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Integrated Guidance and Feedback Control of Underactuated Robotics System in SE(3)

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Abstract

An integrated guidance and feedback control scheme for steering an underactuated vehicle through desired waypoints in three-dimensional space, is developed here. The underactuated vehicle is modeled as a rigid body with four control inputs. These control inputs actuate the three degrees of freedom of rotational motion and one degree of freedom of translational motion in a vehicle body-fixed coordinate frame. This actuation model is appropriate for a wide range of underactuated vehicles including spacecraft with internal attitude actuators, vertical take-off and landing (VTOL) aircraft, fixed-wing multirotor unmanned aerial vehicles (UAVs), maneuverable robotic vehicles, etc. The guidance problem is developed on the special Euclidean group of rigid body motions, SE(3), in the framework ofgeometric mechanics, which represents the vehicle dynamics globally on this configuration manifold. The integrated guidance and control algorithm selects the desired trajectory for the translational motion that passes through the given waypoints, and the desired trajectory for the attitude based on the desired thrust direction to achieve the translational motion trajectory. A feedback control law is then obtained to steer the underactuated vehicle towards the desired trajectories in translation and rotation. This integrated guidance and control scheme takes into account known bounds on control inputs and generates a trajectory that is continuous and at least twice differentiable, which can be implemented with continuous and bounded control inputs. The integrated guidance and feedback control scheme is applied to an underactuated quadcopter UAV to autonomously generate a trajectory through a series of given waypoints in SE(3) and track the desired trajectory in finite time. The overall stability analysis of the feedback system is addressed. Discrete time models for the dynamics and control schemes of the UAV are obtained in the form of Lie group variational integrators using the discrete Lagrange-d’Alembert principle. Almost global asymptotic stability of the feedback system over its state space is shown analytically and verified through numerical simulations.

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References

  1. Aguiar, A.P., Hespanha, J.P.: Position tracking of underactuated vehicles. In: American Control Conference (ACC), 2003, vol. 3. IEEE, pp. 1988–1993 (2003)

  2. Baviskar, A., Feemster, M., Dawson, D., Xian, B.: Tracking control of an underactuated unmanned underwater vehicle. In: American Control Conference (ACC), vol. 2005, pp. 4321–4326. IEEE (2005)

  3. Bhat, S., Bernstein, D.: Finite-time stability of continuous autonomous systems. SIAM J. Control. Optim. 38(3), 751–766 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bloch, A.M., Baillieul, J., Crouch, P., Marsden, J.: Nonholonomic Mechanics and Control, ser. Interdisciplinary Texts in Mathematics. Springer, Verlag (2003)

    Book  Google Scholar 

  5. Bohn, J., Sanyal, A.K.: Almost global finite-time stabilization of rigid body attitude dynamics using rotation matrices International Journal of Robust and Nonlinear Control (2015)

  6. Bonnabel, S., Martin, P., Rouchon, P.: Non-linear symmetry-preserving observers on Lie groups. IEEE Trans. Autom. Control 54(7), 1709–1713 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems, ser. Texts in Applied Mathematics. Springer, Verlag (2004)

    Google Scholar 

  8. Casau, P., Sanfelice, R.G., Cunha, R., Cabecinhas, D., Silvestre, C.: Global trajectory tracking for a class of underactuated vehicles. In: American Control Conference (ACC), 2013. IEEE, pp. 419–424 (2013)

  9. Hua, M., Hamel, T., Morin, P., Samson, C.: Introduction to feedback control of underactuated VTOL vehicles. IEEE Control. Syst. Mag. 33(1), 61–75 (2013)

    Article  MathSciNet  Google Scholar 

  10. Izadi, M., Sanyal, A.K., Barany, E., Viswanathan, S.P.: Rigid body motion estimation based on the Lagrange-d’A,lembert principle. In: 54th IEEE Conference on Decision and Control (CDC), pp. 3699–3704 (2015)

  11. Izadi, M., Sanyal, A.K., Beard, R., Bai, H.: GPS-denied relative motion estimation for fixed-wing UAV using the variational pose estimator. In: 54th IEEE Conference on Decision and Control (CDC), pp. 2152–2157 (2015)

