Tracking and Regulation Control of an Underactuated Surface Vessel with Nonintegrable Dynamics
IntroductionOver the past ten years, the attitude control of rigid body systems has become an active area of research. Among its many applications are the attitude control of rigid aircraft and spacecraft systems (the interested reader is referred to  for a literature review of the many different types of applications). Rigid spacecraft applications are often required to perform highly accurate slewing and/or pointing maneuvers that force the spacecraft to rotate along a relatively large amplitude trajectory. These performance requirements mandate that the control design be predicated on the use of the nonlinear spacecraft model . This nonlinear model is typically represented by: (i) Euler's dynamic equation which is used to describe the time evolution of the angular velocity vector, and (ii) the kinematic equation which relates the time derivatives of the orientation angles to the angular velocity vector. Several kinematic parametrizations exist to represent the orientation angles, including singular, three-parameter representations e.g., the Euler angles, Gibbs vector, Cayley-Rodrigues parameters, and modified Rodrigues parameters) and the nonsingular, four-parameter representation given by the unit quaternion i.e., the Euler parameters). Whereas the three-parameter representations always exhibit singular orientations i.e., the Jacobian matrix in the kinematic equation is singular for some orientations), the unit quaternion globally represents the spacecraft attitude without singularities; however, an additional constraint equation is introduced through the use of the four-parameter representation.
Several solutions to the attitude control problem have been presented in the literature since the early 1970's . In , the authors presented a general attitude control design framework which includes PD, model-based, and adaptive setpoint controllers. Adaptive tracking control schemes based on three-parameter, kinematic representations were presented in [23,25] to compensate for the unknown, spacecraft inertia matrix. In , an adaptive attitude tracking controller based on the unit quaternion was proposed that identified the inertia matrix via periodic command signals. The work of  was later applied to the angular velocity tracking problem in . A suboptimal state feedback controller was developed for the quaternion representation in . In , the authors designed an inverse optimal control law for attitude regulation using the backstepping method for a three-parameter representation. Recently,  presented a variable structure tracking controller using quaternions in the presence of spacecraft inertia uncertainties and external disturbances. In , Costic et al. presented an adaptive control solution to the quaternion-based, attitude tracking control problem that eliminated angular velocity measurements and compensated for parametric uncertainty.
All of the above control strategies assume that the spacecraft is fully actuated (i.e., the strategies required the use of the three independent control inputs). Recently, there has been some interest in designing controllers for the underactuated rigid spacecraft tracking/stabilization problem. In , Crouch provided necessary and sufficient conditions for controllability of a rigid body in the case of one, two, or three independent actuators. In , Byrnes et al. demonstrated that a rigid spacecraft with only two controls cannot be asymptotically stabilized via continuous-state feedback since it does not satisfy Brockett's necessary condition  for smooth feedback stabilizability. In , Morin et al. developed a smooth, time-varying stabilizing controller by using averaging theory. A continuous time-varying/time-periodic switching controller was proposed by Coron and Kerai in . Using averaging theory and Lyapunov control design techniques, Morin and Samson  developed a continuous time varying controller that locally exponentially stabilized the attitude of a rigid spacecraft. Recently, in , Tsiotras et al. proposed a saturated, tracking/stabilizing controller for the kinematic control of an underactuated axisymmetric spacecraft; however, the spin rate on the unactuated axis is required to be zero.
In this paper, we propose a novel, continuous, time-varying, nonlinear tracking controller for the kinematic model of an axisymmetric as well as non-axisymmetric underactuated rigid spacecraft. The control structure is motivated by the Lyapunov-based dynamic oscillator presented in  for wheeled mobile robots. The proposed control approach is novel in that several key characteristics of the quaternion kinematic representation are exploited during the redesign of the control structure originally proposed in . Indeed, it is the fusion of the dynamic oscillator-based approach (i.e., it provides additional design flexibility) and quaternion kinematic representation that facilitates the tracking result for the underactuated spacecraft system. The controller ensures that the spacecraft orientation error is driven to an arbitrarily small neighborhood of zero provided the initial errors are sufficiently small (i.e., the controller guarantees local uniform, ultimately bounded (LUUB) tracking). We also discuss how standard backstepping control techniques can be fused with the kinematic controller to present a solution (i.e., both dynamic and kinematic effects are accounted for) for full-order LUUB\ tracking/regulation of an axisymmetric spacecraft. In contrast to the work presented in , the axisymmetric control strategy is for the full order model; furthermore, it does not impose restrictions on the spin rate of the unactuated axis.
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