# A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. A Variational Vademecum for a Dynamic Obstacle Robotic Problem

#### 2.1. Potential Flow Theory for a Dynamic Obstacle Robotic Problem

#### 2.2. Introducing the Variational Vademecum

## 3. A Progressive Construction of a Variational Vademecum

**Lemma**

**1.**

- (a)
- $\mathrm{span}\phantom{\rule{0.166667em}{0ex}}{\mathcal{M}}_{\le 1}\left({H}_{0}^{1}\left({\mathsf{\Omega}}_{\underline{X}}\right){\otimes}_{a}{L}^{2}\left({\mathsf{\Omega}}_{\underline{P}}\right)\right)$ is dense in ${\mathbf{H}}_{0}.$
- (b)
- It is a cone, that is, if $u\in {\mathcal{M}}_{\le 1}\left({H}_{0}^{1}\left({\mathsf{\Omega}}_{\underline{X}}\right){\otimes}_{a}{L}^{2}\left({\mathsf{\Omega}}_{\underline{P}}\right)\right)$ then $\lambda u\in {\mathcal{M}}_{\le 1}\left({H}_{0}^{1}\left({\mathsf{\Omega}}_{\underline{X}}\right){\otimes}_{a}{L}^{2}\left({\mathsf{\Omega}}_{\underline{P}}\right)\right)$ for all $\lambda \in \mathbb{R}.$
- (c)
- It is a weakly closed set in ${\mathbf{H}}_{0}.$

**Proof.**

- (A1)
- J is Fréchet differentiable, with Fréchet differential ${J}^{\prime}:{\mathbf{H}}_{0}\to {\mathbf{H}}_{0}^{\ast}$;
- (A2)
- J is elliptic;
- (A3)
- ${J}^{\prime}:{\mathbf{H}}_{0}\u27f6{\mathbf{H}}_{0}^{\ast}$ is Lipschitz continuous on bounded sets.

**Definition**

**1**

**.**Since $J:{\mathbf{H}}_{0}\u27f6\mathbb{R}$ satisfies (A1)–(A2) let $u\in {\mathbf{H}}_{0}$ be such that

**Theorem**

**1**

**.**Let $u\in {\mathbf{H}}_{0}$ satisfy (10). Consider a progressive variational vademecum ${\left\{{u}_{m}\right\}}_{m\ge 1}$ over ${\mathcal{M}}_{\le 1}\left({H}_{0}^{1}\left({\mathsf{\Omega}}_{\underline{X}}\right){\otimes}_{a}{L}^{2}\left({\mathsf{\Omega}}_{\underline{P}}\right)\right))$ of $u.$ Then ${\left\{{u}_{m}\right\}}_{m\ge 1},$ converges in ${\mathbf{H}}_{0}$ to $u,$ that is,

- Consider two finite dimensional subspaces ${V}_{1}\subset {H}_{0}^{1}\left({\mathsf{\Omega}}_{\underline{X}}\right)$ and ${V}_{2}\subset {L}^{2}\left({\mathsf{\Omega}}_{\underline{P}}\right).$
- Assume that for each $m\ge 1$ the approximation ${u}_{m}(\underline{X};\underline{P})={\sum}_{n=1}^{m}{v}_{1}^{\left(n\right)}\left(\underline{X}\right){v}_{2}^{\left(n\right)}\left(\underline{P}\right)\in {\mathbf{H}}_{0}$ is known.
- Choose the function ${v}_{1}^{\left(0\right)}\in {V}_{2},$ randomly and let ${U}_{2}^{(m+1)}\subset {V}_{2}$ be a linear subspace such that ${V}_{2}=\mathrm{span}\phantom{\rule{0.166667em}{0ex}}\left\{{v}_{2}^{\left(0\right)}\right\}\oplus {U}_{2}^{(m+1)}.$ Find ${v}_{2}^{\ast}\in {V}_{2}$ be such that$$J({u}_{m}+{v}_{1}^{\left(0\right)}{v}_{2}^{\ast})=\underset{{v}_{2}\in {V}_{2}}{min}J({u}_{m}+{v}_{1}^{\left(0\right)}{v}_{2})$$
- Let ${U}_{1}^{(m+1)}\subset {V}_{1}$ be a linear subspace such that ${V}_{1}={U}_{1}^{(m+1)}\oplus \mathrm{span}\phantom{\rule{0.166667em}{0ex}}\left\{{v}_{2}^{\ast}\right\}.$ Find ${v}_{1}^{\ast}\in {V}_{2}$ be such that$$J({u}_{m}+{v}_{1}^{\ast}{v}_{2}^{\ast})=\underset{{v}_{1}\in {V}_{1}}{min}J({u}_{m}+{v}_{1}{v}_{2}^{\ast})$$
- Repeat steps 3 and 4 just until $J({u}_{m}+{v}_{1}^{\ast}{v}_{2}^{\ast})$ is stabilized. Take ${u}_{m+1}={v}_{1}^{\ast}{v}_{2}^{\ast}$
- If $|J\left({u}_{m}\right)-J\left({u}_{m+1}\right)|<\mathtt{tol}$ then return ${u}_{m+1}.$ Otherwise put $m+2$ and go to step 2.

