Safe trajectory tracking for underactuated vehicles with partially unknown dynamics

Thomas Beckers; Leonardo J. Colombo; Sandra Hirche

doi:10.3934/jgm.2022018

Article Contents

2022, Volume 14, Issue 4: 491-505. Doi: 10.3934/jgm.2022018

This issue Previous Article Next Article Efficient geometric linearization of moving-base rigid robot dynamics

Safe trajectory tracking for underactuated vehicles with partially unknown dynamics

1.
Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA
2.
Centre for Automation and Robotics (CSIC-UPM), Ctra. M300 Campo Real, Km 0,200, Arganda del Rey, Madrid, 28500, Spain
3.
Department of Electrical and Computer Engineering, Technical University of Munich, Munich, 80333, Germany

^*Corresponding author: Thomas Beckers
^*Corresponding author: Thomas Beckers

Received: February 2021

Revised: May 2022

Early access: August 2022

Published: December 2022

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Abstract

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.

Keywords:

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 $

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

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

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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

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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 $

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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

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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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References

[1]	S. A. Al-Hiddabi, Quadrotor control using feedback linearization with dynamic extension, International Symposium on Mechatronics and its Applications, (2009), 1-3.
[2]	K. J. Åström and P. Eykhoff, System identification-a survey, Automatica, 7 (1971), 123-162. doi: 10.1016/0005-1098(71)90059-8.
[3]	F. S. Barbosa, L. Lindemann, D. V. Dimarogonas and J. Tumova, Provably safe control of Lagrangian systems in obstacle-scattered environments, Proc. of the Conference on Decision and Control, (2020), 2056-2061.
[4]	T. Beckers and S. Hirche, Stability of Gaussian process state space models, In Proc. of the European Control Conference, (2016), 2275–2281.
[5]	T. Beckers and S. Hirche, Prediction with approximated Gaussian process dynamical models, IEEE Transactions on Automatic Control (Early Access), (2021).
[6]	T. Beckers, D. Kulić and S. Hirche, Stable Gaussian process based tracking control of Euler-Lagrange systems, Automatica, 103 (2019), 390-397. doi: 10.1016/j.automatica.2019.01.023.
[7]	F. Berkenkamp, A. P. Schoellig and A. Krause, Safe controller optimization for quadrotors with Gaussian processes, In Proc. of the International Conference on Robotics and Automation, (2016), 491–496.
[8]	A. M Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, Springer-Verlag New York, 2015. doi: 10.1007/978-1-4939-3017-3.
[9]	S. Bouabdallah and R. Siegwart, Backstepping and sliding-mode techniques applied to an indoor micro quadrotor, Proc. of the International Conference on Robotics and Automation, (2005), 2247-2252.
[10]	S. L. Brunton and J. N. Kutz, Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control, Cambridge University Press, 2019. doi: 10.1017/9781108380690.
[11]	A. Capone and S. Hirche, Backstepping for partially unknown nonlinear systems using Gaussian processes, IEEE Control Systems Letters, 3 (2019), 416-421.
[12]	A. Capone and S. Hirche, Interval observers for a class of nonlinear systems using Gaussian process models, In Proc. of the European Control Conference, (2019), 1350–1355.
[13]	I.-H. Choi and H.-C. Bang, Adaptive command filtered backstepping tracking controller design for quadrotor unmanned aerial vehicle, Proc. of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 226 (2012), 483-497.
[14]	E. Frazzoli, M. A. Dahleh and E. Feron, Trajectory tracking control design for autonomous helicopters using a backstepping algorithm, In Proc. of the American Control Conference, (2000), 4102–4107.
[15]	M. Greeff and A. P. Schoellig, Exploiting differential flatness for robust learning-based tracking control using Gaussian processes, IEEE Control Systems Letters, 5 (2021), 1121-1126.
[16]	M. K. Helwa, A. Heins and A. P. Schoellig, Provably robust learning-based approach for high-accuracy tracking control of Lagrangian systems, IEEE Robotics and Automation Letters, 4 (2019), 1587-1594.
[17]	Z.-S. Hou and Z. Wang, From model-based control to data-driven control: Survey, classification and perspective, Inform. Sci., 235 (2013), 3-35. doi: 10.1016/j.ins.2012.07.014.
[18]	M. Kobilarov, Trajectory tracking of a class of underactuated systems with external disturbances, In Proc. of the American Control Conference, (2013), 1044–1049.
[19]	D. Lee, H. J. Kim and S. Sastry, Feedback linearization vs. adaptive sliding mode control for a quadrotor helicopter, International Journal of Control, Automation and Systems, 7 (2009), 419-428.
[20]	R. Mahony and T. Hamel, Robust trajectory tracking for a scale model autonomous helicopter, Internat. J. Robust Nonlinear Control, 14 (2004), 1035-1059. doi: 10.1002/rnc.931.
[21]	A. Piazzi and C. G. Lo Bianco, Quintic G²-splines for trajectory planning of autonomous vehicles, In Proc. of the Intelligent Vehicles Symposium, IEEE, (2000), 198–203.
[22]	G. V. Raffo, M. G. Ortega and F. R. Rubio, Backstepping/nonlinear H-$\infty$ control for path tracking of a quadrotor unmanned aerial vehicle, In Proc. of the American Control Conference, (2008), 3356–3361.
[23]	C. E. Rasmussen and C. K. Williams, Gaussian Processes for Machine Learning, MIT Press, Cambridge, MA, 2006.
[24]	M. Reyhanoglu, A. van der Schaft, N. H. McClamroch and I. Kolmanovsky, Dynamics and control of a class of underactuated mechanical systems, IEEE Trans. Automat. Control, 44 (1999), 1663-1671. doi: 10.1109/9.788533.
[25]	F. Scarselli and A. C. Tsoi, Universal approximation using feedforward neural networks: A survey of some existing methods, and some new results, Neural Networks, 11 (1998), 15-37.
[26]	M. W. Spong, Underactuated mechanical systems, In Control Problems in Robotics and Automation, Springer Berlin Heidelberg, (1998), 135–150.
[27]	N. Srinivas, A. Krause, S. M. Kakade and M. W. Seeger, Information-theoretic regret bounds for Gaussian process optimization in the bandit setting, IEEE Transactions on Information Theory, 58 (2012), 3250-3265. doi: 10.1109/TIT.2011.2182033.
[28]	I. Steinwart and A. Christmann, Support Vector Machines, Springer, New York, 2008.
[29]	J. Umlauft and S. Hirche, Feedback linearization based on Gaussian processes with event-triggered online learning, IEEE Trans. Automat. Control, 65 (2020), 4154-4169.
[30]	G. Wahba, Spline Models for Observational Data. SIAM, 1990. doi: 10.1137/1.9781611970128.
[31]	D. H. Wolpert, The lack of a priori distinctions between learning algorithms, Neural Computation, 8 (1996), 1341-1390.