Energy Management and Attitude Control Strategies using Flywheels

1 I n t r o d u c t i o n Space vehicles, typically utilize separate devices to provide energy storage and atti tude control. Conventionally, the energy collected from solar arrays is stored in chemical batteries for use when the spacecraft is in the earth's shadow. Attitude control is usually accomplished through an array of reaction wheels or control moment gyros. In contrast to the standard configuration, a suitable arrangement of four or more flywheels can integrate the energy storage and attitude control functions into a single system, and thereby, reduce the spacecraft's bus mass, volume, cost, and maintenance requirements while maintaining or improving the spacecraft's performance. We will refer to this system as an Integrated Energy Management and Attitude Control (IEMAC) system. Roughly speaking, the integration of these two functions is achieved by decomposing the model of the flywheel array into two separate control problems (i.e., attitude tracking and energy/power tracking, respectively) [5, 6, 12]. As a result, the energy management function can be accomplished without affecting the attitude control function. A comprehensive literature review of the IEMAC concept is presented in [6]. As noted in this work, the IEMAC concept has been investigated since the 1970s [2]; however, the enabling technologies have only recently reached a level of maturity that facilitates on-board evaluation. Also noted in [6] is the fact that most control designs for the IEMAC problem are based on linearization of the spacecraft dynamic equation. With this fact in mind, [12] used the nonlinear dynamic equation presented in [6] along with a Modified Rodrigues Parameters-based kinematic representation to design an att i tude and power tracking control scheme us1This work is supported in part by the U.S. NSF Grants DMI9457967, DMI-9813213, EPS-9630167, ONR Grant N00014-99-10589, a DOC Grant, and an ARO Automotive Center Grant. 3435 ing an array of reaction wheels in an arbitrary non-coplanar configuration. Recently, [5] extended the approach of [12] for flywheels operating in a variable-speed control moment gyro mode. The goal of this paper is to develop nonlinear IEMAC strategies that simultaneously track a desired attitude trajectory and a desired energy/power profile. We consider a spacecraft model in which the flywheels are operated in a reaction wheel mode and the spacecraft attitude kinematics are parameterized by the unit quaternion. Using the backstepping [8] control design framework, we first develop a model-based IEMAC strategy that ensures asymptotic attitude, energy, and power tracking with no energy/power singularities (i.e., when the controller loses the capability of tracking the desired energy or power profile [12]). We then present a second control strategy that actively compensates for parametric uncertainties associated with the spacecraft inertia matrix. This adaptive controller also ensures asymptotic attitude, energy, and power tracking; however, the controller mandates some conditions on the flywheel angular velocity in order to ensure that energy/power singularities are avoided. The proposed IEMAC strategies have the following advantages in comparison to the work of [5, 6, 12]: (i) both controllers ensure tracking of both the desired energy and the power profiles, and hence, allow the spacecraft load requirements to be specified either in terms of energy or power demands, (ii) the model-based controller does not contain energy/power singularities; hence, an additional singularity avoidance scheme [12] is not necessary, (iii) the adaptive controller does not require exact knowledge of the spacecraft inertia, and (iv) the control strategies are not characterized by att i tude singularities since the spacecraft attitude kinematics are modeled using the unit quaternion representation as opposed to the Modified Rodrigues Parameters. The paper is organized as follows. In Section 2, we present the IEMAC system model while the control objective is stated in Section 3. The design of the model-based and adaptive controllers are presented in Sections 4 and 5, respectively. Section 6 contains the concluding remarks. 2 S y s t e m M o d e l We consider a rigid spacecraft with actuators that provide body-fixed torques about a body-fixed reference frame B located at the center of mass of the spacecraft. The bodyfixed torques are applied by an array of N (> 3) flywheels with fixed axis of rotation with respect to B (i.e., reaction wheel-type mode of operation). The dynamic model for the described IEMAC system is given by [7, 12] ]~, = h x J -~ (h A h f ) (1) hs = ~± (2) where h(t) E ~3 is the spacecraft angular momentum with respect to an inertial reference frame Z expressed in B, hf( t) E ~N is the axial angular momentum of the flywheels, J C ~ax3 represents the constant, positive-definite, symmetric spacecraft inertia matrix, A C ~axN is a constant matrix whose columns contain the axial unit vectors of the N flywheels, r f(t) C ~N is the control torque input, and the notation a x, Va = [al,ag.,aa] T, denotes the following skew-symmetric matrix a x A a3 0 a l . (3) --a2 al 0 Note that h(t) is related to Co(t) E ~ a the spacecraft angular velocity with respect to Z expressed in B, by the following equation h = JCo + Ahs , (4) where hs( t ) is related to Cos(t) E NN, the axial angular velocity of the flywheels, by 1 hs = JsCof, (5) The axial moments of inertia of the flywheels, Js C 5}~NxN, is the constant, positive-definite, diagonal matr ix defined as J f ~= d iag ( j s~ , j sg . , . . . , j fN ) (6) where the j f~'s denote the inertia of each flywheel and diag(.) denotes a diagonal matr ix with the enclosed diagonal terms. The kinematics of the spacecraft a t t i tude are represented by the unit quaternion [7] q(t) -~ {qo( t ) ,q( t )} C ~ x N 3 which describes the orientation of B with respect to Z expressed in B. Note that the unit quaternion is subject to the constraint 2 qo + qTq = 1. (7) The differential equation governing the at t i tude kinematics is given by 1 x o = ~(q ~+qo~) (8) qo 1 T q w . (9 ) 2 R e m a r k 1 The pseudo-inverse of the matr ix A defined in (1), denoted by A + E ~Nxa, is given by A+ A AT ( A A T ) -1 = such that A A + = Ia. (10) where Ia denotes the 3 x 3 identi ty matrix. As shown in [10], the above pseudo-inverse satisfies the so-called MoorePenrose Conditions given below A A +A = A A +A A + = A + (A+A) T = A+ A (AA+) T = AA+. (11) l In formulating (5), one must make the modeling assumption that the flywheels spin at a much higher speed than the spacecraft[12], 'i.e., w f (t) >> w(t). 343 In addition, the matrix I N A+A, which projects vectors onto the null space of A, satisfies the .following properties (IN -A+A) (IN -A + A ) = IN -A + A A = 0 I IN -A + A) 1 = IN -A +A IN -A + A) A + = O. (12) where IN denotes the N x N identi ty matrix. 3 C o n t r o l O b j e c t i v e The control objective is to design the control input r f ( t ) in (2) to ensure that: (i) the flywheels track a desired energy/power profile, and (ii) the spacecraft tracks a desired a t t i tude trajectory. We assume tha t the spacecraft attitude, spacecraft angular velocity, and flywheel angular velocities are measurable. Given that the total kinetic energy stored in the N flywheels due to their spin is given by E (t) = ~ r gCof (t)JfCos(t), the desired, stored kinetic energy will be defined as 1 r Ed (t) g -~coSd(t)Jscofd(t ) (13) where cola(t) E ~N represents the desired flywheel angular velocity yet to be specified. We can now quantify the energy storage tracking objective by the kinetic energy tracking error r/E(t) C ~ defined by