A Computationally Inexpensive Optimal Guidance via Radial-Basis-Function Neural Network for Autonomous Soft Landing on Asteroids

<div><p>Optimal guidance is essential for the soft landing task. However, due to its high computational complexities, it is hardly applied to the autonomous guidance. In this paper, a computationally inexpensive optimal guidance algorithm based on the radial basis function neural network (RBFNN) is proposed. The optimization problem of the trajectory for soft landing on asteroids is formulated and transformed into a two-point boundary value problem (TPBVP). Combining the database of initial states with the relative initial co-states, an RBFNN is trained offline. The optimal trajectory of the soft landing is determined rapidly by applying the trained network in the online guidance. The Monte Carlo simulations of soft landing on the Eros433 are performed to demonstrate the effectiveness of the proposed guidance algorithm.</p></div

The terminal error histogram (1000 neurons trained by 9261 sets of data)).

<p>a:position error. b:velocity error. This figure shows the results of the one thousand time simulations with the fast optimal control of 1000 neurons trained by the database of 9261 sets of data. The abscissa is the error range, the ordinate is the statistical probability of the different ranges of the error.</p

The Radial Basis Function Neural Network model.

<p>This figure illustrates the model of the RBFNN. The network has three layers: the input layer, the middle layer and the output layer. The input of the network is passed to all neurons by the input layer. In the middle layer, the neurons of the radial basis function output values based on the inputs. The output of the network is the sum of the outputs of the neurons with different coefficients by the output layer.</p

The structure of the optimal guidance.

<p>This figure illustrates the structure of the optimal guidance. The RBFNN is trained by the database obtained by solving the shooting equations. Then it is applied online to obtain the relative initial co-states quickly. Through the initial co-states, the optimal control can be determined by the analytical equations as Eqs (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137792#pone.0137792.e015" target="_blank">13</a>)–(<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137792#pone.0137792.e017" target="_blank">15</a>).</p

The parameters of the simulation.

<p>The parameters of the simulation.</p

The terminal error histogram (1000 neurons trained by 1331 sets of data)).

<p>a:position error. b:velocity error. This figure shows the results of the one thousand time simulations with the fast optimal control of 1000 neurons trained by the database of 1331 sets of data. The abscissa is the error range, the ordinate is the statistical probability of the different ranges of the error.</p

The conceptual illustration of the spacecraft’s thrust vector.

<p>This figure illustrates the physical relationships of the variables in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137792#pone.0137792.e001" target="_blank">Eq (1)</a>.</p

The contrast of the results.

<p>The contrast of the results.</p