Combinatorics
Related papers: 20
About
Combinatorics is the branch of mathematics concerned with counting, arranging, and analyzing discrete structures — such as graphs, sequences, permutations, and configurations. In robotics and AI, combinatorial reasoning underpins a surprisingly wide range of problems: determining the number of valid configurations a mechanism can take (such as the 40 configurations of a Stewart platform), analyzing graph-based formation structures for multi-robot teams, encoding homotopy classes of paths for motion planning complexity bounds, and reasoning about the arrangement of geometric objects in collision detection and workspace analysis. Combinatorial arguments also appear in computational complexity results, such as proving that sliding-block puzzles and motion planning problems are PSPACE-complete, and in protein loop reconstruction where the space of discrete backbone arrangements must be systematically explored. By providing tools to enumerate, classify, and bound discrete solution spaces, combinatorics helps roboticists understand both what is theoretically possible and what is computationally tractable, making it foundational to algorithm design, mechanism analysis, and multi-agent coordination.
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Top Cited Papers
Singularity analysis of closed-loop kinematic chains
Clément Gosselin, Jorge Angeles
Citations: 1719 • 1990
A fast procedure for computing the distance between complex objects in three-dimensional space
Éric Gilbert, Daniel Johnson, S. Sathiya Keerthi
Citations: 1470 • 1988
Modeling and control of formations of nonholonomic mobile robots
Jaydev P. Desai, J.P. Ostrowski, Vijay Kumar
Citations: 1155 • 2001
Landmarks in graphs
Samir Khuller, Balaji Raghavachari, Azriel Rosenfeld
Citations: 803 • 1996
Applied Interval Analysis With Examples in Parameter and State Estimation, Robust Control and Robotics
Luc Jaulin, Michel Kieffer, Olivier Didrit, Éric Walter
Citations: 754 • 1993
Toward Efficient Trajectory Planning: The Path-Velocity Decomposition
Kamal Kant, Steven W. Zucker
Citations: 753 • 1986
Erratum: Distributed Anonymous Mobile Robots: Formation of Geometric Patterns
I. Suzuki, Masafumi Yamashita
Citations: 631 • 2006
A modal approach to hyper-redundant manipulator kinematics
Gregory S. Chirikjian, Joel W. Burdick
Citations: 584 • 1994
A motion planner for nonholonomic mobile robots
Jean‐Paul Laumond, P. Jacobs, Michel Taïx, Richard M. Murray
Citations: 582 • 1994
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
Singular Configurations of Parallel Manipulators and Grassmann Geometry
Jean‐Pierre Merlet
Citations: 554 • 1989
Stabilisation of infinitesimally rigid formations of multi-robot networks
Laura Krick, Mireille E. Broucke, Bruce A. Francis
Citations: 547 • 2008
New lower bound techniques for robot motion planning problems
John Canny, John H. Reif
Citations: 544 • 1987
Singular configurations of parallel manipulators and grassmann geometry
Jean‐Pierre Merlet
Citations: 499 • 1989
Sub-angstrom accuracy in protein loop reconstruction by robotics-inspired conformational sampling
Daniel J. Mandell, Evangelos A. Coutsias, Tanja Kortemme
Citations: 487 • 2009
Formulation and optimization of cubic polynomial joint trajectories for industrial robots
Choun‐Sea Lin, P.-R. Chang, J. Luh
Citations: 483 • 1983
A fast algorithm for incremental distance calculation
Ming C. Lin, John Canny
Citations: 474 • 2002
On the “piano movers'” problem I. The case of a two‐dimensional rigid polygonal body moving amidst polygonal barriers
Jacob T. Schwartz, Micha Sharir
Citations: 451 • 1983
A new geometric notation for open and closed-loop robots
Wisama Khalil, J. Kleinfinger
Citations: 392 • 1986