On the dynamic Markov-Dubins problemfrom path planning in robotics and biolocomotion to computational anatomy

Resumo

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.

Ficha do documento

Tipo

Artigo científico

Ano

2013

Instituição

Maik Nauka/Interperiodica/Springer

Fonte

Repositório da FGV

Idioma

Inglês

Acesso

Acesso restrito

DOI

10.1134/S1560354713010012

Identificador

oai:repositorio.fgv.br:10438/23314

Palavras-chave

Geometric mechanicsCalculus of variationsMarkov-Dubins problemMatemáticaRobóticaAnatomia