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Safe trajectory tracking for underactuated vehicles with partially unknown dynamics

  • *Corresponding author: Thomas Beckers

    *Corresponding author: Thomas Beckers 
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  • Underactuated vehicles have gained much attention in the recent years due to the increasing amount of aerial and underwater vehicles as well as nanosatellites. The safe tracking control of these vehicles is a substantial aspect for an increasing range of application domains. However, external disturbances and parts of the internal dynamics are often unknown or very time-consuming to model. To overcome this issue, we present a safe tracking control law for underactuated rigid-body dynamics using a learning-based oracle for the prediction of the unknown dynamics. The presented approach guarantees a bounded tracking error with high probability where the bound is explicitly given. With additional assumptions, asymptotic stability of the tracking error is achieved. A numerical example highlights the effectiveness of the proposed learning-based control law.

    Mathematics Subject Classification: Primary: 70Q05, 93E35; Secondary: 70E60, 68T40.


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  • Figure 1.  Underactuated vehicle with full attitude control and a translational force input. The position of the vehicle is described by the vector $ {\boldsymbol{{p}}} $ and the orientation by the matrix $ R $

    Figure 2.  Visualization of the normalized magnitude of the thermal updraft acting on the quadcopter and the recorded training data points (red crosses)

    Figure 3.  Tracking error of the quadcopter with control law (5) without learning (blue) and with our proposed learning-based approach (red)

    Figure 4.  Lyapunov function (solid) converges to a tight set around zero (dashed line) and stays inside this set with high probability as guaranteed by Theorem 3.2

    Figure 5.  Control inputs for the quadcopter. Top: Control input for the thrust $ u $. Bottom: Control input for the torques $ \tau_1, \tau_2, \tau_3 $

    Figure 6.  Ground truth (dashed) of the unknown dynamics and estimates of the GP (solid). Top: Unknown dynamics acting on the $ x_3 $-position of the quadcopter. Bottom: Unknown dynamics acting on the first component of the angular acceleration $ \dot{{\boldsymbol{\omega}}} $ of the quadcopter

    Figure 7.  Left: Three exemplary trajectories of the 30 randomly generated trajectories for the training and test set. Right: $ L^2 $-norm of the tracking error of the test set for the standard control law (left box), the proposed learning-based control law with 375 training points (middle box) and the proposed learning-based control with 750 training points (right box)

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