Steering Law for Two Parallel Variable-Speed Double-Gimbal Control Moment Gyros

Abstract

A CONTROL moment gyro (CMG) is a representative torque producing actuator for the spacecraft attitude control. It generates control torques by changing the momentum direction of a reaction wheel (RW). The most serious difficulty while designing CMG steering laws is the existence of singular configurations, for which CMGs cannot produce torques in a certain direction. Such a CMG singularity problem has been solved by various approaches [1,2].When thewheel speed of the CMG is variable, a variable-speed control moment gyroscope (VSCMG) can be obtained. While investigating VSCMG steering laws, singularity is also the main issue needing to be considered. The VSCMG has three operating modes, including the CMG mode, RW mode, and VSCMG mode. This suggests that a VSCMG has several advantages. For example, it can produce large control torques as a CMG, and it also can produce precision control torques as a RW. Additionally, the VSCMG can be divided into two classes, such as the variable-speed single gimbal control moment gyroscope (SGCMG) and the variable-speed double gimbal control moment gyroscope (DGCMG). Compared with a variable-speed SGCMG, a variable-speed DGCMG can produce a three-dimensional (3-D) torque and has the angular momentum envelope with high efficiency, which contributes to the design of the steering law. However, because the RWmode is not able to generate large torques, there still exist problems when using variable-speed DGCMGs to produce large control torques. For instance, variablespeed DGCMGs may encounter singular gimbal-angle configurations for which the CMG mode cannot generate torques about arbitrary directions, resulting in the large control torque not being produced. Recently, several achievements of the steering law for VSCMGs have been obtained in [3–10]. The VSCMG singular avoidance steering law in [3] is derived from minimizing a cost function based on multiple indices. A null motion, proposed by using a gradient method, is exploited to drive VSCMG gimbal angles away from the CMG singularity [4]. By providing a simplified modification to the singularity measurement function addressed in [4], McMahon and Schaub [5] propose a less complicated method to achieve an appropriate null motion steering law, resulting in a shared similar performance with the previous methods. However, those VSCMG singularity avoidance steering laws [3–7] require controlling both the wheel speed and the gimbal rate, particularly at the same time, and have not considered the respective peculiarities of the CMG and RW modes. To simplify the problem and make VSCMGs operate in the suitable mode, a mode scheduling method is presented in [8]. The practical steering laws in [9,10] are developed through dealing with the combined energy storage and attitude control systems. Despite that, the references identified here provide various studies on the steering law for VSCMGs; all of them are subject to variable-speed SGCMGs. Stevenson and Schaub [11] present a nonlinear control algorithm to implement full-attitude control with single variablespeed DGCMG, but the singularity avoidance steering law for a variable-speed DGCMG cluster has not been discussed. Therefore, there exists a gap in the steering law design for variable-speed DGCMGs. Another actuator type relative to the variable-speedDGCMG is the conventional DGCMGwith a constant wheel speed. Control laws for DGCMGs have been studied by several papers. Kennel presents the vector distribution steering law for orthogonal-mounted and parallelmounted DGCMGs in [12,13], which makes use of redundant degrees of freedom to avoid encountering singular gimbal-angle configurations. The steering law for a DGCMG with null motion, addressed in [14], can make DGCMGs get away from the CMG singularity when it is approaching gimbal locks. Wang et al. [15] point out that the vector distribution steering law in [13] does not have the capability to get away from the CMG singularity, and the weighted indexes cannot vary with the angle motion; hence, it provides the improvement. However, these steering laws for a DGCMGhave not considered the impact of the variablewheel speed, and they are not available to variable-speed DGCMGs. In this Technical Note, the steering law for two parallel-mounted variable-speedDGCMGs by employing themethod based on optimization control theory is investigated. The mode switching method is put forward according to the operating conditions of variable-speed DGCMGs, in which the desired attitude control torque is divided into CMG command torque and RW command torque. Aiming to improve the capability of CMG singularity avoidance, a new singularity measurement function is exploited through analyzing the CMG singularity state of two parallel-mounted variable-speed DGCMGs. Then, the CMG cost function, consisting of singularity indices and CMG energy terms, is established based on the singularity measurement function. To overcome the possibility of wheel speed saturation during the operation of the RW, the RW cost function is established by considering two objectives, including nominally constant wheel speeds and power extraction requirements. Finally, the steering law for two parallel-mounted variable-speed DGCMGs is maintained by minimizing the cost function.