Toric partial density functions and stability of toric varieties

Let $(L, h)\to (X, \omega)$ denote a polarized toric K\"ahler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho}_{tk}:X\to \mathbb{R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along $Y$, for fixed $t>0$ such that $tk\in \mathbb{N}$. We prove the existence of a distributional expansion of $\hat{\rho}_{tk}$ as $k\to \infty$, including the identification of the coefficient of $k^{n-1}$ as a distribution on $X$. This expansion is used to give a direct proof that if $\omega$ has constant scalar curvature, then $(X, L)$ must be slope semi-stable with respect to $Y$. Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.Comment: Accepted by Mathematische Annalen on 13 September 201

CloudGripper: An Open Source Cloud Robotics Testbed for Robotic Manipulation Research, Benchmarking and Data Collection at Scale

We present CloudGripper, an open source cloud robotics testbed, consisting of a scalable, space and cost-efficient design constructed as a rack of 32 small robot arm work cells. Each robot work cell is fully enclosed and features individual lighting, a low-cost custom 5 degree of freedom Cartesian robot arm with an attached parallel jaw gripper and a dual camera setup for experimentation. The system design is focused on continuous operation and features a 10 Gbit/s network connectivity allowing for high throughput remote-controlled experimentation and data collection for robotic manipulation. CloudGripper furthermore is intended to form a community testbed to study the challenges of large scale machine learning and cloud and edge-computing in the context of robotic manipulation. In this work, we describe the mechanical design of the system, its initial software stack and evaluate the repeatability of motions executed by the proposed robot arm design. A local network API throughput and latency analysis is also provided. CloudGripper-Rope-100, a dataset of more than a hundred hours of randomized rope pushing interactions and approximately 4 million camera images is collected and serves as a proof of concept demonstrating data collection capabilities. A project website with more information is available at https://cloudgripper.org.Comment: Under review at IEEE ICRA 202

Quasi-static Soft Fixture Analysis of Rigid and Deformable Objects

We present a sampling-based approach to reasoning about the caging-based manipulation of rigid and a simplified class of deformable 3D objects subject to energy constraints. Towards this end, we propose the notion of soft fixtures extending earlier work on energy-bounded caging to include a broader set of energy function constraints and settings, such as gravitational and elastic potential energy of 3D deformable objects. Previous methods focused on establishing provably correct algorithms to compute lower bounds or analytically exact estimates of escape energy for a very restricted class of known objects with low-dimensional C-spaces, such as planar polygons. We instead propose a practical sampling-based approach that is applicable in higher-dimensional C-spaces but only produces a sequence of upper-bound estimates that, however, appear to converge rapidly to actual escape energy. We present 8 simulation experiments demonstrating the applicability of our approach to various complex quasi-static manipulation scenarios. Quantitative results indicate the effectiveness of our approach in providing upper-bound estimates for escape energy in quasi-static manipulation scenarios. Two real-world experiments also show that the computed normalized escape energy estimates appear to correlate strongly with the probability of escape of an object under randomized pose perturbation.Comment: Paper submitted to ICRA 202

An Efficient and Continuous Voronoi Density Estimator

We introduce a non-parametric density estimator deemed Radial Voronoi Density Estimator (RVDE). RVDE is grounded in the geometry of Voronoi tessellations and as such benefits from local geometric adaptiveness and broad convergence properties. Due to its radial definition RVDE is moreover continuous and computable in linear time with respect to the dataset size. This amends for the main shortcomings of previously studied VDEs, which are highly discontinuous and computationally expensive. We provide a theoretical study of the modes of RVDE as well as an empirical investigation of its performance on high-dimensional data. Results show that RVDE outperforms other non-parametric density estimators, including recently introduced VDEs.Comment: 12 page

BITKOMO: Combining Sampling and Optimization for Fast Convergence in Optimal Motion Planning

Optimal sampling based motion planning and trajectory optimization are two competing frameworks to generate optimal motion plans. Both frameworks have complementary properties: Sampling based planners are typically slow to converge, but provide optimality guarantees. Trajectory optimizers, however, are typically fast to converge, but do not provide global optimality guarantees in nonconvex problems, e.g. scenarios with obstacles. To achieve the best of both worlds, we introduce a new planner, BITKOMO, which integrates the asymptotically optimal Batch Informed Trees (BIT*) planner with the K-Order Markov Optimization (KOMO) trajectory optimization framework. Our planner is anytime and maintains the same asymptotic optimality guarantees provided by BIT*, while also exploiting the fast convergence of the KOMO trajectory optimizer. We experimentally evaluate our planner on manipulation scenarios that involve high dimensional configuration spaces, with up to two 7-DoF manipulators, obstacles and narrow passages. BITKOMO performs better than KOMO by succeeding even when KOMO fails, and it outperforms BIT* in terms of convergence to the optimal solution.Comment: 6 pages, Accepted at IROS 202