# Reaction Forces and Apparent Thrust in Dual Oscillating Control Moment Gyroscopes
## 1 Introduction
Figure 1: The average thrust produced by the gyro scissor operation (from [21] with permission). The purpose of this paper is to provide a detailed explanation of the first observation above, which has led some authors to persistently suspect that [PERSON]'s third law may be violated in the operation of control moment gyroscopes (CMGs). Extensive research has been conducted on control moment gyroscopes (CMGs), with much of the work completed up to 2015 compiled and organized in a book by [PERSON] et al. [26]. Key lessons learned from the use of CMGs on the International Space Station (ISS) were reported in 2010 by three industrial partners, including The Boeing Company [27]. Around the same time, NASA and Boeing also reported on on-orbit propulsion systems and momentum management strategies for the ISS [28]. Scissored-pair control moment gyroscopes (also known as twin gyroscopes) have existed for over a century. [PERSON]'s monorail system [29] employed two counter-rotating flywheels with interlinked gimbals to achieve symmetric stabilization during left or right turns. Like any CMG array, a scissored pair generates attitude control torque by exchanging angular momentum with the host body. During the Skylab era, the Astronaut Maneuvering Research Vehicle utilized scissored pairs for attitude control [30]. These configurations have also been investigated as gyrodampers for large space structures [31]. In robotic applications with single revolute joints, where torque is required along a single axis, the symmetry inherent in a scissored pair enables torque generation without producing significant reaction torques in the robot itself. Such systems are capable of rapidly reorienting a payload with reduced power requirements compared with maneuvering the entire spacecraft [32]. Subsequent studies on this concept were conducted by [PERSON] and [PERSON] [33,34]. A recent layman-friendly explanation of CMGs and their role aboard the International Space Station (ISS) was presented by National Geographic [35]. It emphasizes that the change in orientation caused by gyroscopic axle motion is primarily powered by solar energy, thus eliminating the need for propellant consumption. It is important to note that, despite changes in orientation, the satellite's center of mass remains at a constant distance from the nearest planetary body and is therefore not directly propelled in the radial direction. However, by altering the satellite's orientation, CMGs can indirectly influence its trajectory and position over time, particularly when combined with additional mechanisms that exploit orientation changes, such as gravity-gradient stabilization, solar sails, electrodynamic tethers, or auxiliary thrusters. In the context of spacecraft propulsion, and in addition to the aforementioned methods, electrolysis propulsion [36] involves the use of electrical energy to dissociate water (\(H_{2}O\)) into its constituent elements: hydrogen (\(H_{2}\)) and oxygen (\(O_{2}\)). These gases can subsequently be employed as propellants in a thruster system. Unlike the previously discussed methods, which primarily alter the orientation without affecting the spacecraft's center of mass, electrolysis propulsion results in the expulsion of mass and can thus induce a shift in the spacecraft's center of mass and produce net translational acceleration [37]. In addition to the aforementioned contributions, the team at the Georgia Institute of Technology (USA) has made significant advances in singularity avoidance and steering laws, nonlinear control techniques, fault-tolerant control, and CMG array configurations and dynamics, in addition to comprehensive reviews of CMG technologies [38, 39, 40, 41, 42]. Also, researchers from the United Kingdom have notably contributed to the development of CMG systems, particularly in the context of small satellite platforms and advanced control strategies [43, 44, 45]. Moreover, recent developments from the Republic of Korea [46] and China [47] have primarily focused on attitude control, among other aspects.