Contradictory Postulates of Singularity

Abstract

Modification of rigid body angular momentum permits controlled rotational maneuvers, and one common momentum-exchange actuator contains challenging mathematical singularities that occur when the actuator geometrically aligns perpendicularly to the commanded torque direction. Substantial research has arisen toward singularity avoidance, singularity escape (when avoidance fails), and singularity penetration which permits safe flight through regions of singularity. The latter two in particular, singularity escape and penetration require mathematical calculations of singular and near-singular quantities (very large numbers) using constituent numbers that are sometimes very small. This dichotomy leads to interesting peculiarities in some specific geometries. This short communication critically evaluates three often spoke postulates for defining singularity and the axioms that accompany the postulates. Researchers using disparate postulates arrive at contradictory conclusions about singularities, and we examine these peculiarities, leading to a few conclusions. Singular conditions must never be declared in the abstract without consideration for the commanded maneuver (e.g. the claim “the CMG system is singular”). Seeking the true angular momentum capability at near-planar skew angles, this research concludes that performance prediction is difficult installations at low skew angles should be avoided whenever permissible to enhance abilities of mathematical calculations. It will be shown that maximum momentum performance is easily predicted at very high and very low skew angles, and performance will be shown to be lowest at mid-values of skew angle. Meanwhile, maximum singularity-free performance remains elusive at even modestly low skew-angles.

Development of Deterministic Artificial Intelligence for Unmanned Underwater Vehicles (UUV)

“…Expressing Equations (47) and (50) together in state-variable form (i.e., "state space") yields Equation (51), which may be inverted to solve for the optimal learned values for mass and mass moment of inertia in Equation (52), where the arbitrarily labeled constants are calculated in Equations (47) and (50) using desired states:…”

“…The physical system will be an autonomously piloted (without human intervention) underwater robotic vehicle that has already been well-studied by classical methods [41]. Improvement over the most recent investigation of classical methods [42] was proposed to and funded by the U.S. Office of Naval Research, where the winning proposal [43] contained two parts: deterministic artificial intelligence methods and non-propulsive maneuvering by momentum exchange stemming from a recent lineage [44][45][46][47][48][49][50][51]. While the former is the subject of this manuscript, the latter is neglected here (reserved for a future manuscript).…”

Lasers for Satellite Uplinks and Downlinks

“…Substantial developments in acquisitions, tracking, and fine pointing evolved in the twenty-first century and continue to this day , including substantial development in target acquisition [61,64,65,[67][68][69][70][71][72], target tracking [62,63,75,76,78,81,[83][84][85][86][87][89][90][91], and fine pointing [60,73,74,77,79,80,82,88] at the Naval Postgraduate School, among several other institutions achieving microradian pointing accuracy and closely pursuing nano-radian accuracy.…”

Lasers for Satellite Uplinks and Downlinks

“…Substantial developments in acquisitions, tracking, and fine pointing evolved in the twenty first century and continue to this day including substantial development in target acquisition [61,64,65,[67][68][69][70][71][72], target tracking [62,63,75,76,78,81,[83][84][85][86][87][89][90][91], and fine pointing [60,73,74,77,79,80,82,88] at the Naval Postgraduate School amongst several other institutions achieving microradian pointing accuracy and closely pursuing nano-radian accuracy.…”