Optimal velocity planning based on the solution of the Euler-Lagrange equations with a neural network based velocity regression

Chady Ghnatios; Daniele di Lorenzo; Victor Champaney; Elías Cueto; Francisco Chinesta

doi:10.3934/dcdss.2023080

Article Contents

2024, Volume 17, Issue 7: 2323-2333. Doi: 10.3934/dcdss.2023080

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Optimal velocity planning based on the solution of the Euler-Lagrange equations with a neural network based velocity regression

1.
SKF Chair @ PIMM Laboratory, Arts et Métiers Institute of Technology, 151 Boulevard de l'Hôpital, F-75013 Paris, France
2.
ESI GROUP Chair @ PIMM Laboratory, Arts et Métiers Institute of Technology, 151 Boulevard de l'Hôpital, F-75013 Paris, France
3.
Aragon Institute of Engineering Research, Universidad de Zaragoza, Zaragoza, Spain
4.
CNRS@CREATE LTD, 1 Create Way, 08-01 CREATE Tower, 138602, Singapore

^*Corresponding author: Francisco Chinesta
^*Corresponding author: Francisco Chinesta

Received: March 2023

Revised: March 2023

Early access: April 2023

Published: July 2024

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Abstract

Trajectory optimization is a complex process that includes an infinite number of possibilities and combinations. This work focuses on a particular aspect of the trajectory optimization, related to the optimization of a velocity along a predefined path, with the aim of minimizing power consumption. To tackle the problem, a functional formulation and minimization strategy is developed, by means of the Euler-Lagrange equation. The minimization is later performed using a neural network approach. The strategy is deemed Lagrange-Net, as it is based on the minimization of the energy functional, by the means of Lagrange's equation and neural network approximations.

Keywords:

Mathematics Subject Classification: Primary: 37N35; Secondary: 35G61.

Citation:

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Figure 1. Flow chart for the training of the control network $ \mathcal{H} $, when having access to the analytical form of the cost function $ \mathcal{C} $. The network's weights and biases are named $ \Theta $, while $ v'=\frac{d v}{d s} $

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Figure 2. Neural network solution of the optimal velocity for $ a=1 $, $ b=1 $, $ c=0 $ and $ \lambda=1 $, along with the corresponding finite difference solution for the same number of nodes $ N $ in the domain $ \Gamma $

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Figure 3. Neural network solution of the optimal velocity for $ a=10 $, $ b=10 $, $ c=1 $ and $ \lambda=10 $, along with the corresponding finite difference solution for the same number of nodes $ N $ in the domain $ \Gamma $

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Figure 4. Flow chart for the training of the control network $ \mathcal{H} $, when the cost function $ \mathcal{C} $ is approximated by a regression model $ \mathcal{G} $. The weights and biases of $ \mathcal{H} $ are named $ \Theta $, while $ v'=\frac{d v}{d s} $

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Figure 5. Neural network solution of the optimal velocity for $ a=10 $, $ b=1 $, $ c=1 $ and $ \lambda=10 $, along with the corresponding finite difference solution for the same number of nodes $ N $ in the $ s $ domain. The cost is approximated using the regression $ \mathcal{G} $

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Figure 6. Neural network solution of the optimal velocity for $ a=10 $, $ b=10 $, $ c=1 $ and $ \lambda=10 $, along with the corresponding finite difference solution for the same number of nodes $ N $ in the $ s $ domain. The cost is approximated using the regression $ \mathcal{G} $

Download: Full-size image PowerPoint slide

Figure 7. Flow chart for the training of the control network $ \mathcal{H} $, when the cost function $ \mathcal{C} $ is approximated by a neural network based regression $ \mathcal{G}_2 $. The weights and biases of $ \mathcal{H} $ are named $ \Theta $, while $ v'=\frac{d v}{d s} $

Download: Full-size image PowerPoint slide

Figure 8. Neural network solution of the optimal velocity for $ a=10 $, $ b=1 $, $ c=1 $ and $ \lambda=10 $, along with the corresponding finite difference solution for the same number of nodes $ N $ in the $ s $ domain. The cost is approximated using the neural network surrogate model $ \mathcal{G}_2 $

Download: Full-size image PowerPoint slide

Figure 9. Neural network solution of the optimal velocity for $ a=10 $, $ b=10 $, $ c=1 $ and $ \lambda=10 $, along with the corresponding finite difference solution for the same number of nodes $ N $ in the $ s $ domain. The cost is approximated using the neural network surrogate model $ \mathcal{G}_2 $

Download: Full-size image PowerPoint slide

Table 1. Structure of the deep convolution/convolution transpose neural network used for the fitting of $\mathcal{H}$. "elu" stands for the exponential linear unit, while "linear" stands for linear activation or no activation function.

Layers	Shape	activation
1	Input data layer, shape $ (N\times 1) $	No activation
2	Reshape layer into a 3D Tensor of shape $ (N\times 1\times 1) $	No activation
3	2D convolution, 30 filters, kernel $ (5\times 1) $, strides $ (5\times 1) $, no padding	elu
4	Flatten layer	No activation
5	Dense fully connected layer, $ 1200 $ units	elu
6	Reshape layer into a 3D Tensor of shape $ (120\times 10\times 1) $	No activation
7	2D convolution transpose, 60 filters, kernel $ (5\times 1) $, strides $ (5\times 1) $, no padding	elu
8	2D convolution transpose, 30 filters, kernel $ (5\times 1) $, strides $ (5\times 1) $, no padding	elu
9	2D convolution transpose, 5 filters, kernel $ (4\times 1) $, strides $ (4\times 1) $, no padding	linear
10	Flatten layer	No activation
11	Dense fully connected layer, $ N $ units	linear

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