## 4. Navigation Example

- Evaluate the abacus at the point i defined by the current configuration given by the parameters ${\underline{P}}_{i}$. This evaluation will give the solution of the Laplace’s equation for any position $\underline{X}$ of the domain and for the current set of parameters: $u(\underline{X};{\underline{P}}_{i})$.
- Evaluate the gradient of the solution $(\underline{X};{\underline{P}}_{i})$ in order to define the streamline.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Reif, J.H. Complexity of the Mover’s Problem and Generalizations. In Proceedings of the IEEE ymposium on Foundations of Computer Science, San Juan, PR, USA, 29–31 October 1976; pp. 421–427. [Google Scholar]
- González, D.; Pérez, J.; Milanés, V.; Nashashibi, F. A Review of Motion Planning Techniques for Automated Vehicles. IEEE Trans. Ntelligent Transp. Syst.
**2016**, 17, 1135–1145. [Google Scholar] [CrossRef] - Kavraki, L.E.; LaValle, S. Chapter 5. Motion Planning; Khatib, S., Ed.; Handbook of Robotics; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Rimon, E.; Koditschek, D. Exact robot navigation using artificial potential functions. IEEE Trans. Robot. Autom.
**1992**, 8, 501–518. [Google Scholar] [CrossRef] [Green Version] - Khatib, O. Real-time obstacle avoidance for manipulators and mobile robots. Int. J. Robot Res.
**1986**, 5, 90–98. [Google Scholar] [CrossRef] - Zhachmanoglou, E.; Thoe, D.W. Introduction to Partial Differential Equations with Applications; Dover Publications, Inc.: New York, NY, USA, 1986. [Google Scholar]
- Kim, J.; Khosla, P. Real-time obstacle avoidance using harmonic potencial functions. IEEE Trans. Robot. Autom.
**1992**, 8, 338–349. [Google Scholar] [CrossRef] [Green Version] - Connolly, C.I.; Grupen, R. The Application of Harmonic functions to Robotics. J. Robot. Syst.
**1993**, 10, 931–946. [Google Scholar] [CrossRef] [Green Version] - Canny, J.F. The Complexity of Robot Motion Planning; MIT Press: Cambridge, UK, 1988. [Google Scholar]
- Garrido, S.; Moreno, L.; Blanco, D.; Martín, M.F. Robotic Motion Using Harmonic Functions and Finite Elements. J. Intell. Robot Syst.
**2010**, 59, 57–73. [Google Scholar] [CrossRef] [Green Version] - Connolly, C.I.; Burns, J.B.; Weiss, R. Path planning using Laplace’s equation. In Proceedings of the IEEE International Conference on Robotics and Automation, Cincinnati, OH, USA, 13–18 May 1990; pp. 2102–2106. [Google Scholar]
- Saudi, A.; Sulaiman, J. Path Planing for mobile robots using 4EGSOR via Nine-Point Laplacian (4EGSOR9L) Iterative method. Int. J. Comput. Appl.
**2012**, 53, 38–42. [Google Scholar] - Saudi, A.; Sulaiman, J.; Ahmad, H.M.H. Robot Path Planing with EGSOR Iterative Method using Laplacian Behaviour-Based Control (LBBC). In Proceedings of the 5th International Conference on Intelligent Systems, Modelling and Simulation, Langkawi, Malaysia, 27–29 January 2014; pp. 87–91. [Google Scholar]
- Yan, W.; Bai, X.; Peng, X.; Zuo, L.; Dai, J. The routing problem of autonomous underwater vehicles in ocean currents. In Proceedings of the MTS/IEEE Conference OCEANS’14, Taipei, Taiwan, 7–10 April 2014; pp. 1–6. [Google Scholar]
- Bai, X.; Yan, W.; Cao, M.; Xue, D. Distributed multi-vehicle task assignment in a time-invariant drift field with obstacles. IET Control Theory Appl.
