T HE problem of designing fast detumbling maneuvers for a spacecraft with restricted actuation torques arises inmany space applications. Satellites are often equipped with an attitude control system (ACS), which usually has two modes of operation: detumbling and stabilization. In detumbling mode, the ACS controller is responsible for dumping the initial angular velocity from when the satellite was in the idle mode; this situation occurs when the satellite is released from the launch vehicle or when power is so low that ACS has to be turned off to save power [1]. Only after this phase can the stabilization mode be activated to control the orientation of the satellite to align antennas toward the Earth. Reaction wheels are commonly used as the actuators of the ACSs. Satellites frequently need fast maneuvers [2], minimum time maneuvers, while the speed is restricted owing to a low level capacity of the actuators. Hence, planning optimal maneuvers is highly desired. Alternatively, spin-stabilized spacecraft allow simple attitude maneuvers without the need for complex control systems. The spacecraft can be spun up around its axisymmetric axis to stabilize the orientation of the vehicle axis through the gyroscopic effect. This method is also widely used to stabilize the final stage of a launch vehicle [3]. However, when a spacecraft is in the tumblingmotion, its angular velocity is not parallel to the axisymmetric axis. Therefore, the objective in deployment of a spin-stabilized spacecraft [4] is to bring the spacecraft form the tumblingmotion to the statewherein the spacecraft spins around a single axis. The detumbling or passivation of a satellite is also required before on-orbit servicing of the satellite or its retrieval [5]. For such a mission, an orbital maneuvering vehicle can be used to apply torques to the target satellite for removing any relative velocity [5,6]. Also, the vehicle can be equipped by an articulating arm with a grappling device on it that could be driven to capture a tumbling target satellite and then detumble it [7,8]. After capture of an uncontrolled tumbling satellite by a spacemanipulator, the satellite should be brought to rest in minimum time. Again, the restriction of the manipulator end effector to provide “braking torques” for the fast maneuvers motivates an optimal trajectory planning. Optimal detumbling ofmultibody systems has been considered for spacecraft possessing appendages, such as a robotic manipulator, with well-controlled motion relative to the spacecraft to achieve detumbling [9–13].However, because of the complexity of dynamics of space manipulators, only a numerical solution or intensive trialand-error procedures have been found for the optimization problem. The problem of time-optimal detumbling control of rigid spacecraft is formulated as a nonlinear programming and solved numerically by using an iterative procedure in [14], while nonoptimal control approaches have been reported in [15–17]. There also exists a fair amount of research done on the time-optimal reorientation control, rest to rest, of rigid spacecraft [18–20] and a survey, for example, can be found in [21]. This paper presents a closed-form solution for time-optimal detumbling control of a rigid spacecraft with the constraint on maintaining the Euclidean norm of the braking torques below a prescribed value. Thefinal angular rate can be specified as zero or any vector parallel to the eigenaxis. The optimal control theory and Pontryagin’s principle are applied to derive the optimal solution, which is not only easy to implement but also gives a great deal of insight. First, it will be shown that for our particular optimal control problem, the system costates and states are related by a nonlinear but static function. Subsequently, the optimal control law is explicitly derived in the form of a nonlinear state feedback. The magnitude of the system angular momentum is found to be linearly deceasing with time, leading to a simple expression for the terminal time needed for control implementation. Furthermore, the time-optimal controller is extended for the case of nonzero terminal velocity. To this end, the time-optimal control technique is applied to derive a nonlinear feedback control law which can drive a tumbling axisymmertic spacecraft to a final spin-stabilized state. Finally, the time-optimal detumbling technique is illustrated through numerical examples.
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