# Reaction Forces and Apparent Thrust in Dual Oscillating Control Moment Gyroscopes
## 1 Introduction  
Over the past century, humanity's aspiration for interstellar travel has sparked considerable interest in alternatives to traditional rocket propulsion (e.g., [1, 2]). Despite advances in technology, launching even a small spacecraft still necessitates the use of large and powerful launch vehicles to provide sufficient propulsion, an inefficiency that demands a solution. At the turn of the 21 st century, two major initiatives addressed this challenge: Project Greenglow at BAE Systems [3, 4, 5, 6, 7] and NASA's Breakthrough Propulsion Physics program [8, 9, 10, 11]. The findings of the latter were compiled into a comprehensive 740-page book [12]. Alternative propulsion methods have been classified into much more than 20 categories--including Hall thrusters, the Casimir effect, and the EM-Drive [11, 12]--one of which is known as _inertial propulsion_. The concept of inertial propulsion dates back to the early 1930s in Italy and Russia and later gained attention in the United States with the controversial \"Dean drive\" [13, 14], which proposed that two contra-rotating masses could produce unidirectional motion. In the mid-1970s, [PERSON], Professor of Electrical Engineering at Imperial College London, demonstrated that a heavy gyroscope could be lifted with surprising ease [15, 16, 17]. Most historical efforts related to inertial propulsion have been documented in a comprehensive review [18], while a more recent study has shown that continuous propulsion cannot be expected from rotating mass particles alone [19]. However, previous studies have not thoroughly explored the potential of gyroscope-based inertial propulsion. While BAE Systems and NASA have made their alternative propulsion projects publicly available, the same cannot be said for Boeing Co. (Arlington, VA, USA). In October 2014, senior engineer [PERSON] received permission from The Boeing Company to publicly disclose information on their control moment gyroscope (CMG) research. This disclosure took place in August 2015 and provided historical insight into Boeing's CMG work [20]. As he noted in his authorized presentation, _\"The work started back in the 1960s and continued into the 1990s. The presenter got involved with it in 1995 when he took over operations of the Guidance, Navigation, and Controls (GN&C) lab at the Boeing Kent (WA) Space Center. This lab and the building that housed it were badly damaged in the 2001 Seattle earthquake and later demolished.\"_[20]. At the same conference, the same author presented a second paper [21], from which Figure 1 was derived with his kind permission. It illustrates the sawtooth-shaped input torquing rate waveform (referred to as \"scissoring\") used to generate a pulsed output force. He claimed that by applying torque rapidly in one direction and slowly in the other, a pulsed (average) output force is produced, with a similar waveform to that observed in systems involving rotating masses. Furthermore, between 2017 and 2020, a series of four conference papers documented the development and testing of several alternative prototypes [22; 23; 24; 25]. Each prototype featured a wheeled cart equipped with a differential control moment gyroscope (CMG). In these designs, the pair of CMGs did not rotate continuously through a full circle but instead oscillated in a contra-rotating manner within the range of \([-45^{\circ},45^{\circ}]\). To the best of our understanding, the main conclusions are as follows:  
* When the intended thrust was upward, the average reaction force at two (front end) out of the four cart wheels was also directed upward (thus claiming _net thrust_, in violation of [PERSON]'s third law), consistent with the behavior shown in Figure 1.
* When the intended thrust was horizontal, the oscillation of the eccentricity resulted in smooth, continuous rolling of the cart along the ground (video-recorded in [25]). This effect was attributed to frictional forces similar to those involved in human locomotion.