Spectral Properties of the Normalized Rigidity Matrix for Triangular Formations
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Bibliographic record
Abstract
This work establishes properties of the normalized rigidity matrix in two- and three-dimensional spaces. The upper bound of the normalized rigidity matrix singular values is derived for minimally and infinitesimally rigid frameworks in two- and three-dimensional spaces. We prove that the transformation of a framework does not affect the normalized rigidity matrix properties. The largest minimum singular value of the normalized rigidity matrix for a rigid framework of three agents in two-dimensional space is given as well as necessary and sufficient conditions to reach that value. These results can be used in stability analysis and control design of a distance-based formation control. The numerical simulation for multi-agent systems in two-dimensional space illustrates the theoretical results. Moreover, a real-time simulation is provided to demonstrate the spectral properties of the normalized rigidity matrix.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it