  12. Kobilarov, M.: Trajectory tracking of a class of underactuated systems with external disturbances. In: American Control Conference (ACC), 2013, pp. 1044–1049 (2013)

  13. Lee, T., Leok, M., McClamroch, N.H.: Dynamics of connected rigid bodies in a perfect fluid. IEEE, 408–413 (2009)

  14. Lee, T., Leok, M., McClamroch, N.H.: Geometric tracking control of a quadrotor UAV on SE(3). In: 49th IEEE Conference on Decision and Control (CDC). IEEE, pp. 5420–5425 (2010)

  15. Leishman, R.C., McLain, T.W., Beard, R.W.: Relative navigation approach for vision-based aerial GPS-denied navigation. J. Intell. Robot. Syst. 74(1-2), 97–111 (2014)

    Article  Google Scholar 

  16. Leonard, N.E.: Periodic forcing, dynamics and control of underactuated spacecraft and underwater vehicles. In: 34th IEEE Conference on Decision and Control (CDC), vol. 4. IEEE, pp. 3980–3985 (1995)

  17. Mellinger, D., Michael, N., Kumar, V.: Trajectory generation and control for precise aggressive maneuvers with quadrotors. Int. J. Robot. Res. 31(5), 664–674 (2012)

    Article  Google Scholar 

  18. Nordkvist, N., Sanyal, A.K.: A lie group variational integrator for rigid body motion in SE(3) with applications to underwater vehicle dynamics. In: 49th IEEE Conference on Decision and Control (CDC), pp. 5414–5419 (2010)

  19. Pucci, D., Hamel, T., Morin, P., Samson, C.: Nonlinear feedback control of axisymmetric aerial vehicles. Automatica 53, 72–78 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rehbinder, H., Ghosh, B.K.: Pose estimation using line-based dynamic vision and inertial sensors. IEEE Trans. Autom. Control 48(2), 186–199 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sanyal, A., Holguin, L., Viswanathan, S.: Guidance and control for spacecraft autonomous chasing and close proximity maneuvers (2012)

  22. Sanyal, A., Nordkvist, N., Chyba, M.: An almost global tracking control scheme for maneuverable autonomous vehicles and its discretization. IEEE Trans. Autom. Control 56(2), 457–462 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tabuada, P., Lima, P.: Position tracking for underactuated rigid bodies on SE(3), ISR Internal Report RT-401-2000, Tech Rep. (2000)

  24. Tayebi, A., McGilvray, S.: Attitude stabilization of a VTOL quadrotor aircraft. IEEE Trans. Control Syst. Technol. 14(3), 562–571 (2006)

    Article  Google Scholar 

  25. Varadarajan, V.S.: Lie Groups, Lie Algebras, and Their Representations. Springer Verlag, New York (1984)

    Book  MATH  Google Scholar 

  26. Vasconcelos, J.F., Cunha, R., Silvestre, C., Oliveira, P.: A nonlinear position and attitude observer on SE(3) using landmark measurements. Syst. Control Lett. 59(3), 155–166 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors wish to thank Taeyoung Lee for helpful discussions on control of quadcopter UAVs, and with Jan Bohn for useful inputs on finite-time stable control on the Lie group of rigid body motions.

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Authors and Affiliations

  1. Akrobotix LLC, Syracuse, NY, 13202, USA

    Sasi Prabhakaran Viswanathan

  2. Faculty of Mechanical and Aerospace Engineering at Syracuse University, Syracuse, NY, 13244, USA

    Amit K. Sanyal

  3. New Mexico State University, Las Cruces, NM, USA

    Ehsan Samiei

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  1. Sasi Prabhakaran Viswanathan
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  2. Amit K. Sanyal
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  3. Ehsan Samiei
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Correspondence to Amit K. Sanyal.

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Viswanathan, S.P., Sanyal, A.K. & Samiei, E. Integrated Guidance and Feedback Control of Underactuated Robotics System in SE(3). J Intell Robot Syst 89, 251–263 (2018). https://doi.org/10.1007/s10846-017-0547-0

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  • DOI: https://doi.org/10.1007/s10846-017-0547-0

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