Sampling-based Certified Planning with Graphs of Convex Sets

Reactive task-space planners such as Bug2 operate with fixed Cartesian step sizes and are unaware of the manipulator's joint-angle limits. When the Jacobian is poorly conditioned, even small Cartesian steps can demand joint changes that exceed admissible bounds; clipping the joints to their limits causes tracking drift and can prevent goal reaching entirely. We address this by computing, at each planning step, the largest Cartesian hyperrectangle that is \emph{certifiably reachable} under joint displacement bounds. Using a second-order polynomial approximation of the inverse kinematics and the S-procedure, we formulate a small semidefinite program whose solution yields the certified half-width~$λ^\star$. An equivalent bisection procedure exploiting the quadratic structure solves the certification in sub-millisecond time. Integrating this certificate with Bug2 yields a planner whose step size adapts to local kinematic conditioning. In a statistical evaluation over 94 adversarial scenarios spanning six joint-limit settings, the SOS-verified planner achieves \emph{zero} joint-limit violations with a 100% goal-reaching rate, whereas a standard Bug2 planner violates joint limits in 6--11% of steps and fails to reach the goal in up to 18% of scenarios.

Collision-free path planning in cluttered, real-world environments relies on a representation of the collision-free space, and existing representations broadly fall into two categories. Explicit representations, such as unions of convex sets, can be plugged into optimization-based planners as hard collision-free constraints, but their parameters scale poorly with configuration-space dimension. Implicit representations, by contrast, are flexible and scale well to complex geometries, yet typically lack such guarantees. We bridge this gap with ILD (Invertible Latent Decomposition), a framework that jointly learns an invertible mapping and a union of explicit convex polytopes in the resulting latent space. Planning is carried out over these latent convex sets, and the invertible mapping decodes the resulting paths back to the original configuration space while preserving feasibility with respect to the refined explicit safe regions. We further propose Visibility-Guided Sampling (VGS) to keep the convex sets connected for path planning. Across 2D navigation, 6-DoF, and 14-DoF manipulation environments, ILD achieves broader coverage, better inter-set connectivity, and higher path-planning success rates than prior baselines, with zero observed false positives after test-time refinement. On a 14-DoF bimanual manipulator, we further demonstrate real-time collision-free planning, with test-time refinement adapting to scene-geometry changes during real-world deployment on a single 6-DoF arm.

Motion planning problems such as collision-free navigation and contact-rich manipulation can be naturally formulated as optimization problems that couple discrete decisions with continuous trajectories. The Graphs of Convex Sets (GCS) framework offers a practical solution to these problems. It represents discrete decisions as nodes of a graph and encodes continuous trajectories in the edges connecting them. However, the resulting optimization subproblems can become computationally prohibitive for online replanning. In this work, we propose a learning-based strategy to mitigate this limitation. Specifically, we replace the costly convex relaxation step required by nominal GCS with a single forward pass through a Graph Attention Network that predicts a set of highly probable candidate paths through the graph. A lightweight ranking network then orders these candidates by their estimated trajectory cost. Evaluating them in this order, we terminate our search early while still recovering a near-optimal motion plan. We validate the resulting pipeline across diverse robotic tasks, including collision-free motion planning for a 3D quadrotor and a 7-DoF manipulator, and planning through contact for planar pushing. Across both convex and non-convex cost and constraint settings, our approach yields up to two orders of magnitude speedup over nominal GCS while maintaining a 100% success rate, at the cost of some suboptimality in the recovered solutions. Code implementations and video demonstrations can be found at https://neural-gcs.github.io/.