Continuous Vector Fields Research Articles

The infinitesimal earthquake theorem for vector fields on the circle

We prove that any continuous vector field on a circle is the extension in a suitable sense, of a unique infinitesimal earthquake of the hyperbolic plane. Furthermore, we obtain other extension results when the vector field is assumed only to be upper or lower semicontinuous. This leads to a generalization of Kerckhoff’s and Gardiner’s infinitesimal earthquake theorems to a broader setting, using a completely novel approach. The proof is based on the geometry of the dual of Minkowski three-space, also called Half-pipe three-geometry. In this way, we obtain a simple characterization of Zygmund vector fields on the circle in terms of width of convex hulls.

Smooth Bijective Projection in a High-order Shell

We propose a new structure called a higher-order shell, which is composed of a set of triangular prisms. Each triangular prism is enveloped by three Bézier triangles (top, middle, and bottom) and three side surfaces, each of which is trimmed from a bilinear surface. Moreover, we define a continuous vector field to smoothly and bijectively transfer attributes between two surfaces inside the shell. Since the higher-order shell has several hard construction constraints, we apply an interior-point strategy to robustly and automatically construct a high-order shell for an input mesh. Specifically, the strategy starts from a valid linear shell with a small thickness. Then, the shell is optimized until the specified thickness is reached, where explicit checks ensure that the constraints are always satisfied. We extensively test our method on more than 8300 models, demonstrating its robustness and performance. Compared to state-of-the-art methods, our bijective projection is smoother, and the space between the shell and input mesh is more uniform.

Quantifying Landscape-Flux via Single-Cell Transcriptomics Uncovers the Underlying Mechanism of Cell Cycle.

Recent developments in single-cell sequencing technology enable the acquisition of entire transcriptome data. Understanding the underlying mechanism and identifying the driving force of transcriptional regulation governing cell function directly from these data remains challenging. This study reconstructs a continuous vector field of the cell cycle based on discrete single-cell RNA velocity to quantify the single-cell global nonequilibrium dynamic landscape-flux. It reveals that large fluctuations disrupt the global landscape and genetic perturbations alter landscape-flux, thus identifying key genes in maintaining cell cycle dynamics and predicting associated functional effects. Additionally, it quantifies the fundamental energy cost of the cell cycle initiation and unveils that sustaining the cell cycle requires curl flux and dissipation to maintain the oscillatory phase coherence. This study enables the inference of the cell cycle gene regulatory networks directly from the single-cell transcriptomic data, including the feedback mechanisms and interaction intensity. This provides a golden opportunity to experimentally verify the landscape-flux theory and also obtain its associated quantifications. It also offers a unique framework for combining the landscape-flux theory and single-cell high-through sequencing experiments for understanding the underlying mechanisms of the cell cycle and can be extended to other nonequilibrium biological processes, such as differentiation development and disease pathogenesis.

Finite-in-time flocking of the thermodynamic Cucker–Smale model

<abstract><p>We illustrate finite-in-time flocking in the thermodynamic Cucker–Smale (TCS) model. First, we extend the original TCS model to allow for a continuous vector field with a locally Lipschitz continuity. Then, within this system, we derive appropriate dissipative inequalities concerning the position-velocity-temperature using several preparatory estimates. Subsequently, based on initial data and system parameters, we formulate sufficient conditions to guarantee the desired finite-time flocking in each case where the communication weight conditions are divided into two scenarios: one with a positive lower bound and another with nonnegativity and monotonicity. Finally, we provide several numerical simulations and compare them with the analytical results.</p></abstract>

Shrink & Morph: 3D-Printed Self-Shaping Shells Actuated by a Shape Memory Effect

While 3D printing enables the customization and home fabrication of a wide range of shapes, fabricating freeform thin-shells remains challenging. As layers misalign with the curvature, they incur structural deficiencies, while the curved shells require large support structures, typically using more material than the part itself. We present a computational framework for optimizing the internal structure of 3D printed plates such that they morph into a desired freeform shell when heated. This exploits the shrinkage effect of thermoplastics such as PLA, which store internal stresses along the deposition directions. These stresses get released when the material is heated again above its glass transition temperature, causing an anisotropic deformation that induces curvature. Our inverse design method takes as input a freeform surface and finds an optimized set of deposition trajectories in each layer such that their anisotropic shrinkage deforms the plate into the prescribed surface geometry. We optimize for a continuous vector field that varies across the plate and within its thickness. The algorithm then extracts a set of deposition trajectories from the vector field in order to fabricate the flat plates on standard FFF printers. We validate our algorithm on freeform, doubly-curved surfaces.

