# First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Navigation Functions

**Definition**

**1.**

- 1.
- Analytic on the interior of $\mathcal{F}$,
- 2.
- Polar on $\mathcal{F}$ with a minimum at an interior point ${q}_{d}\in \circ \mathcal{F}$,
- 3.
- Admissible on $\mathcal{F}$,
- 4.
- Morse on Ω.

#### 2.2. Semi-Analytical Approximation of the Green’s Function for the Laplacian Operator

#### 2.2.1. Green’s Function

#### 2.2.2. Green’s Function from Conformal Mappings

**Definition**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

#### 2.2.3. Harmonic Function Exterior to Non-Overlapping Disks

#### 2.3. First Order Variation of the Harmonic Solution

## 3. Results and Discussion

#### 3.1. Numerical Validation of the Green’s Function Approximation

#### 3.1.1. Convergence of the Method

#### 3.1.1.1. Reconstruction of the Harmonic Solution for Constant Boundary Conditions

#### 3.1.1.2. Reconstruction for Non-Constant Boundary Conditions

#### 3.1.2. Time Complexity

#### 3.1.3. Possible Applications of the Solution

#### 3.1.3.1. Green’s Function of Sphere Worlds

#### 3.2. Numerical Validation of the Hadamard Variation of the Green’s Function

- by varying the center of the obstacle disc, which is representative of the effect of a locally constant wind on an aircraft.
- and by varying the radius of the obstacle disc.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Khatib, O. Real-time obstacle avoidance for manipulators and mobile robots. Int. J. Robot. Res.
**1986**, 5, 90–98. [Google Scholar] [CrossRef] - Kim, D.H.; Wang, H.; Shin, S. Decentralized control of autonomous swarm systems using artificial potential functions: Analytical design guidelines. J. Intell. Robot. Syst.
**2006**, 45, 369–394. [Google Scholar] [CrossRef] - Zeghal, K. Vers une Théorie de la Coordination D’actions. Application à la Navigation Aérienne. Ph.D. Thesis, Université Pierre et Marie Curie, Paris, France, 1994. [Google Scholar]
- Ge, S.S.; Cui, Y.J. Dynamic motion planning for mobile robots using potential field method. Auton. Robots
**2002**, 13, 207–222. [Google Scholar] [CrossRef] - Dimarogonas, D.V.; Kyriakopoulos, K.J.; Theodorakatos, D. Totally distributed motion control of sphere world multi-agent systems using decentralized navigation functions. In Proceedings of the 2006 IEEE International Conference on Robotics and Automation, Orlando, FL, USA, 15–19 May 2006; pp. 2430–2435. [Google Scholar]
- Rimon, E.; Koditschek, D.E. Exact robot navigation using artificial potential functions. IEEE Trans. Robot. Autom.
**1992**, 8, 501–518. [Google Scholar] [CrossRef] [Green Version] - Guys, L. Planification de Trajectoires d’Avions sans Conflit: Fonctions Biharmoniques et Fonction de Navigation Harmonique. Ph.D. Thesis, Université Toulouse 3 Paul Sabatier, Toulouse, France, 2014. [Google Scholar]
- Pêtrès, C.; Romero-Ramirez, M.-A.; Plumet, F. Reactive path planning for autonomous sailboat. In Proceedings of the 2011 15th International Conference on Advanced Robotics (ICAR), Tallinn, Estonia, 20–23 June 2011; pp. 112–117. [Google Scholar]
- Romano, M.; Virgili-Llop, J.; Zappulla, R., II. Near-optimal real-time spacecraft guidance and control using harmonic potential functions and a modified rrt. In Proceedings of the 27th AAS/AIAA Spaceflight Mechanics Meeting, San Antonio, TX, USA, 6–9 February 2017. [Google Scholar]
- Hacohen, S.; Shoval, S.; Shvalb, N. Applying probability navigation function in dynamic uncertain environments. Robot. Auton. Syst.
**2017**, 87, 237–246. [Google Scholar] [CrossRef] - Hadamard, J.; Fréchet, M. Leçons Sur le Calcul des Variations; A. Hermann et fils: Paris, France, 1910; Volume 1. [Google Scholar]
- Jordan, K.E.; Richter, G.R.; Sheng, P. An efficient numerical evaluation of the green’s function for the helmholtz operator on periodic structures. J. Comput. Phys.
