Novel coordinates for nonlinear multibody motion with applications to spacecraft dynamics and control.

摘要

Novel sets of attitude coordinates called the Stereographic Parameters (SPs) and configuration quasivelocity coordinates called the Eigenfactor Quasivelocities (EQVs) are discussed. The SPs are generated through stereographic projections of the Euler parameter constraint hypersphere onto hyperplanes. SP sets are non-unique and have distinct alternate sets referred to as shadow sets. They abide by the same differential kinematic equation, but generally display a different singular behavior. Explicit expressions are developed that map the original SP set to the shadow set and thus avoid any singularities. Both symmetric SPs such as the classical and Modified Rodrigues Parameters (MRPs), as well as asymmetric SPs are discussed. A globally asymptotically stable MRP feedback law which tracks any reference trajectory is presented. Both unsaturated and saturated control cases are discussed. Further, an MRP costate switching condition is developed that allows both original and shadow MRPs to be used simultaneously in optimal control problems.; The Lagrange equations of motion in terms of the n-dimensional EQV vector are developed. The EQV formulation has an identity mass matrix which results in no matrix inverse being taken in numerical simulations. An explicit expression is presented that incorporates Pfaffian non-holonomic constraints into the EQV formulation without increasing the system order. Unfortunately, the use of EQV in numerical simulations only proved beneficial in selected cases. Generally the computational burden proved too high. However, the EQVs are found to be valuable when used as velocity feedback coordinates. EQV feedback laws have an exponentially decaying kinetic energy, superior performance to traditional state velocity feedback laws and are found to decouple the motion of multi-link robotic systems.; The equations of motion and steering laws of spacecraft containing Variable Speed Control Moment Gyroscopes (VSCMGs) are developed. Contrary to classical CMG steering laws, VSCMG steering laws derived herein do not encounter singularities. Both gimbal angle velocity and acceleration based steering laws are presented. Since gimbaling is more efficient energy consumption wise, the steering law presented only utilizes reaction wheel modes close to classical CMG singularities. The reaction wheel mode also assists in driving the gimbals to preferred angles using null motion.