# Chaotic behaviors of an underactuated system in the trajectory tracking task

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## Abstract

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.

## Keywords

Underactuated systems Nonlinear control Chaos Pseudoinverse Computed torque## 1 Introduction

### 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 [10]. 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 [11]. 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 [9]. Free-joint manipulator with one actuated and one unactuated joint was successfully controlled by using periodic inputs to obtain desired trajectories [12]. Complex chaotic behaviors were also observed in planar five-bar closed-chain mechanism [8]. 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

*m*and inertia \(I_{ C}\) in the center of mass (point

*C*). Coordinates

*x*(

*t*) and

*y*(

*t*) describe its position; \(\varphi (t)\) denotes the angle between the object symmetry line and the

*X*axis of the global coordinate system \(O_{XY}\). The vector of force \(\overrightarrow{\varvec{F}}\) acts on the object in a point away from point

*C*by distance

*a*. Force \(\overrightarrow{\varvec{F}}\) as a system’s input is described by its magnitude

*f*(

*t*) and direction angle \(\beta (t)\). Constant drag coefficients

*c*and \(c_{\varphi }\) are used. The equations of motion for the system are as follows

*R*and velocity parameter \(\theta \). Third component of Eq. (9) refers to desired tangential orientation of the object with reference to the trajectory. Substitution of desired trajectory functions (9) and the system’s description matrices into algorithm proposed by Eq. (2) results with control functions. The most interesting behavior of tracking errors occurs in the situation of bounded inputs. Limitation of the force magnitude

*f*(

*t*) results in maximum possible velocity of the system. Limitation of the force direction \(\beta (t)\) may cause irregular behaviors.

### 2.2 Bifurcation diagrams

*x*,

*y*and \(\varphi \) tracking errors. Limitation parameter \(\beta _\mathrm{max}\) of the force direction \(\beta (t)\) was chosen as a bifurcation parameter. Only the most interesting region of parameters is presented in these pictures. For \(\beta _\mathrm{max}>19.07^{\circ }\) region of periodic motion is observed. Example steady-state trajectory, phase portraits and Poincare maps are presented in Fig. 3. After a bifurcation at \(\beta _\mathrm{max}=19.07^{\circ }\), period doubling effect is visible only on the

*x*tracking error phase portrait and Poincare map. Trajectory period stays unchanged, but trajectory loses symmetry in relation to

*x*axis. Second bifurcation occurs at \(\beta _\mathrm{max}=17.85^{\circ }\). For its lower values period doubling for all tracking errors occurs (Fig. 4b, c, d), which involves trajectory qualitative change (Fig. 4a). Another bifurcation happens at \(17.59^{\circ }\) and \(17.50^{\circ }\). Then, a region of irregular motion starts. Figure 5a presents the system’s behavior for \(\beta _\mathrm{max}=17.4^{\circ }\). One can observe some one-dimensional sets on the Poincare maps. The system’s behavior for \(\beta _\mathrm{max}=17.2^{\circ }\) (Fig. 5b) presents interesting shapes on the Poincare maps—probably fractals. The last two presented examples give us probability of chaotic motion, which needs proof by proper values of Lyapunov exponents and shapes of Fourier spectrum. Region of \(\beta _\mathrm{max}\in (16.85^{\circ },16.96^{\circ })\) presents a situation of periodic motion with period five times longer than for \(\beta _\mathrm{max}>19.07^{\circ }\) (Fig. 6a). Starting from \(\beta _\mathrm{max}=16.85^{\circ }\) bifurcations occur up to the region of irregular motion that ends with qualitative change of solution at \(\beta _\mathrm{max}=16.65^{\circ }\) (Fig. 6b). Further decrease of \(\beta _\mathrm{max}\) is not reasonable because of a lack of similarities between the desired (tracked) trajectory and real trajectory.

### 2.3 Lyapunov exponents

### 2.4 Fourier spectrum

## 3 Conclusions

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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