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Electromagnetic Formation Flight Progress Report: March 2003

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Electromagnetic Formation Flight Progress Report: March 2003 Electromagnetic Formation Flight Progress Report: March 2003 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 OVERVIEW Description of the Effort The Massachusetts Institute of Technology Space Systems Lab (MIT SSL) and the Lockheed Martin Advanced Technology Center (ATC) are collaborating to explore the potential for an Electro-Magnetic Formation Flight (EMFF) system applicable to Earth-orbiting satellites flying in close formation. Progress Overview At MIT, work on EMFF has been pursued on two fronts: the MIT conceive, design, implement and operate (CDIO) class, and the MIT SSL research group. This report summarizes recent progress made in the MIT SSL research group with regards to analyzing the dynamics of a two-spacecraft electromagnetic formation flying system. It is a follow-up to the work presented in the October 2002 progress report, which introduced the nonlinear and linearized dynamics of such a system. In this report, we summarize our recent findings about the controllability and stability of this system. Introduction 1 1. Introduction In this report, we sumarize the dynamics of the electromagnetic formation flying (EMFF) system of spacecraft introduced in the October 2002 progress report [1], and then continue by presenting results from recent controllability and stability analyses performed on simi­ lar EMFF systems. In particular, we consider a formation of two spacecraft under no external forces and rotating about a common origin. Each spacecraft contains a specified configuration of fixed electromagnets (EM) and reaction wheels (RW) for use as position and attitude actuators. We begin in Section 2 by reviewing the results presented in [1]. In Sections 3 and 4, respectively, we define and respond to certain questions pertaining to the controlla bility and stability of the system at hand. We then summarize our results and draw con­ clusions in Section 5. 2. Background We begin by reviewing the linearized dynamics of a system similar to those considered in this exercise. Reference [1] describes the dynamics of a two-spacecraft EMFF system, where each spacecraft has a single fixed electromagnet, nominally pointed along the line of sight between the two spacecraft. Each spacecraft also has a single reaction wheel, nominally oriented perpendicular to the plane of rotation of the two spacecraft. Such a system has nine degrees of freedom (or eighteen state variables): T ∆ ∆φ ∆ψ ∆α ∆α ∆α ∆β ∆β ∆β x = r 1 2 3 1 2 3 (2.1) where ∆r, ∆φ, ∆ψ are the relative displacements of the vehicles in curvilinear coordi­ ∆α , ∆α , ∆α nates depicted in Figures 1 and 2, 1 2 3 are the Euler angles of the first space­ craft (denoted spacecraft “A”) about its body-fixed z, y, and x axes, respectively, and ∆β , ∆β , ∆β 1 2 3 are the corresponding Euler angles of the second spacecraft (denoted spacecraft “B”) about its body-fixed coordinate axes. Background 2 Z ˆ ˆ eψ ez eˆ eˆ eˆ x y φ ˆ er Spacecraft A r ψ X Y φ Spacecraft B Figure 1 Geometry of Two-Spacecraft Array Piece of an imaginary sphere centered at ˆ the origin of the eψ X, Y, Z frame ˆ er eˆ φ Spacecraft A r Z X Y Figure 2 Local Curvilinear Coordinate Frame on Spacecraft A The linearized equations of motion for this system were presented in second-order form in [1] as: Background 3 ∆·· 1 0 0 0 0 0 0 0 0 r ∆φ·· 0 r0 0 0 0 0 0 0 0 ·· 00 r 0 0 0 0 0 0 ∆ψ 0 ∆α·· 00 0 I , 0 0 0 0 0 1 zz s ∆α·· 00 0 0 Irr, s + Irr , w 0 0 0 0 2 ·· 00 0 0 0 Irr, s + Irr , w 0 0 0 ∆α3 ·· 00 0 0 0 0 Izz, s 0 0 ∆β 1 00 0 0 0 0 0 I + I 0 ·· rr, s rr , w ∆β2 0 0 0 0 0 0 0 0 I ,+ I , ·· rr s rr w ∆β3 φ· 0 –2r0 0 0 0 0 0 0 0 0 ∆r· · ∆φ· 2φ0 