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 TechnologyTWO-SPACECRAFT NONLINEAR EQUATIONS OF MOTION, INCLUDING GYROSTIFFENING Nomenclature:A Coil Cross-Sectional Area i Current Running Through Electromagnetic Coil [A] .Irr,s Spacecraft Mass-Moment of Inertia about Radial Axes [kg m2] .Irr,w Reaction Wheel Mass-Moment of Inertia about Radial Axes [kg m2] .Izz,s Spacecraft Mass-Moment of Inertia about Spin Axis [kg m2] .Izz,w Reaction Wheel Mass-Moment of Inertia about Spin Axis [kg m2] Fr, Fφ , Fψ 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 Tr, Tφ , Tψ Torques on Spacecraft about Local r, φ , ψ Frame Tx, Ty, Tz 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 αi ith Euler Angle of Spacecraft A βi ith Euler Angle of Spacecraft B Ωz,w Constant Spin Rate of RW µ Magnetic Moment of Coil [A m2] 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: 1In 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 eˆ r eˆφ eˆψ eˆ xeˆ y eˆ 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 theeˆ r vec-System Description 2tor always aligns with the position vector, r , of spacecraft A relative to the origin of the global frame. The x, y, z frame, in contrast, is fixed to spacecraft A; it rotates with the body relative to the r, φ, ψ frame. We now define the relative orientations of the two spacecraft using Euler angles. The Euler angles of spacecraft A are α1, α2, and α3, which represent sequential rotations about the body-fixed z, y, and x axes, respectively. Similarly, the orientation of spacecraft B is defined by the Euler angles β1, β2, and β3, which represent three sequential rotations about a body-fixed frame on B that is nominally aligned with the r, φ, ψ frame on spacecraft A. The nominal orientation of each spacecraft is such that the x, y, z frame aligns with the r, φ, ψ frame. In the following sections, we consider perturbations from this nominal orienta-tion; in other words, we consider the dynamics of the x, y, z frame rotating relative to the r, φ, ψ frame. With the variables defined so far and the constraints on the position of spacecraft B: rB = rA = r, = φA + π = φπ, = ψA = Ψ (2.1) + ψBφB we have defined 18 state variables that make up the state vector, x: T x = r φΨα1 α2 α3 β1 β2 β· β2 · β1 · β3 · α2 · α1 · α · Ψ · φ· r3 (2.2) .3 In this analysis, we consider that spacecraft A and B each contain a single electromagnetic ˆeis in its nominal orientation). The magnetic moment of the electromagnet on spacecraft A is defined as: dipole oriented along the body-fixed x-axis (and thus aligned with when the spacecraft r µˆA = µAe = ˆ x nAiAAAex (2.3) where nA is the number of times the conductor is wrapped around to form the electromag-netic coil, iA is the current running through the coil, and AA is the cross-sectional area of System Description 3the coil system. The magnitude of the magnetic moment, µA, is assumed constant in this analysis, although its direction, e, rotates with the spacecraft. ˆ x The magnetic moment of the electromagnet on spacecraft B is defined similarly and points along the local body-fixed x-axis on spacecraft B. For this analysis, we assume the same geometry for the coils on both spacecraft, so that: nB = nA = n, AB = AA = A (2.4) However, the currents iA and iB are unique and depend on the dynamics and closed-loop control of the system. Finally, we assume that each spacecraft contains a reaction wheel (RW) whose spin axis is aligned with the body-fixed z-axis. Each RW is spinning at a constant rate, Ωz,w , neces-sary to conserve the angular momentum of the spinning array. In other words, the angular momentum stored in the two RWs is equal and opposite to the angular momentum of the two-spacecraft array. Nominally the two spacecraft would assume a circular trajectory in =the global X, Y plane ( ψ 0) with a constant angular velocity, · φ = · φ .0 In this case, the conservation of angular momentum is expressed as: Izz w , Ω zw, 2· φ= mr0 (2.5) 0 where Izz,w is the RW mass-moment of inertia about its spin axis, m is the mass of each spacecraft, and r0 is the nominal array radius. 3. Nonlinear Equations of Motion 3.1 Translational Equations The translational equations of motion for spacecraft A describe the motion of its center of mass with respect to the global coordinate frame. They