Underactuated system
The dynamic system described by a set of second-order ODE in form
$$\begin{aligned} \ddot{\varvec{q}}(t)=\varvec{f}_{1}+\varvec{f}_{2}\varvec{u}(t) \end{aligned}$$
(1)
with generalized coordinates \(\varvec{q}(t)=\left[ q_{1},q_{2},\ldots ,q_{n}\right] \) and inputs \(\varvec{u}(t)=\left[ u_{1},u_{2},\ldots ,u_{w}\right] \) is called underactuated if the unbounded control inputs cannot produce accelerations \(\ddot{\varvec{q}}\) in an arbitrary direction. This could be verified by the condition of \(\mathrm {rank}\varvec{f}_{2}<\mathrm {dim}\varvec{q}\). The situation when there is a fewer number of inputs than the number of degrees of freedom is called trivial underactuation. There are many systems with the underactuated property, e.g., acrobot, pendubot, cart-pole, the beam-and-ball, inertia-wheel pendulum, airplane and multicopter, hovercraft, surface vessel. A recent review of underactuated systems and their control is presented widely in [7].
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.
Trajectory tracking task
The most popular method of tracking control of underactuated systems with the input coupling effect is based on the change of variables that converts this problem into noncoupled [4]. Then, some accelerations are separately controlled by inputs and some stay without control (selective control method). This contribution focuses on a new method of full-state control, based on the pseudoinverse operation combined with a computed torque technique and PD (proportional-derivative) feedback presented in [5]. The inputs are then proposed as
$$\begin{aligned} \varvec{\varvec{\tau }}(t)=\varvec{f}_{2}^{+}\left( \ddot{\varvec{d}}(t)-\varvec{f}_{1}+\varvec{K}_{D}\dot{\varvec{e}}(t)+\varvec{K}_{P}\varvec{e}(t)\right) \end{aligned}$$
(2)
where \(\varvec{f}_{2}^{+}\) is a pseudoinverse of \(\varvec{f}_{2}\), \(\varvec{d}(t)\) is a vector of desired trajectory functions, \(\varvec{K}_{D}\) and \(\varvec{K}_{P}\) are diagonal matrices of positive coefficients, and \(\varvec{e}(t)=\varvec{d}(t)-\varvec{q}(t)\) is a vector of tracking errors. Moore–Penrose pseudoinverse defined by Tikhonov’s regularization can be calculated with formula
$$\begin{aligned} \varvec{f}_{2}^{+}=\underset{\delta \rightarrow 0}{\lim }(\varvec{f}_{2}^{T}\varvec{f}_{2}-\delta \varvec{I})^{-1}\varvec{f}_{2}^{T} \end{aligned}$$
(3)
Substitution of the control method proposed by Eq. (2) instead of \(\varvec{u}(t)\) in system of equations of motion (1) yields the errors dynamic equation
$$\begin{aligned} \varvec{\ddot{e}}+\varvec{f_{2}}\varvec{f}_{\varvec{2}}^{\varvec{+}}\varvec{K_{D}\dot{e}}+\varvec{f_{2}f_{2}^{+}K_{P}e}=\left( \varvec{f_{2}f_{2}^{+}}-\varvec{I}\right) \left( \varvec{f_{1}}-\varvec{\ddot{d}}\right) \end{aligned}$$
(4)
Analytic and numerical analysis of Eq. (4) can ensure stability of the system while tracking desired trajectory. This equation is also useful during tuning process of PD gains.
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.