Chaotic behaviors of an underactuated system in the trajectory tracking task
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This contribution presents some nonlinear behaviors of an underactuated mechanical system under the trajectory tracking task. Presented hovercraft model is fully controlled by the computed torque algorithm with the pseudoinverse operation and proportional-derivative feedback. General form of errors dynamic equation gives possibility to analyze their behavior. These errors present irregular behaviors because of the input force limitations. Positive values of highest Lyapunov exponent and Fourier spectrum shape prove chaotic behavior of the system.
KeywordsUnderactuated systems Nonlinear control Chaos Pseudoinverse Computed torque
1.1 Underactuated system
1.2 Input coupling
One of the insufficiently studied problems related to underactuated systems is the input coupling. The system described by Eq. 1 has input coupling if at least one input acts on at least two accelerations . This situation causes big problems in the trajectory tracking task—generalized coordinates cannot be controlled separately.
1.3 Trajectory tracking task
1.4 Chaotic behaviors of underactuated systems
Researchers show that some physical underactuated systems behave in a chaotic manner. One of these systems is a tethered satellite system . In such a system, the mother satellite is treated as a rigid body, subsatellite as a point of mass and tether as an inelastic massless beam. The whole system moves in Kepler elliptical orbit. These ones can be controlled to convert chaotic motions into periodic motions using time-delay autosynchronization method. A time-delay feedback control strategy can also be used to stabilize unstable periodic orbits of a two-link planar manipulator . Free-joint manipulator with one actuated and one unactuated joint was successfully controlled by using periodic inputs to obtain desired trajectories . Complex chaotic behaviors were also observed in planar five-bar closed-chain mechanism . In previous research, various types of underactuated hovercrafts were analyzed in terms of nonlinear control for trajectory tracking [1, 2, 3] but without input coupling effect and bifurcation analysis.
This research focuses on chaotic behaviors of an underactuated system controlled with novel full-state control method with pseudoinverse operation, where irregular behaviors stand from the input’s limitation, not from dynamics and control.
2 Nonlinear behaviors of a hovercraft model
2.1 Model formulation
2.2 Bifurcation diagrams
2.3 Lyapunov exponents
2.4 Fourier spectrum
In this contribution nonlinear behaviors of an underactuated hovercraft model were presented. Trajectory tracking method, based on the Moore–Penrose pseudoinverse operation combined with a computed torque technique and PD feedback, gives the possibility to control full-system configuration (hovercraft position and rotation). It was presented on an eight-shaped trajectory that the limitation of force direction can change system behavior—from periodic solution close to the desired trajectory, through a sequence of bifurcation, up to chaotic situations. These chaotic behaviors were verified by the Lyapunov exponent values and Fourier spectra of time series. There is a possibility to use the Lyapunov exponents for system’s behavior prognosis, but online operation could be complicated due to a long evaluation time.
Example presented in this paper shows that the limitation of driving force direction may cause not only huge tracking errors but also irregular behaviors of underactuated systems. Systems controlled to track periodic trajectories with relatively high speed, like tethered satellite system, are particularly vulnerable to these behaviors. Irregular behaviors of the presented hovercraft model stand from the limitations of input, not from dynamics and control.
Future research could be focused on general analysis of error dynamics Eq. (4) and 6DoF underactuated models of airplanes and rockets in control task. Influence of system damping properties onto chaotic behavior and its existence should be analyzed.
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