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

  • *Corresponding author: Francisco Chinesta

    *Corresponding author: Francisco Chinesta
Abstract Full Text(HTML) Figure(9) / Table(1) Related Papers Cited by
  • 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.

    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} $

    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 $

    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 $

    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} $

    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} $

    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} $

    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} $

    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 $

    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 $

    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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    [3] G. De Filippis, B. Lenzo, A. Sorniotti, K. Sannen, J. De Smet and P. Gruber, On the energy efficiency of electric vehicles with multiple motors, in: 2016 IEEE Vehicle Power and Propulsion Conference (VPPC), (2016), 1-6. doi: 10.1109/VPPC. 2016.7791737.
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    [7] S. Karaman and E. Frazzoli, Sampling-based algorithms for optimal motion planning, The International Journal of Robotics Research, 30 (2011), 846-894. 
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    [10] M. RaissiP. Perdikaris and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, 378 (2019), 686-707.  doi: 10.1016/j.jcp.2018.10.045.
    [11] M. SachenbacherM. LeuckerA. Artmeier and J. Haselmayr, Efficient energy-optimal routing for electric vehicles, AAAI, 25 (2011), 1402-1407.  doi: 10.1609/aaai.v25i1.7803.
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