Laplace operator
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The Laplace operator (also called the Laplacian, denoted ∇²) is a second-order differential operator that computes the divergence of the gradient of a scalar or vector field, essentially measuring how a quantity at a point differs from its local average. In robotics and AI, it appears across a remarkably wide range of applications. In path and motion planning, harmonic potential fields derived from Laplace's equation guide robots smoothly from start to goal while avoiding obstacles. In multi-robot systems, the graph Laplacian — a matrix encoding network topology — is central to consensus algorithms, formation control, connectivity maintenance, and distributed optimization, where its eigenvalues quantify how well-connected and coordinated a swarm is. In machine learning, Laplacian-based methods such as Laplacian Eigenmaps and Laplacian Support Vector Machines enable dimensionality reduction, terrain classification, and semi-supervised learning by preserving geometric structure in data. The Laplace operator matters because it provides a unified mathematical foundation connecting physical field theory, graph theory, and spectral analysis, enabling elegant and computationally tractable solutions to coordination, navigation, and perception problems in autonomous systems.
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Laplacian Sheep: A Hybrid, Stop-Go Policy for Leader-Based Containment Control
Giancarlo Ferrari‐Trecate, Magnus Egerstedt, Annalisa Buffa, M. Ji
Citations: 136 • 2006
Distributed formation maneuver control by manipulating the complex Laplacian
Héctor García de Marina
Citations: 37 • 2021
Distributed optimization for a class of uncertain MIMO nonlinear multi-agent systems with arbitrary relative degree
Ranran Li, Guang‐Hong Yang
Citations: 34 • 2019
Global Motion Planning using a Laplacian Potential Field.
Keisuke Sato
Citations: 28 • 1993
Constrained distributed algebraic connectivity maximization in robotic networks
Andrea Simonetto, Tamás Keviczky, Robert Babuška
Citations: 25 • 2013
On distributed maximization of algebraic connectivity in robotic networks
Andrea Simonetto, Tamás Keviczky, Robert Babuška
Citations: 22 • 2011
Laplacian Support Vector Machine for Vibration-Based Robotic Terrain Classification
Wenlei Shi, Zerui Li, Wenjun Lv, Yuping Wu, Ji Chang, Xiaochuan Li
Citations: 22 • 2020
Laplacian-Based Consensus on Spatial Computers
Nelson Elhage, Jacob Beal
Citations: 20 • 2012
Enforcing biconnectivity in multi-robot systems
Mehran Zareh, Lorenzo Sabattini, Cristian Secchi
Citations: 19 • 2016
PATH PLANNING SIMULATION USING HARMONIC POTENTIAL FIELDS THROUGH FOUR POINT-EDGSOR METHOD VIA 9-POINT LAPLACIAN
Azali Saudi, Jumat Sulaiman
Citations: 18 • 2016
Distributed Nash equilibrium seeking for high-order integrator dynamics subject to disturbances of unknown bounds
Xiongnan He, Jie Huang
Citations: 18 • 2023
Bilateral control of the degree of connectivity in multiple mobile-robot teleoperation
Cristian Secchi, Antonio Franchi, HH Bülthoff, Paolo Robuffo Giordano
Citations: 17 • 2013
Spatial adaption of robot trajectories based on laplacian trajectory editing
Thomas Nierhoff, Sandra Hirche, Yoshihiko Nakamura
Citations: 17 • 2015
Robot Path Planning Using Four Point-Explicit Group Via Nine-Point Laplacian (4EG9L) Iterative Method
Azali Saudi, Jumat Sulaiman
Citations: 17 • 2012
Oscillatory Group-Bipartite Consensus in a Swarm of Robots With Multiple Oscillatory Leaders
Jun Liu, Shaorong Xie, Hengyu Li
Citations: 17 • 2022
Network Connectivity Maintenance via Nonsmooth Control Barrier Functions
Pio Ong, Beatrice Capelli, Lorenzo Sabattini, Jorge Cortés
Citations: 14 • 2021
Distributed Laplacian Eigenvalue and Eigenvector Estimation in Multi-robot Systems
Mehran Zareh, Lorenzo Sabattini, Cristian Secchi
Citations: 14 • 2018
Dynamic facial expression recognition using Laplacian Eigenmaps-based manifold learning
Bogdan Raducanu, Fadi Dornaika
Citations: 12 • 2010
Fast trajectory replanning using Laplacian mesh optimization
Thomas Nierhoff, Sandra Hirche
Citations: 11 • 2012
Semisupervised Location Awareness in Wireless Sensor Networks Using Laplacian Support Vector Regression
Jaehyun Yoo, H. Jin Kim
Citations: 11 • 2014