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Efficient continuous curvature path generation with pseudo-parametrized algebraic splines

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Abstract

This paper investigates a method to achieve computationally efficient path generation and tracking, suitable for manned operations. To provide an adequate level of comfort to the passengers, path geometric continuity of the second order (\(G^2\)) is used. Thus, generalized 3D \(G^2\) Dubins paths are proposed as a template, later approximated with algebraic polynomial splines of the third order. This allows to leverage the strengths of both Dubins paths and spline-based methods. The proposed splines pseudo-parametrization avoids explicit numerical evaluation of elliptic integrals, while achieving a good approximation of arc-length parametrization. The method is first applied to planar maneuvers, and then extended to complete 3D ones, such as climbing turns. The strategy leads to facilitated 4D planning, that can be either velocity or time-based, re-planning is a local problem, and path tracking convergence is enhanced. The technique has been demonstrated on a certifiable fly-by-wire platform both in laboratory tests and in flights, including landings. Flight results will be shown and discussed at the end of the paper.

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Abbreviations

\(\chi\), \(\gamma\) :

Aircraft’s ground track and flight path angles

\(\phi\), \(\theta\) :

Aircraft’s bank and pitch angles

p, q :

Aircraft’s bank and pitch rates

u :

Non-normalized spline parameter

r :

Pseudo-normalized spline parameter

\(\bar{r}_k\) :

Root at timestep k

\(\bar{r}^n\) :

Candidate root at iteration n

\(r_1\) :

Candidate root for quadratic minimization

\(\varvec{S}(u)\) :

A non arc-length parametrized spline

\(\varvec{\tilde{S}}(r)\) :

A pseudo-parametrized spline

\(x_s\), \(y_s\), \(z_s\) :

Spline’s components along coordinate x,y or z

\(\varvec{p}\) :

A point in 3D space

\(\varvec{\tau }\) :

Tangent vector in 3D space

\(\kappa\) :

Curve curvature

l(u):

Spline arc length

\(\hat{L}\) :

Estimated spline length

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Acknowledgements

Part of the work executed for this paper has been funded by the FlySmart project, conducted under the LuFo program, and financed by the German Bundesministeriums für Wirtschaft und Technologie. The authors would like to thank the partners in this project, namely Airbus Defence and Space for the management, the Institute for Aircraft Systems of the University of Stuttgart for providing the FBW platform, and Diamond Aircraft for conducting the flight tests.

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  1. Institute of Flight Mechanics and Control, University of Stuttgart, 70569, Stuttgart, Germany

    Federico Pinchetti, Alexander Joos & Walter Fichter

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  1. Federico Pinchetti
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  2. Alexander Joos
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  3. Walter Fichter
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Correspondence to Federico Pinchetti.

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Pinchetti, F., Joos, A. & Fichter, W. Efficient continuous curvature path generation with pseudo-parametrized algebraic splines. CEAS Aeronaut J 9, 557–570 (2018). https://doi.org/10.1007/s13272-018-0306-3

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