**2019**, 13, 2886–2893. [Google Scholar] [CrossRef] - Bai, X.; Yan, W.; Ge, S.S.; Cao, M. An integrated multi-population genetic algorithm for multi-vehicle task assignment in a drift field. Inf. Sci.
**2018**, 453, 227–238. [Google Scholar] [CrossRef] - Falcó, A.; Nouy, A. Proper Generalized Decomposition for Nonlinear Convex Problems in Tensor Banach Spaces. Numer. Math.
**2012**, 121, 503–530. [Google Scholar] [CrossRef] [Green Version] - Chinesta, F.; Leygue, A.; Bordeu, F.; Aguado, J.V.; Cueto, E.; Gonzalez, D.; Alfaro, I.; Ammar, A.; Huerta, A. PGD-Based computational vademecum for efficient Design, Optimization and Control. Arch. Comput. Methods Eng.
**2013**, 20, 31–49. [Google Scholar] [CrossRef] - Chinesta, F.; Keunings, R.; Leygue, A. The Proper Generalized Decomposition for Advanced Numerical Simulations. A Primer; Springer Briefs in Applied Science and Technology; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Falcó, A.; Montés, N.; Chinesta, F.; Hilario, L.; Mora, M.C. On the Existence of a Progressive Variational Vademecum based on the Proper Generalized Decomposition for a Class of Elliptic Parameterized Problems. J. Comput. Appl. Math.
**2018**, 330, 1093–1107. [Google Scholar] [CrossRef] - Montés, N.; Chinesta, F.; Falcó, A.; Mora, M.C.; Hilario, L.; Rosillo, N. PGD-based Method for mobile robot applications. In Proceedings of the Congress on Numerical Methods in Engineering CMN2017, Valencia, Spain, 3–5 July 2017. [Google Scholar]
- Montés, N.; Chinesta, F.; Falcó, A.; Mora, M.C.; Hilario, L.; Nadal, E.; Duval, J.L. A PGD- based Method for Robot Global Path Planning: A Primer. In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), Prague, Czech Republic, 29–31 July 2019; pp. 31–39. [Google Scholar]
- Domenech, L.; Falcó, A.; García, V.; Sánchez, F. Towards a 2.5D geometric model in mold filling simulation. J. Comput. Appl. Math.
**2016**, 291, 183–196. [Google Scholar] [CrossRef] - Falcó, A.; Nouy, A. A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. J. Math. Anal. Appl.
**2011**, 376, 469–480. [Google Scholar] [CrossRef] [Green Version] - Gingras, D.; Dupuis, E.; Payre, G.; Lafontaine, J. Path Planning Based on Fluid mechanics for mobile robots used Unstructured Terrain models. In Proceedings of the IEEE International Conference on Robotics and Automation, Anchorage, AK, USA, 3–8 May 2010. [Google Scholar]
- Cancès, E.; Ehrlacher, V.; Lelièvre, T. Convergence of a greedy algorithm for high-dimensional convex nonlinear problems, Mathematical Models and Methods. Appl. Sci.
**2011**, 21, 2433–2467. [Google Scholar] - Falcó, A.; Hackbusch, W. On minimal subspaces in tensor representations. Found. Comput. Math.
**2012**, 12, 765–803. [Google Scholar] [CrossRef] [Green Version] - Canuto, C.; Urban, K. Adaptive optimization of convex functionals in banach spaces. SIAM J. Numer. Anal.
**2005**, 42, 2043–2075. [Google Scholar] [CrossRef] - Ammar, A.; Chinesta, F.; Falcó, A. On the convergence of a Greedy Rank-One Update Algorithm for a class of Linear Systems. Arch. Comput. Methods Eng.
**2010**, 17, 473–486. [Google Scholar] [CrossRef]

**Figure 1.**PGD reconstruction to obtain the set of path planning trajectories in a dynamic environment where the black hole represents a dynamic obstacle.

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**MDPI and ACS Style**

Falcó, A.; Hilario, L.; Montés, N.; Mora, M.C.; Nadal, E.
A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition. *Mathematics* **2020**, *8*, 2245.
https://doi.org/10.3390/math8122245

**AMA Style**

Falcó A, Hilario L, Montés N, Mora MC, Nadal E.
A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition. *Mathematics*. 2020; 8(12):2245.
https://doi.org/10.3390/math8122245

**Chicago/Turabian Style**

Falcó, Antonio, Lucía Hilario, Nicolás Montés, Marta C. Mora, and Enrique Nadal.
2020. "A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition" *Mathematics* 8, no. 12: 2245.
https://doi.org/10.3390/math8122245