Singularity Analysis of a Three-Leg Six-Degree-of-Freedom Parallel Platform Mechanism Based on Grassmann Line Geometry
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Bibliographic record
Abstract
This paper addresses the determination of the singularity loci of a six-degree-of-freedom spatial parallel platform mechanism of a new type that can be statically balanced. The mechanism consists of a base and a mobile platform that are connected by three legs using five-bar linkages. A general formulation of the Jacobian matrix is first derived that allows one to determine the Plücker vectors associated with the six input angles of the architecture. The linear dependencies between the corresponding lines are studied using Grassmann line geometry, and the singular configurations are presented using simple geometric rules. It is shown that most of the singular configurations of the three-leg six-degree-of-freedom parallel manipulator can be reduced to the generation of a general linear complex. Expressions describing all the corresponding singularities are then obtained in closed form. Thus, it is shown that for a given orientation of the mobile platform, the singularity locus corresponding to the general complex is a quadratic surface (i.e., either a hyperbolic, a parabolic, or an elliptic cylinder) oriented along the z-axis. Finally, three-dimensional representations that show the intersection between the singularity loci and the constant-orientation workspace of the mechanism are given.
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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 | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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