Andrew Niu | Search arXiv

arXiv2026

Becoming a Fruit Ninja: Real-Time Probabilistic Kinodynamic Planning for Manipulator Projectile Interception

Projectile interception is a challenging dynamic manipulation problem. Intercepting a thrown object with a robot arm requires reaching a point on the object's path as the object passes through it. Slicing also fixes the blade's velocity and orientation at contact. The goal is therefore a subset of the states of the robot and arrival times that moves as the object falls, and the arm must reach it within its actuator limits in milliseconds. We present FRUITNINJA, an anytime sampling-based planner that grows a tree on the GPU in batches toward the interception manifold. Each edge is an exact cubic whose travel time is found by a parallel search against the arm's dynamics, so every edge satisfies the actuator limits. Plans are ranked by a risk-aware objective over the uncertainty in the object's position and the arm's arrival time. We evaluate on a Franka Research 3 against six baselines in a calibrated real-time simulator, where FRUITNINJA cuts 96.7% of tosses in the open and 68.3% among five obstacles, versus the best baseline's 68.3% and 35.0% respectively.

cs.RO↗

arXiv2026

Conformal Rigidity of Graphs: Subdifferentials and Orbit-Isometries

A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $λ_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge weights minimize the largest eigenvalue $λ_n(w)$ over all normalized edge weights; $G$ is conformally rigid if it is lower or upper conformally rigid. This paper establishes a new framework for conformal rigidity through the language of subdifferentials, unifying the variational perspective on eigenvalue optimization with the geometry of edge-isometric spectral embeddings, which are known to characterize conformal rigidity. This subdifferential framework lends itself naturally to techniques of symmetry reduction that motivate the notion of an orbit-isometric embedding - a weaker condition than edge-isometry that accounts for the symmetries of $G$ while remaining sufficient for conformal rigidity. The notion opens the door to tools from representation theory: for a large class of graphs, including all vertex-transitive ones, we show that conformal rigidity is certified by a single eigenvector, resolving an open question and explaining the conformal rigidity of previously unexplained graphs. This extra structure enables a new, algebraically exact certification method for conformal rigidity, bypassing the numerical difficulties of prior approaches. In many cases, the problem reduces to a check of linear feasibility, and in general, to solving a system of quadratic equations via Gröbner bases.

math.CO↗