JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 27, No. 4, July–August 2004 Solar Sail Attitude Control and Dynamics, Part 2 Bong Wie ∗ Arizona State University, Tempe, Arizona 85287-6106 Dynamical models and attitude control concepts are developed for the purpose of sailcraft attitude control systems design. Particular emphasis is placed on a two-axis gimbaled control boom to counter the significant solar-pressure disturbance torque caused by an uncertain offset between the center of mass and center of pressure. Controlling sailcraft attitude by sail shifting and tilting is also investigated. A flight experiment in a geostationary orbit for the purpose of validating the principle of solar sailing is also proposed. A 40 × 40 m, 160-kg sailcraft in an Earth-centered elliptic orbit, with a nominal solar-pressure force of 0.01 N, an uncertain center-of-mass/center-of- pressure offset of ±0.1 m, and moments of inertia of (6000, 3000, 3000) kg · m 2 , is studied to illustrate the various concepts and principles involved in dynamic modeling and attitude control design. I. Introduction T HE technical challenges and issues associated with solar sail- ing, sail membranes and booms, sail packaging, boom deploy- ment, and attitude control are discussed in Refs. 1 and 2 for a sail flight validation experiment previously proposed for the New Mil- lennium Program Space Technology 7 (NMP ST7). A 40 × 40 m sailcraft configuration and its system architecture, as illustrated in Fig. 1, have been developed by the Jet Propulsion Laboratory (JPL) and AEC-Able Engineering for the NMP ST7 flight validation mis- sion. A significant feature of this baseline ST7 sailcraft is the use of a two-axis gimbaled control boom, instead of control vanes, for propellantless sail-attitude control. Although the proposed solar sail mission was not selected as an actual flight validation mission of the NMP ST7, a 20-m scaled model of the baseline ST7 sailcraft is currently under development by NASA and AEC-Able Engineering for a ground validation experiment in 2005 (Ref. 3). A sail-attitude- control system employing a two-axis gimbaled control boom is also being further developed for a ground validation experiment on AEC- Able’s 20-m sail in 2005 through the NASA In-Space Propulsion Solar Sail Program (Ref. 3). In this paper, a 40 × 40 m, 160-kg sailcraft with a nominal solar-pressure force of 0.01 N, an uncertain center-of-mass/center- of-pressure (cm/cp) offset of ±0.1 m, and moments of inertia of (6000, 3000, 3000) kg · m 2 is further studied to illustrate the various concepts and principles involved in dynamic modeling and attitude- control design. Particular emphasis is placed on various control- design options for countering the significant solar-pressure distur- bance torque caused by an uncertain cm/cp offset. An overview of solar sail attitude control issues, as well as a simple spin-stabilization approach, was presented in Part 1 (Ref. 4). The remainder of this paper is outlined as follows. Section II presents dynamic models and an attitude control problem for a sail- craft equipped with a gimbaled control boom and control vanes. In Sec. III, a gimbaled thrust-vector control (TVC) design problem is formulated for a sailcraft with a two-axis gimbaled control boom, and preliminary TVC design results are presented. In Sec. IV, a sail- craft controlled by shifting and tilting sail panels is investigated, and a flight experiment in a geostationary orbit for the purpose of validating the principle of solar sailing is also described. Received 3 June 2002; presented as Paper 2002-4573 at the AIAA Guid- ance, Navigation, and Control Conference, Monterey, CA, 2–6 August 2002; revision received 25 August 2003; accepted for publication 3 November 2003. Copyright c  2003 by the American Institute of Aeronautics and As- tronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/04 $10.00 in correspondence with the CCC. ∗ Professor, Department of Mechanical & Aerospace Engineering; bong.wie@asu.edu. Associate Fellow AIAA. II. Sailcraft with a Gimbaled Control Boom and Control Vanes As discussed in Part 1 (Ref. 4), one method of controlling the attitude of a three-axis stabilized sailcraft is to change its center-of- mass location relative to its center-of-pressure location. This can be achieved by articulating a control boom with a tip-mounted mass. Another method is to employ small reflective control vanes mounted at the spar tips. A dynamic model of a generic three-axis stabilized sailcraft with such tip-mounted vanes and a control boom, as illus- trated in Fig. 2, is developed here. The complexity of the modeling and control problem inherent in even such a simple rigid sail, but with a moving mass, will be discussed. The problem of a rigid spacecraft with internal moving mass was first investigated in the early 1960s. For spacecraft dynamical prob- lems with internal moving mass, one may choose the composite center of mass of the total system as a reference point for the equa- tions of motion. This formulation leads to a time-varying inertia matrix of the main rigid body, because the reference point is not fixed at the main body as the internal mass moves relative to the main body. On the other hand, one may choose the center of mass of the main body as the reference point, which leads to a constant inertia matrix of the main body relative to the reference point, but results in complex equations of motion. In this paper, the second approach, choosing the center of mass of the main body as the reference point, is employed. A. Dynamical Equations of Motion Consider an ideal sailcraft model consisting of a rigid sail subsys- tem of mass m s and a payload/bus of mass m p located at the end of a massless control boom of length l , as shown in Fig. 2. The origin of the body-fixed reference frame (x , y , z ) is located at point O, which is assumed to be the center of mass of a rigid sail subsystem of mass m s . The position vector of the payload/bus mass from the reference point O is expressed as r = x i + y j + z k = l e r = l (cos φi + sin φ cos θ j + sin φ sin θ k) (1) where φ is the boom tilt angle and θ is the boom azimuth angle relative to the sailcraft body axes (x , y , z ). These two gimbal angles can be considered as control inputs; however, the gimbal dynamics needs to be included later for detailed control design. There are four different ways to model the attitude dynamics of a spacecraft with a moving mass. In general, the four different angular momentum equations are given by ˙ H o + ˙ R o × m ˙ r c = M o (2a) ˙ h o + r c × ma o = M o (2b) ˙ H c + r c × ma c = M o (2c) ˙ H c = M c (2d) 536