Pure mathematics
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Pure mathematics encompasses the rigorous study of abstract structures—including algebra, geometry, topology, analysis, and number theory—developed for logical completeness rather than immediate application. In robotics and AI, these mathematical foundations are indispensable: Lie group theory enables elegant representations of rigid body motion and robot kinematics; differential geometry and topology underpin motion planning on manifolds and configuration spaces; algebraic methods solve calibration problems and polynomial systems arising in sensor fusion; and tools like Morse theory, Laplace's equation, and topological complexity guide path planning algorithms. Concepts such as nonholonomic mechanics, Koopman operators, and generalized inverses directly shape how robots are modeled, controlled, and coordinated. Pure mathematics matters because it provides the precise language and proven theorems that give engineers confidence in their algorithms—ensuring correctness, stability, and generality across diverse robotic systems. Without these abstract foundations, many practical advances in motion planning, state estimation, multi-robot coordination, and geometric computing would lack the rigorous grounding necessary to guarantee reliable real-world performance.
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Structural properties and classification of kinematic and dynamic models of wheeled mobile robots
G. Campion, Georges Bastin, Brigitte d’Andréa-Novel
Citations: 1162 • 1996
Nonholonomic Mechanics and Control
AM Bloch, Bernard Brogliato
Citations: 1123 • 2004
On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds
Jacob T. Schwartz, Micha Sharir
Citations: 807 • 1983
The Invariant Extended Kalman Filter as a Stable Observer
Axel Barrau, Silvère Bonnabel
Citations: 588 • 2016
Some algebraic and geometric computations in PSPACE
John Canny
Citations: 581 • 1988
Robot navigation functions on manifolds with boundary
Daniel E. Koditschek, Elon Rimon
Citations: 565 • 1990
Robot sensor calibration: solving AX=XB on the Euclidean group
F.C. Park, B.J. Martin
Citations: 551 • 1994
LEDA: A platform for combinatorial and geometric computing
Citations: 489 • 2000
On the Parametrization of the Three-Dimensional Rotation Group
John Stuelpnagel
Citations: 479 • 1964
Path planning using Laplace's equation
Christopher I. Connolly, J. Brian Burns, Richard Weiss
Citations: 478 • 2002
Stable adaptive observers for nonlinear time-varying systems
Georges Bastin, M. Gevers
Citations: 477 • 1988
APPLICATION OF THEOREM PROVING TO PROBLEM SOLVING**This research is a part of Project Defender and was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by Rome Air Development Center under Contracts AF 30(602)-4147 and F30602-69-C-0056.††This preprint is a preliminary version and is subject to modification prior to publication.
Cordell Green
Citations: 463 • 1981
Coordination and Geometric Optimization via Distributed Dynamical Systems
Jorge Cortés, Francesco Bullo
Citations: 450 • 2005
The Lie group of rigid body displacements, a fundamental tool for mechanism design
JM Hervé
Citations: 425 • 1999
Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability
Jorge Cortés
Citations: 401 • 2009
Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses
Arthur Albert
Citations: 395 • 1969
Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry
Leo Dorst, Daniel Fontijne, Stephen Mann
Citations: 390 • 2007
Geometric control of mechanical systems : modeling, analysis, and design for simple mechanical control systems
Francesco Bullo, Andrew D. Lewis
Citations: 383 • 2005
Topological Complexity of Motion Planning
Michael Färber
Citations: 380 • 2003
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Rida T. Farouki
Citations: 364 • 2008