Gravity Modeling for Variable Fidelity Environments

Abstract

Aerospace simulations can model worlds, such as the Earth, with differing levels of fidelity. The simulation may represent the world as a plane, a sphere, an ellipsoid, or a highorder closed surface. The world may or may not rotate. The user may select lower fidelity models based on computational limits, a need for simplified analysis, or comparison to other data. However, the user will also wish to retain a close semblance of behavior to the real world. The effects of gravity on objects are an important component of modeling real-world behavior. Engineers generally equate the term gravity with the observed free-fall acceleration. However, free-fall acceleration is not equal to all observers. To observers on the surface of a rotating world, free-fall acceleration is the sum of gravitational attraction and the centrifugal acceleration due to the world's rotation. On the other hand, free-fall acceleration equals gravitational attraction to an observer in inertial space. Surface-observed simulations (e.g. aircraft), which use non-rotating world models, may choose to model observed free fall acceleration as the "gravity" term; such a model actually combines gravitational attraction with centrifugal acceleration due to the Earth's rotation. However, this modeling choice invites confusion as one evolves the simulation to higher fidelity world models or adds inertial observers. Care must be taken to model gravity in concert with the world model to avoid denigrating the fidelity of modeling observed free fall. The paper will go into greater depth on gravity modeling and the physical disparities and synergies that arise when coupling specific gravity models with world models.

Gravity Modeling Effects on Surface-Interacting Vehicles in Supersonic Flight

“…2 For a symmetric ellipsoid mass, the even zonal harmonic † Constant gravitation better predicts surface gravity by a small fraction when applied to the geodetic unit vector. 2 However, constant gravitation produces lower trajectory errors when applied to the geocentric unit vector.…”

“…3 To model the Earth as a sphere, this study uses the radius of a sphere with equal surface area to the WGS84 ellipsoid because it minimizes error in the mean surface gravity. 2 Vehicles in surface interacting simulations are bound to the world by gravitation. Thus, only the world's motion relative to its center of mass, i.e.…”

Further Investigations of Gravity Modeling on Surface Interacting Vehicle Simulations

“…2 For a symmetric ellipsoid mass, the even zonal harmonic coefficients (Cn where n is even) are functions of the world's mass (M), equatorial radius (a), and moments of inertia (Ixx, Izz) as shown in Equation 5: 6,7 I − I 3e…”

“…However, its error profile against the WGS84 gravity formula shows improved fidelity over the mean gravity (which is a physical property) . 2 The gravitation constant selected for this model is the mean gravitation, 9.82023 m/s 2 (32.219 ft/s 2 ).…”

“…Some pairs produce degraded fidelity. 2 This study examines the actual performance of paired world and gravity or gravitation models ("world-gravity pairs") in a simulation using the Earth. The choice of world-gravity pairs is based on prior analytical work in Reference 2; the pairs are described in section II.C.…”