Taking into account flexibility in attitude control1

Large structures in space do deform in a non negligible manner. The effect that deformations have on the dynamic of a spacecraft might be captured by complex non linear mathematical models. In a recent work the authors developed a non coupled set of ordinary differential equations describing the full non linear dynamic of a flexible spacecraft equipped with a system of next generation fly-wheels. These equations are here exploited to test the robustness of a “Velocity Based” steering law to flexible dynamic. As the “Velocity Based” steering law for a VSCMG device is designed under the hypothesis of rigid dynamic, ”spill-over” effects are visible. Even though vibrations are suppressed quite rapidly, the ”spill-over” introduces a tracking error that has to be accounted for. As a result, a minimum attitude acquisition time is found for a selected satellite configuration, faster manoeuvres are not possible as flexible modes would be excited in such a way as to make the spacecraft lose the required pointing accuracy. Introduction In recent years satellite platforms have increased their performances in attitude maneuverability and precision. Among the attitude actuators flywheels are certainly the most used for precision pointing. These wheels store angular momentum and exchange it with the satellite platform causing the spacecraft to reorient. Control Moment Gyros (CMG) and Variable Speed Control Moment Gyros (VSCMG) are such devices that have been recently given a fair attention due to their capability of exerting greater torques. The problem of designing a control steering law for a spacecraft equipped with these modern attitude actuators has been faced by several authors. A VSCMG device is a wheel gimballed to the spacecraft main structure that has the ability to rotate around the gimbal axis as well as chaniging its wheel speed. These control devices allow to avoid the singular configurations that affect CMG. When a singular configuration approaches, a properly chosen spin axis angular velocity variation may in-fact still generate the required torque. The problem of tracking a given 1This work is dedicated to the memory of Prof. Chiara Valente. attitude history using these devices has been faced and solved in some recent works by Shaub et al. [1], Vadali et al. [2] and Tsiotras et al. [3] using a Lyapunov approach and considering the gimbal and wheel speed as controls. In these articles a “Velocity Based” steering law is developed that exploit in an optimal way the redundant degrees of freedom available with CMG and VSCMG. In this work, the equations describing the motion of a multi-flexible body, mainly taken from [4] and [5], are used to test the robustness of this velocity based steering law to flexible dynamic. The steering law In the works [1, 3, 2] the problem of attitude and power tracking for satellites equipped with Control Moment Gyros or variable Speed Control Moment Gyros is faced. By a clever handling of Lyapunov theory the problem is there solved leading to a steering law based upon the rigid body dynamic equations: JT ω̇+∑k Ygk γ̈kgk +∑k I w sk Ω̇ksk +∑k I w sk Ωk γ̇ktk+ +∑k(Ysk −Ytk)γ̇k(tks T k + skt T k )ω+ω JT ω+ +∑k Ygk γ̇kω gk +∑k Iw sk Ωkω sk = g (1) where ω is the spacecraft angular velocity projected in body axes, γ j are the angular positions of the n gimbal plus wheel devices and Ω j are the wheels spin velocities. The above equations, equivalent to those used in [1, 3, 2], may be derived from [5, 4] by neglecting the flexible terms. We here briefly recall the developments of the above mentioned steering law. Kinematics relations for the whole spacecraft given in terms of Modified Rodriguez Parameters (MRP) have to be added to Eq. (1) in order to have a complete set of motion equations. Let us introduce ωd and σd respectively as the tracking signal for the angular velocity and for the MRP, equation (1) together with the kinematics relation can be written in the compact form: f(ẋ,x,u) = 0 where x = [ωe,σe], u = [γ̇ j,Ω̇ j], ωe = ω−ωd and σe is the MRP that overlaps the actual body frame to the desired one. Following the development in [3] a Lyapunov function V (ωe,σe) may be introduced and the following relation may be obtained by imposing that V is negative definite during the spacecraft motion: Bγ̈+Cγ̇+DΩ̇ = Lrm (2) where Lrm, B, C and D are functions of the tracking signal, of the state and of the gimbal angles and the wheels speed. Eq. (2) may be regarded to as a set of 3 relations between the 2n unknown quantities γ̇ j,Ω̇ j. By neglecting the term in γ̈ (excessive gimbal acceleration are carefully avoided as result in infeasible steering laws) Eq.(2) can be rewritten as Qu = Lrm. (3) As these equations are not sufficient to determine all of our unknowns as soon as n > 1, a selection between all the controls time histories that make V negative definite has to be made. This might be done in several ways reminding that gimbal accelerations have to be kept small and that singular configurations have to be avoided. The result is a “Velocity Based” steering law (the name reminds us that gimbal accelerations have been neglected), capable (in theory) to perfectly track any given signal. It has however to be noted that, as soon as the signal becomes too demanding, the spacecraft may excite its flexible modes and its dynamic may therefore part from a rigid one. This may lead to an inaccurate tracking and to undesirable effect. The rest of this paper is focused on this last issue. Equation of flexible dynamic are taken from [5] and [4], these equations have been developed by taking care to follow both the methods dealing with multibody dynamic and those dealing with VSCMG dynamic. As some structural invariants are involved in this general formulation, the following section will select and define a test case and will evaluate such invariants. Spacecraft’s test configuration A simple geometrical configuration has been used in order to obtain the necessary structural invariants by means of analytical formulas (a preliminary FEM analysis is otherwise necessary whenever the structure considered is complex). Let us consider a spacecraft with a 6m long flexible boom mounted as shown in figure 1. A tip mass is also considered. The body frame Fb is centered in the center of mass G of the undeformed satellite and has the x axis aligned with the undeformed boom. The boom root has a distance of . 6 meters from the point G. Considering the standard modal analysis for a cantilever beam with tip mass the following normal modes may be evaluated: φi(x) = 1 ci √ L {