Fair curves for motion planning

Brunnett, Guido; Kiefer, Jorg; Wendt, Jörg

doi:10.1504/ijvd.1999.005579

Cited by 2 publications

(3 citation statements)

References 4 publications

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“…Although Markov-Dubins path has been extensively studied both theoretically and practically, finding as wide range of applications as the path planning of drones (or uninhabited aerial vehicles) and robots, and the tunnelling in underground mines, to the author's knowledge, its generalization to interpolation has not been studied in its entirety or true form yet. Brunnett, Kiefer and Wendt [8] consider Problem (P) (but not in the form we pose it here) with a bound on the average, rather than the pointwise, curvature, and propose an algorithm for finding what they call a "Dubins spline" with the following ad hoc steps: (Step 1) Estimate/guess the missing velocity directions at the interior nodes, (Step 2) Find a Markov-Dubins path of type CSC (but not of type CCC or a subset thereof) between each two consecutive nodes and (Step 3) Update the estimated missing velocity directions at the interior nodes using some nonlinear optimization procedure to minimize the overall length of the spline; [if some stopping criterion is not satisfied, then] go to Step 2. In [8], not only the problem that is solved is different from Problem (P), but also the theory and pertaining analysis are not adequately covered. Similar ad hoc approaches ultimately leading to interpolants which are suboptimal solutions, of problems related to Problem (P), can be found in the literature; see, for example, [20,29,37].…”

Section: Interpolation and The Markov-dubins Problemmentioning

confidence: 99%

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Markov–Dubins interpolating curves

Kaya

2019

Comput Optim Appl

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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 (C) and straight-line (S) 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. In particular, any feasible interpolant with arc types of CSC in each stage is proved to be stationary, i.e., critical. We propose a numerical method for computing Markov-Dubins interpolating paths. We illustrate the theory and the numerical approach by four qualitatively different examples.

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Section: Interpolation and The Markov-dubins Problemmentioning

confidence: 99%

“…Relatively more recently, Goaoc, Kim and Lazard [18] studied Problem (P), by using the same assumption that was also used in [8]: the optimal path between any two consecutive nodes will be of type CSC. To guarantee this, every two consecutive nodes are assumed to be placed farther than 4/a units apart.…”

Section: Interpolation and The Markov-dubins Problemmentioning

confidence: 99%

Markov–Dubins interpolating curves

Kaya

2019

Comput Optim Appl

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“…Wang et al [20] study the B-spline interproximation with different energy forms and parametrization techniques. Brunnett [21] proposes two variational models of fair curves for motion planning. The geometric Hermite curves with minimal strain energy are investigated by Yong and Cheng [22].…”

Section: Related Workmentioning

confidence: 99%

Geometric construction of energy-minimizing Béezier curves

Wang

Chen

2011

Sci. China Inf. Sci.

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Modeling energy-minimizing curves have many applications and are a basic problem of Geometric Modeling. In this paper, we propose the method for geometric design of energy-minimizing Bézier curves. Firstly, the necessary and sufficient condition on the control points for Bézier curves to have minimal internal energy is derived. Based on this condition, we propose the geometric constructions of three kinds of Bézier curves with minimal internal energy including stretch energy, strain energy and jerk energy. Given some control points, the other control points can be determined as the linear combination of the given control points. We compare the three kinds of energy-minimizing Bézier curves via curvature combs and curvature plots, and present the collinear properties of quartic energy-minimizing Bézier curves. We also compare the proposed method with previous methods on efficiency and accuracy. Finally, several applications of the curve generation technique, such as curve interpolation with geometric constraints and modeling of circle-like curves are discussed.

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Fair curves for motion planning

Cited by 2 publications

References 4 publications

Markov–Dubins interpolating curves

Markov–Dubins interpolating curves

Geometric construction of energy-minimizing Béezier curves

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