Tensorial-formulation-based method for modeling of flight dynamics with generalized coordinates

Abstract

For the sake of minimizing fuel consumption, a rocket typically keeps flying in or near a plane specified by the Earth's center, launch pad, and orbit-insertion point, where the plane is called the Reference Flight Plane (RFP) here. To facilitate trajectory analysis, a series of generalized position and velocity coordinates are defined based on the RFP. To develop the Ordinary Differential Equations (ODEs) of the generalized coordinates, a novel dynamics modeling method based on tensor theory is proposed. As the first step of the new method, the rotating Earth is taken as the reference frame and a flight dynamics model described by orthogonal coordinates is developed. Then, according to the definitions of the generalized coordinates, two special curvilinear coordinate systems are put forward, and the relationships between the curvilinear and orthogonal coordinate systems are found. Finally, skillfully using the relationships as well as tensor theory, the dynamics model for the orthogonal coordinates are transformed into that for the generalized coordinates. The benefit of the new method is that it can derive the ODEs of multiple coordinates simultaneously because it is entirely based on matrix operations. By contrast, the traditional modeling methods from Newtonian mechanics and analytical mechanics need to use the infinitesimal approach to develop the equation of each generalized coordinate one by one, which may result in a large amount of repeated derivation. Therefore, the new method has higher efficiency in developing flight dynamics model described by generalized coordinates, and helps to prevent mistakes in derivation.