On the Jacobian matrix of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si67.svg" display="inline" id="d1e1044"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo> <mml:mi>R</mml:mi> <mml:mi>P</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> and triangular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si355.svg" display="inline" id="d1e1058"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo> <mml:mi>U</mml:mi> <mml:mi>P</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> linkages
Bibliographic record
Abstract
The Jacobian matrix of a linkage relates actuated velocities to its end-effector velocity. Deriving this matrix for serial mechanisms is straightforward, but usually more challenging for parallel linkages. The velocity equation of these linkages typically comprises two distinct Jacobian matrices. An additional step required to establish the velocity of the end-effector of the linkage as a function of velocities of its actuators consists in inverting the first matrix and multiplying the result with the second, yielding a unique matrix which is here referred to as the Jacobian matrix. The previous inversion is almost universally conducted numerically except for the simplest of linkages. Here, we demonstrate that, even for non-trivial linkages such as the 3 − R P S and triangular 6 − U P S this inversion can also be done analytically. Similar to serial linkages, the Jacobian matrix is presented as a series of twists, each reflecting the impact of an actuator on the end-effector’s motion. Their geometric interpretation is provided and studied. Finally, how the method applies to other topologies of mechanisms is also discussed. • The 3-RPS and 6-UPS linkages’ Jacobian matrices are given analytically. • The final Jacobian matrix is expressed column-wise as twists. • Numerical examples are presented to support the effectiveness of the method. • The approach is extendable to other limited-DOF and hybrid mechanisms.
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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 | 0.009 | 0.004 |
| Meta-epidemiology (narrow) | 0.004 | 0.007 |
| Meta-epidemiology (broad) | 0.002 | 0.006 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.005 | 0.004 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.007 | 0.007 |
| Research integrity | 0.008 | 0.007 |
| Insufficient payload (model declined to judge) | 0.086 | 0.004 |
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; both teacher heads agree on what is shown here.
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".