On the dynamic Markov-Dubins problem: From path planning in robotics and biolocomotion to computational anatomy
Bibliographic record
Abstract
Andrei Andreyevich Markov proposed in 1889 the problem (solved by Dubins in 1957) of finding the twice continuously differentiable (arc length parameterized) curve with bounded curvature, of minimum length, connecting two unit vectors at two arbitrary points in the plane. In this note we consider the following variant, which we call the dynamic Markov-Dubins problem (dM-D): to find the time-optimal C 2 trajectory connecting two velocity vectors having possibly different norms. The control is given by a force whose norm is bounded. The acceleration may have a tangential component, and corners are allowed, provided the velocity vanishes there. We show that for almost all the two vectors boundary value conditions, the optimization problem has a smooth solution. We suggest some research directions for the dM-D problem on Riemannian manifolds, in particular we would like to know what happens if the underlying geodesic problem is completely integrable. Path planning in robotics and aviation should be the usual applications, and we suggest a pursuit problem in biolocomotion. Finally, we suggest a somewhat unexpected application to “dynamic imaging science”. Short time processes (in medicine and biology, in environment sciences, geophysics, even social sciences?) can be thought as tangent vectors. The time needed to connect two processes via a dynamic Markov-Dubins problem provides a notion of distance. Statistical methods could then be employed for classification purposes using a training set.
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".