**1986**, 63, 222–235. [Google Scholar] [CrossRef] - Beck, J.V.; Cole, K.D.; Haji-Sheikh, A.; Litkouhi, B. Heat Conduction Using Green’s Functions; Hemisphere Publishing Corporation: London, UK, 1992. [Google Scholar]
- Bertoluzza, S.; Decoene, A.; Lacouture, L.; Martin, S. Local error estimates of the finite element method for an elliptic problem with a dirac source term. arXiv, 2015; arXiv:1505.03032. [Google Scholar] [CrossRef]
- Sadybekov, M.A.; Torebek, B.T.; Turmetov, B.K. Representation of the green’s function of the exterior neumann problem for the laplace operator. Sib. Math. J.
**2017**, 58, 153–158. [Google Scholar] [CrossRef] - Crowdy, D.; Marshall, J. Green’s functions for laplace’s equation in multiply connected domains. IMA J. Appl. Math.
**2007**, 72, 78–301. [Google Scholar] [CrossRef] - Crowdy, D.G.; Kropf, E.H.; Green, C.C.; Nasser, M.M.S. The schottky–klein prime function: A theoretical and computational tool for applications. IMA J. Appl. Math.
**2016**, 81, 589–628. [Google Scholar] [CrossRef] - Koditschek, D.E.; Rimon, E. Robot navigation functions on manifolds with boundary. Adv. Appl. Math.
**1990**, 11, 412–442. [Google Scholar] [CrossRef] - Roussos, G.P.; Dimarogonas, D.V.; Kyriakopoulos, K.J. 3d navigation and collision avoidance for a non-holonomic vehicle. In Proceedings of the 2008 American Control Conference, Seattle, WA, USA, 11–13 June 2008; pp. 3512–3517. [Google Scholar]
- Delgado, L.; Prats, X. Fuel consumption assessment for speed variation concepts during the cruise phase. In Proceedings of the Conference on Air Traffic Management (ATM) Economics, Belgrade, Serbia, 10 September 2009. [Google Scholar]
- Rimon, E.; Koditschek, D.E. Exact robot navigation using cost functions: The case of distinct spherical boundaries in e/sup n. In Proceedings of the 1988 IEEE International Conference on Robotics and Automation, Philadelphia, PA, USA, 24–29 April 1988; pp. 1791–1796. [Google Scholar]
- Maeda, F.-Y.; Suzuki, N. The integrability of superharmonic functions on lipschitz domains. Bull. Lond. Math. Soc.
**1989**, 21, 270–278. [Google Scholar] [CrossRef] - Nevanlinna, R.; Behnke, H.; Grauert, H. Analytic Functions; Springer: Berlin/Heidelberg, Germany, 1970; Volume 3. [Google Scholar]
- Henrici, P. Applied and Computational Complex Analysis; John Wiley & Sons: Hoboken, NJ, USA, 1993; Volume 3. [Google Scholar]
- Krantz, S.G. Handbook of Complex Variables; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Trefethen, L.N. Ten digit algorithms. Presented at the A. R. Mitchell Lecture at the 2005 Dundee Biennial Conference on Numerical Analysis, Dundee, UK, 28 June–1 July 2005. [Google Scholar]
- Lagrange, J.L. Mécanique Analytique; Mallet-Bachelier: Paris, France, 1853; Volume 1. [Google Scholar]
- Peetre, J. On Hadamard’s variational formula. J. Differ. Equ.
**1980**, 36, 335–346. [Google Scholar] [CrossRef] - Suzuki, T.; Tsuchiya, T. First and second hadamard variational formulae of the green function for general domain perturbations. J. Math. Soc. Japan
**2016**, 68, 1389–1419. [Google Scholar] [CrossRef] - Santos, I.; Puechmorel, S. Stochastic Navigation Function Implementation by a Semi-Analytical Algorithm. Available online: https://hal.inria.fr/hal-01869860 (accessed on 9 September 2018).
- Jones, E.; Oliphant, T.; Peterson, P. SciPy: Open Source Scientific Tools for Python. 2001. Available online: https://www.scipy.org/ (accessed on 9 September 2018).
- Needham, T. Visual Complex Analysis; Oxford University Press: Oxford, UK, 1998. [Google Scholar]
- Fletcher, R. Practical Methods of Optimization; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Stargate Consortium. The Stargate Project. Available online: https://stargate.recherche.enac.fr/ (accessed on 9 September 2018).
- Guys, L.; Puechmorel, S.; Lapasset, L.; Amodei, L.; Maréchal, P. Génération automatique de trajectoires aériennes sans conflit à l’aide de fonctions biharmoniques. In Proceedings of the ROADEF 2012, 13ème Congrès Annuel de la Société Française de Recherche Opérationnelle et d’Aide à la Décision, Angers, France, 11–13 April 2012. [Google Scholar]