0 0 0 0 0 0 0 0 · 0 0 0 0 0 0 0 0 0 ∆ψ 0 0 0 0 0 0 0 0 0 ∆α· 1 2 · 0 0 0 0 0 mr φ 0 0 0 ∆α· (2.2) + 0 0 2 2φ· ∆α· 0 0 0 0 –mr0 0 0 0 0 0 3 · 0 0 0 0 0 0 0 0 0 ∆β 1 2φ· ∆β· 0 0 0 0 0 0 0 0 mr0 0 2 2 · ∆β· 0 0 0 0 0 0 0 –mr φ 0 3 0 0 · 2 –5φ0 00 0 000 00 ∆r c2 c2 00 000 c 00c 00 ∆φ 1 1 00 0 0 ∆ψ · 2 00r φ 0 –c 00 –c 0 00 0 0 ∆µ 0 0 1 1 ∆α A 1 000–2c 00–c 00 00K T 0 ∆µ 0 0 ∆α B + 2 = 000 0–2c 00 –c 0 00 0 0 ∆ 0 0 ∆α iRW, A 3 00 0 0 000 0 000 00 ∆i ∆β 00 0K RW, B 1 T 000 –c0 002– c0 00 ∆β 00 0 0 000 0 –c 00 – 2c 0 2 0 0 ∆β 00 0 0 000 0 000 00 3 where: 2 · 2 · 2 –mr φ –r φ µ ≡ 0 0 ≡ 0 0 ≡ φ· 3 0 c0 ------------------ , c1 ------------­ - , and c2 – 0 ------------------- (2.3) 3 2 π 3 32 mr0 KT is the reaction wheel torque constant, and all remaining values are defined in [1]. Controllability Analysis 4 We recognize that, rather than deriving the linearized dynamics of the similar sys tems considered in the following sections from first principles, we can simply modify Equation 2.2 by removing the appropriate degrees of freedom from the dynamic matrices and altering the actuator coefficient matrix as necessary. 3. Controllability Analysis We begin by considering the linearized dynamics of two vehicles in two dimensions (a plane), where the total angular momentum of the system is zero (wheels plus spacecraft), but the rotational angular momentum of the two vehicles about a common origin is non­ zero. Part a) In this case, each spacecraft has a single dipole, fixed to the spacecraft and nominally pointed along the line of sight to the other spacecraft. Each spacecraft also has a single reaction wheel, oriented perpendicular to the plane of rotation of the spacecraft. Also, we assume the vehicles are free to rotate about any axis passing through their center of mass (any “central” axis), and are not necessarily constrained to rotate about axes per­ pendicular to the plane of rotation. Hence each spacecraft has three rotational degrees of freedom. After analyzing this case, and before moving on to Part b, we will also consider a simplified version, in which rotations are constrained to axes perpendicular to the sys- tem’s rotational plane, and each spacecraft thus has only one rotational degree of freedom. With these assumptions, we recognize that the only difference between this system and the system in [1] is that in the present system, the two spacecraft do not leave the nominal hor­ izontal plane of rotation (the X-Y plane in Figure 1). Therefore, the system does not have a degree of freedom associated with the latitude, ∆ψ , and we modify Equation 2.2 by strik­ ing the third column of each square “coefficient” matrix, as well as the third row of the entire matrix equation. This results in the following modified equations of motion: Controllability Analysis 5 ∆·· 1000 0 0 0 0 r ·· 0 r 0 0 0 0 0 0 ∆φ 0 ∆α·· 00 I , 0 0 0 0 0 1 zz s ∆α·· 00 0 Irr, s + Irr , w 0 0 0 0 2 ·· 00 0 0 I ,+ I , 0 0 0 ∆α rr s rr w 3 ·· 00 0 0 0 Izz, s 0 0 ∆β 1 0000 00I + I 0 ·· rr, s rr , w ∆β2 0000 000I , + I , ·· rr s rr w ∆β3 φ· 0 –2r0 0 0 0 0 0 0 0 ∆r· · · 2φ0 0 0 0 0 0 0 0 ∆φ ∆α· 0 0 0 0 0 0 0 0 1 2φ· · 0 0 0 0 mr0 0 0 0 ∆α2 (3.1) + 0 2 · · 0 0 0 –mr φ 0 0 0 0 ∆α3 0 0 · 0 0 0 0 0 0 0 0 ∆β 1 2 · · 0 0 0 0 0 0 0 mr φ ∆β2 0 0 · 2φ· ∆β3 0 0 0 0 0 0 –mr0 0 0 2 φ· ∆r –5 0 0 0 0 0 0 0 0 c2 c2 00 ∆φ 00c 1 00c 1 00 00 0 0 ∆α ∆µ 1 00K 0 A 00– 2 c0 00– c0 00 T ∆α ∆µ 2 00 0 0 B + 000 – 2c0 00 –c0 0 = ∆α ∆ 000 00 0 00 3 00 0 0 iRW, A ∆β 1 00 0K ∆i 00– c0 00– 2 c0 00 T RW, B ∆β 2 00 0 0 000 –c0 00 – 2c0 0 ∆β 00 0 0 000 00 0 00 3 Equation 3.1 is in second order form: Mx·· + Cx· + Kx = Fu (3.2) where: T x = ∆ ∆φ ∆α ∆α ∆α ∆β ∆β ∆β (3.3) r 1 2 3 1 2 3 T u = ∆µ ∆µ ∆ ∆ (3.4) A B iRW, A iRW, B Controllability Analysis 6 and M, C, K, and F are the appropriate matrix coefficients.
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