Algebraic Input-output Angle Equation Derivation Algorithm for the Six Distinct Angle Pairings in Arbitrary Planar 4R Linkages
Bibliographic record
Abstract
This paper describes a generalised algorithm that can be applied to any single degree of freedom parallel kinematic chain to determine the algebraic polynomial that represents the input-output equation relating any pair of distinct angles between any pair of links in the kinematic chain. There are six such algebraic polynomials for an arbitrary four-bar linkage. The algorithm consists of assigning standard Denavit- Hartenberg coordinate systems and parameters to the open kinematic chain. The open chain is conceptually closed by equating the forward kinematic transformation that maps coordinates of points in the “end-effector” coordinate system to the relatively non-moving base coordinate system to the identity matrix. The resulting transformation is mapped to Study soma coordinates wherein the twist and joint angles have been converted to tangent half-angle parameters. Elimination theory is then applied to the soma coordinates revealing a single algebraic polynomial in terms of the link lengths and the desired angle pair. Example applications are discussed for continuous approximate synthesis, mobility classification, and the design parameter space.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.018 | 0.007 |
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 source (direct Gemma or distilled Codex), 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".