Stability analysis of a hybrid DC-DC buck converter model using dissipation inequality and convex optimization

The stability analysis of a DC-DC buck converter is a challenging problem due to the hybrid systems characteristic of its dynamics. Such a challenge arises from the buck converter operation which depends upon the ON/OFF logical transitions of its electronic switch component to correspondingly activate different continuous vector fields of the converter’s temporal dynamics. This paper presents a sum of squares (SOS) polynomial optimization approach for stability analysis of a hybrid model of buck converter which explicitly takes into account the converter’s electronic switching behavior. The proposed method first transforms the converter’s hybrid dynamics model into an equivalent polynomial differential algebraic equation (DAE) model. An SOS programming algorithm is then proposed to computationally prove the stability of the obtained DAE model using Lyapunov’s stability concept. Based on simulation results, it was found that the proposed method requires only 8.5 seconds for proving the stability of a buck converter model. In contrast, exhaustive simulations based on numerical integration scheme require 15.6 seconds to evaluate the stability of the same model. These results thus show the effectiveness of the proposed method as it can prove the converter stability in shorter computational times without requiring exhaustive simulations using numerical integration.

Continuous Vector Fields for Precise Cable-Guided Landing of Tethered UAVs

This letter presents a method for the precise cable-guided landing of tethered drones. The proposed approach is based on a continuous velocity vector field that relies on the tether variables only. We assume that the vehicle is attached to a landing platform through a tensioned and free-of-collision cable whose length and angles are measured by a tether management system. We do not assume that the cable is taut. The method is suitable for highly degraded environments, where traditional landing strategies based on GNSS and vision are not feasible, such as dark underground mines and caves. This letter shows that, besides being easy to implement and adjust, the proposed vector field is proven to correctly guide the vehicle to its desired landing position. Experiments with a quad-rotor vehicle landing on static and moving platforms illustrate the precision, robustness, and efficacy of the method.

A new self-adaptive iterative method for variational inclusion problems on Hadamard manifolds with applications

The objective of this work is to design a new iterative method based on Armijo’s type-modified extragradient method for solving the inclusion problem $$\varvec{(A+B)^{-1}(0)}$$ , where $$\varvec{A}$$ is a maximal monotone vector field and $$\varvec{B}$$ is a continuous monotone vector field. The proposed method requires one projection at each iteration, reducing the cost of computational viewpoint and improving the convergence rate. A convergence theorem is established for the proposed extragradient method, significantly improving existing results. We provide concrete examples of Hadamard manifolds and convergency for numerical confirmation. Moreover, we demonstrate convergence results for the variational inequality problems in which the vector field’s monotonicity can be removed.

The curved uncut chip thickness model: A general geometric model for mechanistic cutting force predictions

The curved uncut chip thickness model is introduced to predict the cutting forces for general uncut chip geometries using the mechanistic approach. Classical geometric models assume that the cutting force is distributed along straight elementary sections of the uncut chip area, which has limited physical validity, but makes mathematical treatments easier for simple cases. The new model assumes that the flow of the material on the contact area of the tool is given by a continuous vector field, according to which the curved uncut chip thickness is measured. The cutting force is distributed along these paths, which leads to a mathematically unique and consistent solution for regular and complex cutting edge geometries. These curved paths can be generated by basic mechanical models, which mimic the more realistic motion of the chip segments along the rake face, without the need of explicit time-consuming cutting simulations. The presented computational procedure generalizes cutting force prediction based on geometric parameters, orthogonal cutting data and the orthogonal to oblique transformations only. The effectiveness of the model for various cutting edge geometries (e.g., thread turning inserts) under extreme cutting conditions is presented in case studies, laboratory and industrial experiments.

Smooth collision avoidance for the formation control of first order multi-agent systems

This work addresses collision avoidance in the formation control of a group of mobile robots with first-order dynamics perturbed by lateral and longitudinal slipping parameters. A Generalized Proportional–Integral Observer (GPIO) is designed to estimate these perturbations. Then, an Active Disturbance Rejection Control (ADRC) is proposed to solve the well-known formation control avoiding collisions among the agents. The control strategy only depends on the agents’ position measurements. On the other hand, Continuous Repulsive Vector Fields (C-RVFs) are developed to avoid collisions among the agents. For this purpose, a parameter depending on the inter-robot distance is developed to scale the RVFs properly. By proposing C-RVFs, the chattering is eliminated when using Discontinuous RVFs (D-RVFs). Numerical simulations and real-time experiments illustrate the agents’ performance when they are at risk of collision.

Neural Joint Space Implicit Signed Distance Functions for Reactive Robot Manipulator Control

In this letter, we present an approach for learning a neural implicit signed distance function expressed in joint space coordinates, that efficiently computes distance-to-collisions for arbitrary robotic manipulator configurations. Computing such distances is a long standing problem in robotics as approximate representations of the robot and environment geometry can lead to overly conservative constraints, numerical instabilities and expensive computations – limiting real-time reactive control and task success. Leveraging GPU parallelization and the differentiable nature of the proposed distance function allows for fast calculation of gradients with respect to the neural network inputs, providing a continuous repulsive vector field directly in joint space. We show that the learned high-resolution collision representation can be used to achieve real-time reactive control by i) formulating it as a collision-avoidance constraint for a quadratic programming (QP) inverse kinematics (IK), and ii) introducing it as a collision cost in a sampling-based joint space model predictive controller (MPC). For a reaching benchmark task with a 7DoF robot and dynamic obstacles intentionally obstructing the robot's path we achieve average 250 Hz control frequency with QP-IK and 92 Hz with MPC, showing an accelerated performance of 15% for QP-IK and 40% for MPC over baseline distance computation techniques.

