Smooth Feedback Construction Over Spherical Polytopes
Bibliographic record
Abstract
In this work, we investigate the partitioning and control problems on the 2− sphere, the set of all unit vectors in ${{\mathbb{R}}^3}$. Specifically, we present a spherical-polytope-based partitioning for the 2− sphere and then propose a novel approach to construct a feedback control law over a given set of spherical polytopes. Instead of designing the control law directly on the sphere, we propose a smooth atlas on it based on the gnomonic projection. We further show that the gnomonic map projects the spherical polytopes to Euclidean polytopes. Moreover, the kinematics evolving on a spherical polytope can be transformed via feedback into a single integrator in the Euclidean space. Thanks to these properties, control algorithms that were originally developed for polytopes in Euclidean spaces can now be applied to spherical polytopes on the 2 –sphere. We conclude this paper by showing a control construction on the sphere with cluttered obstacles.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".