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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
• 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:

• 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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