Markov-Dubins Interpolating Curves
C. Yal\c{c}{\i}n Kaya

TL;DR
This paper extends the Markov--Dubins problem to include passing through multiple points, formulates it as an optimal control problem, analyzes solution structures, and proposes a numerical method for path computation.
Contribution
It introduces the Markov--Dubins interpolation problem, analyzes the structure of optimal solutions, including abnormal cases, and develops a numerical approach for path planning.
Findings
Interpolants are composed of circular and straight segments.
Abnormal solutions exist but cannot contain straight segments.
Feasible $CSC$-type solutions are stationary and critical.
Abstract
A realistic generalization of the Markov--Dubins problem, which is concerned with finding the shortest planar curve of constrained curvature joining two points with prescribed tangents, is the requirement that the curve passes through a number of prescribed intermediate points/nodes. We refer to this generalization as the Markov--Dubins interpolation problem. We formulate this interpolation problem as an optimal control problem and obtain results about the structure of its solution using optimal control theory. The Markov--Dubins interpolants consist of a concatenation of circular () and straight-line () segments. Abnormal interpolating curves are shown to exist and characterized; however, if the interpolating curve contains a straight-line segment then it cannot be abnormal. We derive results about the stationarity, or criticality, of the feasible solutions of certain structure.…
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Computational Geometry and Mesh Generation · Polynomial and algebraic computation
