A Multi-Lattice Framework for Robotics: Navigating the Complexity-Efficiency Pareto Frontier

Abstract

1. Title A Multi-Lattice Framework for Robotics: Navigating the Complexity-Efficiency Pareto Frontier This version has been withdrawn by the author for further refinement. Please refer to the updated 6D proof in Record. 2. Abstract / Description This preprint introduces a unified theoretical framework for selecting $N$-dimensional mathematical lattices ($A_n$, $E_n$) to optimize robotic control manifolds. While randomized sampling methods are standard, they often lack geometric interpretability. We propose a Pareto-optimal approach that balances computational complexity—scaled as $O(K \log K)$—against kinematic efficiency. Key Contributions: The Dimensionality Frontier: A formal model identifying the "Efficiency Decay Zone" when lattice dimensionality exceeds system degrees of freedom. 5D Restorative Law: Empirical evidence of stability constraints using a 109-node $A_5$ lattice. 6D Robotics-Only Proof: Validation of the 72-node $E_6$ lattice on an ABB IRB 4600 industrial manipulator, demonstrating provable singularity avoidance and high manipulability ($\kappa \approx 4.5$) without supplementary software heuristics. This work establishes structured lattices as a fundamental primitive for reliable, singularity-robust robotic design. Primary: Robotics, Lattice-Based Planning, $E_6$ Lattice, Kinematic Singularities. Secondary: Pareto Optimization, ABB IRB 4600, Computational Geometry, Manipulability. 4. Metadata Fields (Zenodo Specific) Upload Type: Publication -> Preprint Publication Date: 2026-01-06 License: Creative Commons Attribution 4.0 International (standard for open research). Related Identifiers: If you have uploaded the code or dataset separately, use the "Is supplemented by" field to link to your other DOIs: 5-DoF Data: 10.5281/zenodo.18147633 6-DoF Data: 10.5281/zenodo.18148976 5. Final "Pro Tip" In the description, explicitly mention: "This research identifies $E_6$ as the optimality peak for 6-DoF systems." This direct statement helps other researchers cite your work as the definitive source for why $E_6$ is preferred over $E_7$ or $E_8$ for standard manipulators.