nro7_200210

nro7_200210 - Electromagnetic Formation Flight Progress...

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Electromagnetic Formation Flight Progress Report: October 2002 Submitted to: Lt. Col. John Comtois Technical Scientific Officer National Reconnaissance Office Contract Number: NRO-000-02-C0387-CLIN0001 MIT WBS Element: 6893087 Submitted by: Prof. David W. Miller Space Systems Laboratory Massachusetts Institute of Technology
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T WO -S PACECRAFT N ONLINEAR E QUATIONS OF M OTION , I NCLUDING G YROSTIFFENING Nomenclature: A Coil Cross-Sectional Area i Current Running Through Electromagnetic Coil [A] . I rr,s Spacecraft Mass-Moment of Inertia about Radial Axes [kg m 2 ] . I rr,w Reaction Wheel Mass-Moment of Inertia about Radial Axes [kg m 2 ] . I zz,s Spacecraft Mass-Moment of Inertia about Spin Axis [kg m 2 ] . I zz,w Reaction Wheel Mass-Moment of Inertia about Spin Axis [kg m 2 ] F r , F φ , F ψ Forces on Spacecraft m Spacecraft Mass n Number of Conductor Wraps around Electromagnet r Position Vector of Spacecraft A [m] r , φ , ψ Position Coordinates of Spacecraft A RW Reaction Wheel T r , T φ , T ψ Torques on Spacecraft about Local r , φ , ψ Frame T x , T y , T z Torques on Spacecraft about Body-Fixed x , y , z Frame x State Vector x , y , z Local Body-Fixed Coordinates on Spacecraft A X , Y , Z Global Coordinates α i i th Euler Angle of Spacecraft A β i i th Euler Angle of Spacecraft B z,w Constant Spin Rate of RW µ Magnetic Moment of Coil [A m 2 ] 1. Introduction The goal of this work is to define the nonlinear equations of motion for a two-spacecraft formation flying array undergoing a steady-state spin maneuver. While these equations will capture the nonlinear dynamics of the system being considered, they will be linear- ized for purposes of control design and stability analysis. Once a controller has been designed using the linearized design model of the dynamics, the original nonlinear equa- tions may serve as an evaluation model for simulating the closed-loop behavior of the nonlinear system. Nomenclature: 1
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In the following section, we define the geometry of the system being considered. In Section 3, the nonlinear equations of motion are presented, and in Section 4, the equations are linearized. 2. System Description The two-spacecraft array being considered is depicted in Figure 2.1. The X , Y , Z coordi- nate frame represents a global, non-rotating frame whose origin lies at the center of mass of the two-spacecraft array. The first spacecraft, denoted as “spacecraft A,” lies at coordi- nates r , φ , ψ . Since the global frame’s origin coincides with the array’s center of mass, and we are considering the two spacecraft to be identical in mass and geometry, the second spacecraft, denoted as “spacecraft B,” lies at coordinates r , φ + π , ψ (or equivalently r , φ , ψ + π ). r Z Y φ ψ e ˆ r e ˆ φ e ˆ ψ e ˆ x e ˆ y e ˆ z X Spacecraft A Spacecraft B Figure 2.1 Geometry of Two-Spacecraft Array While the X , Y , Z frame represents a global frame, the r , φ , ψ frame represents a local frame whose origin lies at the center of mass of spacecraft A. The r , φ , ψ frame is not fixed to the body in that it does not rotate or “tilt” with the spacecraft. Notice the e ˆ r vec- System Description 2
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This note was uploaded on 11/08/2011 for the course AERO 16.810 taught by Professor Olivierdeweck during the Winter '07 term at MIT.

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nro7_200210 - Electromagnetic Formation Flight Progress...

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