**Figure 1.**A sphere world bounded by an outer circle of center ${c}_{1}$ and three non overlapping inner circles of centers ${c}_{2}$, ${c}_{3}$ and ${c}_{4}$, respectively.

**Figure 2.**Mapping of the sphere world $\mathsf{\Omega}$ by ${T}_{{z}_{0}}$. ${\mathsf{\Omega}}^{T}$ is the space exterior to three circles.

**Figure 3.**$log|1-{\int}_{\partial \mathsf{\Omega}}\frac{\partial g}{\partial n}(x,y)\phantom{\rule{4pt}{0ex}}\mathrm{d}y|$ with $x=-2$ and a fixed N for values of ${n}_{c}$ ranging from 2 to 64 in the configuration described in Section 3.1.1.1. The x-axis is the number of collocation points ${n}_{c}$.

**Figure 4.**Domain on which an exact solution is computed. The outer boundary is the unit circle. The inner boundary is the circle of center ${c}_{2}=0.4$ and of radius ${r}_{2}=0.4$. The Dirichlet boundary conditions used are such that the value is ${c}_{1}=0$ on the inner boundary and ${r}_{1}=1$ on the outer boundary.

**Figure 5.**$log{\u03f5}_{1}$ with our method based on the Green’s function compared to that obtained with the finite elements method. (

**a**) $log{\u03f5}_{1}$ as a function of ${n}_{c}$ for a fixed number of terms in the Laurent expansion $N=2$, $N=5$ or $N=10$; (

**b**) $log{\u03f5}_{1}$ as a function of the number of mesh vertices for the finite element method with Lagrangian elements of order 2.

**Figure 6.**Normalized execution as a function of ${n}_{c}$ for a fixed number of terms in the Laurent expansion $N=2$, $N=5$, or $N=10$.

**Figure 7.**Approximation g of the Green’s function on unit disc with obstacle of radius $0.1$ centered at $-0.2-0.5i$ and destination disc of radius $0.2$ centered at $0.5i$.

**Figure 8.**Navigation functions with trajectories from starting points $40-40i$, $35-55i$, $50-25i$, $-80-40i$ and $-80+50i$ represented.

**Figure 9.**Relative error of the variation of the Green’s function deduced from the Hadamard variation for a perturbation of the radius of the inner disc (displacement component $\overrightarrow{\delta c}$ is null). The x-axis is the logarithm of the inverse of the step by which the inner disc is modified.

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**MDPI and ACS Style**

Santos, I.; Puechmorel, S.; Dufour, G.
First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World. *Math. Comput. Appl.* **2018**, *23*, 48.
https://doi.org/10.3390/mca23030048

**AMA Style**

Santos I, Puechmorel S, Dufour G.
First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World. *Mathematical and Computational Applications*. 2018; 23(3):48.
https://doi.org/10.3390/mca23030048

**Chicago/Turabian Style**

Santos, Isabelle, Stéphane Puechmorel, and Guillaume Dufour.
2018. "First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World" *Mathematical and Computational Applications* 23, no. 3: 48.
https://doi.org/10.3390/mca23030048