Non-Euclidean Contraction Theory for Robust Nonlinear Stability

In this article, we study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _\infty$</tex-math></inline-formula> norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields, and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide incremental input-to-state stability and finite input-state gain properties for contracting systems, and a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system.

Contact-centric deformation learning

We propose a novel method to machine-learn highly detailed, nonlinear contact deformations for real-time dynamic simulation. We depart from previous deformation-learning strategies, and model contact deformations in a contact-centric manner. This strategy shows excellent generalization with respect to the object's configuration space, and it allows for simple and accurate learning. We complement the contact-centric learning strategy with two additional key ingredients: learning a continuous vector field of contact deformations, instead of a discrete approximation; and sparsifying the mapping between the contact configuration and contact deformations. These two ingredients further contribute to the accuracy, efficiency, and generalization of the method. We integrate our learning-based contact deformation model with subspace dynamics, showing real-time dynamic simulations with fine contact deformation detail.

There Exist Transitive Piecewise Smooth Vector Fields on $$\mathbb {S}^2$$ but Not Robustly Transitive

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $\S^2$. Piecewise-smooth vector fields, on the other hand, may present non-trivial recurrence even on $\S^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $\S^2$ is proved, see Theorem \ref{teorema-principal}. We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as {\it sliding region} and {\it escaping region}. More precisely, Theorem \ref{main:transitivity} states that, under the presence of transitivity, trajectories must interchange between sliding and escaping regions through tangency points. In addition, we prove that every transitive piecewise-smooth vector field is neither robustly transitive nor structural stable on $\S^2$, see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem \ref{main:general} addressing non-robustness on general compact two-dimensional manifolds.

Resource-aware synchronized path following of multiple unmanned surface vehicles with experiments: A cooperative vector field approach

This paper presents a cooperative vector field approach for synchronized path following of multiple autonomous surface vehicles (USVs) with limited communication resources. First, a cooperative vector field is proposed by introducing a path coordination design into a three-dimensional vector field. Based on the cooperative vector field, concise guidance laws of surge speed and angular rate are derived for synchronized path following. In order to reduce communication times among USVs, an event-triggered cooperative vector field is next proposed. Specifically, a predictor is devised to predict the current path variables by using the last transmitting ones, and an event-triggered mechanism is developed based on the predicted path variables. Then, a self-triggered cooperative vector field is developed to avoid continuous listening. A self-triggered mechanism is designed such that each USV predicts its next triggering time and broadcasts it to its neighbors at the current triggering time. The stability of the three cooperative vector fields is analyzed, and Zeno behavior is excluded. Finally, both simulation and experiment results are provided to illustrate the efficacy of the proposed continuous, event-triggered, and self-triggered cooperative vector fields for resource-aware synchronized path following.

Lie brackets of nonsmooth vector fields and commutation of their flows

It is well-known that the flows generated by two smooth vector fields commute, if the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of their Lie bracket in the sense of almost everywhere equality. We show that this cannot be extended to general a.e. differentiable vector fields admitting a.e. unique flows. We show, however, that the extension holds when one field is Lipschitz continuous and the other one is merely Sobolev regular (but admitting a regular Lagrangian flow).

Well-posedness of an interaction model on Riemannian manifolds

&lt;p style='text-indent:20px;'&gt;We investigate a model for collective behaviour with intrinsic interactions on smooth Riemannian manifolds. For regular interaction potentials, we establish the local well-posedness of measure-valued solutions defined via optimal mass transport. We also extend our result to the global well-posedness of solutions for manifolds with nonpositive bounded sectional curvature. The core concept underlying the proofs is that of Lipschitz continuous vector fields in the sense of parallel transport.&lt;/p&gt;

The Hairy Ball problem is PPAD-complete

The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of (a) computing an approximate zero is PPAD-complete, and (b) computing an exact zero is FIXP-hard. We also consider the Hairy Ball Theorem on toroidal instead of spherical domains and show that the approximate problem remains PPAD-complete. On a conceptual level, our PPAD-membership results are particularly interesting, because they heavily rely on the investigation of multiple-source variants of End-of-Line, the canonical PPAD-complete problem. Our results on these new End-of-Line variants are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the Imbalance problem defined by Beame et al. in 1998.

A directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space where p is larger than the space dimension, under the assumption that the so-called “forward-backward integral curves” associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable. Additionally, for such vector fields it is possible to prove almost everywhere uniqueness of (standard) integral curves, which also implies uniqueness of positive measure solutions to the continuity equation with absolutely continuous initial datum.

New Tseng\u2019s extragradient methods for pseudomonotone variational inequality problems in Hadamard manifolds

We propose Tseng’s extragradient methods for finding a solution of variational inequality problems associated with pseudomonotone vector fields in Hadamard manifolds. Under standard assumptions such as pseudomonotone and Lipschitz continuous vector fields, we prove that any sequence generated by the proposed methods converges to a solution of variational inequality problem, whenever it exits. Moreover, we give some numerical experiments to illustrate our main results.