. We contribute two key theoretical insights; to our knowledge this is the first theoretical analysis of semidefinite relaxations for collision-free motion planning. First, we show that solving the convex relaxation is equivalent to solving, to global optimality, a related motion planning problem in a potentially higher-dimensional space. This geometric interpretation yields necessary and sufficient conditions for tightness, and a clear intuition for when the relaxation is loose. Second, we show that the relaxation admits a symmetry reduction that makes it significantly smaller than one might expect, with positive semidefinite cone sizes that scale linearly with the polynomial degree and are independent of the ambient dimension. The resulting relaxation is 10 to 100 times faster than direct nonlinear programming transcriptions solved with SNOPT and IPOPT, exhibits significantly lower variance in solve times, and reliably finds a locally optimal path for the original problem. We demonstrate its effectiveness as a convex steering function in an RRT planner for minimum-snap quadrotor planning with $C^4$ continuous trajectories.

Explore related subjects

cs.ROcs.SYeess.SY

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bernhard Paus Graesdal, Alexandre Amice, Pablo A. Parrilo, Russ Tedrake. 2026-06-12. Semidefinite Relaxations for Collision-Free Motion Planning. https://arxiv.org/abs/2606.14063

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Correct-by-Construction Vision-based Pose Estimation using Geometric Generative Models

We consider the problem of vision-based pose estimation for autonomous systems. While deep neural networks have been successfully used for vision-based tasks, they inherently lack provable guarantees on the correctness of their output, which is crucial for safety-critical applications. We present a framework for designing certifiable neural networks (NNs) for perception-based pose estimation that integrates physics-driven modeling with learning-based estimation. The proposed framework begins by leveraging the known geometry of planar objects commonly found in the environment, such as traffic signs and runway markings, referred to as target objects. At its core, it introduces a geometric generative model (GGM), a neural-network-like model whose parameters are derived from the image formation process of a target object observed by a camera. Once designed, the GGM can be used to train NN-based pose estimators with certified guarantees in terms of their estimation errors. We first demonstrate this framework in uncluttered environments, where the target object is the only object present in the camera's field of view. We extend this using ideas from NN reachability analysis to design certified object NN that can detect the presence of the target object in cluttered environments. Subsequently, the framework consolidates the certified object detector with the certified pose estimator to design a multi-stage perception pipeline that generalizes the proposed approach to cluttered environments, while maintaining its certified guarantees. We evaluate the proposed framework using both synthetic and real images of various planar objects commonly encountered by autonomous vehicles. Using images captured by an event-based camera, we show that the trained encoder can effectively estimate the camera pose relative to a traffic sign in accordance with the certified bound provided by the framework.

cs.RO↗
arXiv2026

HydroMap: Probabilistic Water Surface Elevation Mapping for Semantic Scene Representation in Inland Waterways

Autonomous surface vehicles operating in inland waterways require a persistent representation of both surrounding structures and the water surface. LiDAR-based simultaneous localization and mapping often produces sparse or missing water returns, leaving this operational surface absent from the reconstructed scene. We propose HydroMap, an odometry-decoupled framework that reconstructs water surface elevation from stereo observations and integrates it with the structural map. Per-frame water points form joint cell observations with propagated stereo and pose uncertainty, and successive observations are fused into a persistent probabilistic elevation map. Semantic map conversion then combines the elevation map with structural geometry in a unified 2.5D representation of water, boundaries, structures, and overhead regions. On the Pohang Canal and Leuven Vaart datasets, the elevation RMSE remains below 5 cm relative to LiDAR references expressed in the same map frame. The elevation and semantic maps are published at 2 Hz and 1 Hz, respectively. HydroMap thereby complements LiDAR maps with a persistent representation of the water surface for downstream navigation in inland waterways.

cs.RO↗
arXiv2026

Density-Driven Area Coverage for Nonholonomic Multi-Robot Systems with Safety Guarantee

Density-Driven Optimal Control (D2OC) provides a principled approach to distributing multi-robot teams over non-uniform spatial distributions. Applying D2OC to nonholonomic robots, however, creates a gap between safety constraints imposed on a reference motion and the physical inputs that determine the actual robot motion. We address this issue by enforcing the safety constraint directly on the robot's physical inputs while preserving the density-driven coverage objective. The proposed framework combines D2OC with a control barrier function safety filter through a feedback-linearizing look-ahead point, allowing safety and actuator limits to be considered together during control. We further derive a safety margin that accounts for the look-ahead geometry, robot footprint, and motion during each control interval. Simulation results show that the proposed method maintains the required physical separation while achieving coverage performance comparable to a conventional reference-tracking approach, which can satisfy safety on the reference motion yet violate the corresponding physical clearance. Experiments on multiple nonholonomic robots in the Robotarium further demonstrate safe execution while driving the robots toward the desired spatial distribution. These results show that enforcing safety directly on the physical inputs can eliminate the mismatch between safety certification and physical robot motion in density-driven multi-robot coverage.

cs.RO↗