Research Papers

A Control Design Method for Underactuated Mechanical Systems Using High-Frequency Inputs

[+] Author and Article Information
Sevak Tahmasian

Department of Engineering Science
and Mechanics,
Virginia Tech,
Blacksburg, VA 24061
e-mail: sevakt@vt.edu

Craig A. Woolsey

Department of Aerospace
and Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: cwoolsey@vt.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 3, 2015; final manuscript received January 12, 2015; published online March 4, 2015. Editor: Joseph Beaman.

J. Dyn. Sys., Meas., Control 137(7), 071004 (Jul 01, 2015) (11 pages) Paper No: DS-15-1003; doi: 10.1115/1.4029627 History: Received January 03, 2015; Revised January 12, 2015; Online March 04, 2015

This paper presents a control design technique which enables approximate reference trajectory tracking for a class of underactuated mechanical systems. The control law comprises two terms. The first involves feedback of the trajectory tracking error in the actuated coordinates. Building on the concept of vibrational control, the second term imposes high-frequency periodic inputs that are modulated by the tracking error in the unactuated coordinates. Under appropriate conditions on the system structure and the commanded trajectory, and with sufficient separation between the time scales of the vibrational forcing and the commanded trajectory, the approach provides convergence in both the actuated and unactuated coordinates. The procedure is first described for a two degree-of-freedom (DOF) system with one input. Generalizing to higher-dimensional, underactuated systems, the approach is then applied to a 4DOF system with two inputs. A final example involves control of a rigid plate that is flapping in a uniform flow, a 3DOF system with one input. More general applications include biomimetic locomotion systems, such as underwater vehicles with articulating fins and flapping wing micro-air vehicles.

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Brockett, R. W., 1983, “Asymptotic Stability and Feedback Stabilization,” Differential Geometric Control Theory, R. W.Brockett, R. S.Millman, and H. J.Sussmann, eds., Birkhäuser, Boston, MA, pp. 181–191.
Meerkov, S. M., 1980, “Principle of Vibrational Control: Theory and Applications,” IEEE Trans. Autom. Control, 25(4), pp. 755–762. [CrossRef]
Bellman, R. E., Bentsman, J., and Meerkov, S. M., 1986, “Vibrational Control of Nonlinear Systems: Vibrational Stabilizability,” IEEE Trans. Autom. Control, 31(8), pp. 710–716. [CrossRef]
Bellman, R. E., Bentsman, J., and Meerkov, S. M., 1986, “Vibrational Control of Nonlinear Systems: Vibrational Controllability and Transient Behavior,” IEEE Trans. Autom. Control, 31(8), pp. 717–724. [CrossRef]
Sastry, S., 1999, Nonlinear Systems: Analysis, Stability, and Control (Interdisciplinary Applied Mathematics), Springer, New York.
Bullo, F., 2002, “Averaging and Vibrational Control of Mechanical Systems,” SIAM J. Control Optim., 41(2), pp. 542–562. [CrossRef]
Bullo, F., and Lewis, A. D., 2005, Geometric Control of Mechanical Systems (Texts in Applied Mathematics), Springer, New York.
Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences), Springer, New York.
Sanders, J. A., and Verhulst, F., 1985, Averaging Methods in Nonlinear Dynamical Systems (Applied Mathematical Sciences), Springer, New York.
Morgansen, K. A., Vela, P. A., and Burdick, J. W., 2002, “Trajectory Stabilization for a Planar Carangiform Robot Fish,” Proceedings of the IEEE International Conference on Robotics and Automation, pp. 756–762.
Vela, P. A., and Burdick, J. W., 2003, “Control of Underactuated Mechanical Systems With Drift Using Higher-Order Averaging Theory,” Proceedings of the IEEE Conference on Decision and Control, pp. 3111–3117.
Vela, P. A., 2003, “Averaging and Control of Nonlinear Systems (With Application to Biomimetic Locomotion),” Ph.D. thesis, California Institute of Technology, Pasadena, CA.
Sanyal, A. K., Bloch, A. M., and McClamroch, N. H., 2005, “Control of Mechanical Systems With Cyclic Coordinates Using Higher Order Averaging,” Proceedings of the IEEE Conference on Decision and Control, pp. 6835–6840.
Weibel, S., and Baillieul, J., 1998, “Averaging and Energy Methods for Robust Open-Loop Control of Mechanical Systems,” Essays on Mathematical Robotics, J.Baillieul, S. S.Sastry, and H. J.Sussmann, eds., Springer, New York, pp. 203–269.
Martinez, S., Cortes, J., and Bullo, F., 2003, “Analysis and Design of Oscillatory Control Systems,” IEEE Trans. Autom. Control, 48(7), pp. 1164–1177. [CrossRef]
Tsakalis, K. S., and Ioannou, P. A., 1993, Linear Time-Varying Systems, Prentice-Hall, Upper Saddle River, NJ.
Ogata, K., 2010, Modern Control Engineering, Prentice Hall, Upper Saddle River, NJ.
Greenwood, D. T., 2003, Advanced Dynamics, Cambridge University Press, Cambridge, UK.
Tahmasian, S., Taha, H. E., and Woolsey, C. A., 2013, “Control of Underactuated Mechanical Systems Using High Frequency Input,” Proceedings of the IEEE American Control Conference, pp. 603–608.


Grahic Jump Location
Fig. 1

Time history of actuated coordinates θ1 and θ2

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Fig. 2

Time history of unactuated coordinates θ3 and θ4

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Fig. 3

Notation for the oscillating wing

Grahic Jump Location
Fig. 4

Aerodynamic and control forces on the flapping device

Grahic Jump Location
Fig. 6

Variations of y with respect to time in hovering motion using modified controller

Grahic Jump Location
Fig. 7

Variations of y, z, and θ following a harmonic trajectory with ω=50 rad/s

Grahic Jump Location
Fig. 8

Variations of y, z, and θ following a harmonic trajectory with ω=17 rad/s

Grahic Jump Location
Fig. 5

Variations of y, z, and θ with respect to time in hovering motion



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