Mixed Control Moment Gyro and Momentum Wheel Attitude Control Strategies

AAS03-558 Mixed Control Moment Gyro and Momentum Wheel Attitude Control Strategies C. Eugene Skelton II and Christopher D. Hall Department of Aerospace & Ocean Engineering Virginia Polytechnic Institute & State University Blacksburg, Virginia 24061 This paper develops control strategies that use single-gimbal control moment gyros for coarse large angle slew maneuvering and momentum wheels for precise control and error reduction due to initial conditions. Lyapunov and feedback linearization control laws are developed. Numerical simulations of the derived mixed control moment gyro and momentum wheel control laws are evaluated. INTRODUCTION A spacecraft can use either internal or external actuators to control its attitude. Generally external actuators such as thrusters are used for large fast slewing maneuvers. However thrusters are not ideal for precision attitude control due to their discontinuous nature. Internal actuators can be momentum exchange devices, such as momentum wheels and control moment gyros, or non-moving devices like magnetic torquers. A momentum wheel is a high inertia flywheel mounted on a motor that can provide one axis control by changing the relative momentum of the vehicle. Momentum wheels can perform precise maneuvers and maintain attitude; however the varying wheel speeds tend to excite structural dynamics. Momentum wheels require torques proportionate to the desired output torque which makes them non-ideal for rapid slewing. A control moment gyro has a flywheel mounted on a motor that spins at a constant relative speed. The flywheel and motor are mounted to a gimbal motor that can rotate about an axis perpendicular to the spin axis of the flywheel. Control moment gyros (CMGs) can be double-gimballed or single-gimballed [1]. For the purpose of this paper a CMG refers to a single gimballed CMG. A control moment gyro can create high torques for slewing, that are dependant on the flywheel speed, inertia, and the gimbal angle rate, based on momentum exchange. Many control laws have been developed for control moment gyro applications such as the International Space Station [2, 3]. The disadvantage of CMGs is that certain sets of the gimbal angles can cause singularities where the CMG cluster is unable to provide torque about a given axis. Many papers have explored control laws that have various methods of avoiding CMG singularities including use of gimballed momentum wheels [4] or implementing optimization techniques to steer away from singularity configurations. Oh and Vadali [5] presented a Lyapunov control law for attitude and velocity control with CMGs. They argued that commanding gimbal velocities were ideal for attitude control of the body in order to take advantage of the so called torque amplification factor, rather than commanding gimbal accelerations which lead to undesirable gimbal rates. In Ref [6] Ford presented an extension of Ref. [5] by including the possibility of gimballed momentum wheels in the notation that we use in this research. Schaubet.al [7] presented an extension of the control law of Ref. [5] by changing the attitude representation from Euler Parameters to Modified Rodrigues Parameters. Both changes Graduate Research Assistant, cskelton@vt.edu. Student Member ASME. Professor, cdhall@vt.edu. Associate Fellow AIAA, Member AAS. 1

were made for singularity avoidance. The advantage of Ford s equations are that the actual motor torques are explicit for ease of implementation. Control laws developed to use thrusters for coarse slew control and momentum wheels for precise control or to reduce errors due to initial conditions have been proven to be effective [8]. This paper develops control strategies that use single-gimbal control moment gyros for coarse large slew maneuvering and momentum wheels for precise control and error reduction due to initial conditions. Mixing control moment gyros and momentum wheels for attitude control is believed to be a novel approach. The governing equations for a rigid body with control moment gyros and momentum wheels are presented much like that of Ref. [6]. Lyapunov control laws are developed from the governing equations. Feedback linearization techniques are used to transform the state equations and then choose a control input to cancel out the nonlinearities of the system dynamics. Simulations of the derived mixed control moment gyro and momentum wheel control laws are evaluated. The goal of this paper is to develop control laws that use control moment gyros and momentum wheels for large slew maneuvers. EQUATIONS OF MOTION The governing equations of a rigid body with gimballed momentum wheels have been developed in Ref. [6]. These equations are easily used to represent a body with any combination of control moment gyros, momentum wheels, and gimballed momentum wheels (GMW). We modify the equations slightly to represent a rigid body with only control moment gyros and momentum wheels. The governing equations represent the absolute angular momentum of the body about its center of mass, the absolute angular momenta of the actuators about their control axes, the gimbal angles, and the Modified Rodrigues Parameters from the body fixed frame to the inertial frame. These are the equations that are used to derive the mixed control laws. In this section we briefly review the equations derived by Ford [6] and the notation. Refs. [4, 6] contain complete derivations of these equations. We then use Ford s notation to present the governing equations for a rigid body with CMGs and MWs. GMW Equations of Motion The equations that describe the motion of a rigid body with N GMWs are where ḣ = ω h + g e (1) ḣ swa = g w (2) ḣ ga = diag((i t (I sg + I sw ))A T s ω I sw Ω)(A T t ω) + g g (3) δ = I 1 g h ga A T g ω (4) h = (I B + A t I t A T t + A s I sg A T s )ω + A s h swa + A g h ga (5) h swa = I sw A T s ω + A s Ω (6) h ga = I g A T g ω + I g δ (7) The absolute angular momentum of the system is h. The skew symmetric matrix is defined as ω = 0 ω 3 ω 2 ω 3 0 ω 1 (9) ω 2 ω 1 0 The external torques on the system are g e. The absolute axial angular momentum of the GMW flywheel is h swa and the torque on the flywheel about the same axis is g w. Equation 3 describes (8) 2

the change in absolute angular momentum of the GMW about its gimbal axis. The matrices I s, I t, and I g are diagonal matrices that represent the inertias of the GMWs about their spin, transverse, and gimbal axes, respectively. The spin axis inertia matrix is split into I sw and I sg to represent the inertia of the flywheel and GMW frame, respectively. The angular velocity of between the body-fixed reference frame and the inertial reference frame is ω. The relative speeds of the momentum wheels are Ω, and the gimbal angle rates are δ, where both are N 1. A matrix containing all of the gimbal angles is δ. The torques that the gimbal motors exert on the GMWs are g g. The matrices A s, A t, and A g contain the orientation of the GMWs spin, transverse, and gimbal axes, respectively. Note that A s and A t change with gimbal angle, while A g remains constant regardless of gimbal angle. The inertia of the body including transverse inertias of the momentum wheels are I B. CMG and Momentum Wheel Equations of Motion We wish to use this notation to define the equations of motion for a rigid body with N control moment gyros, and M momentum wheels. The momentum wheels are defined in a similar manner to the GMWs. Each momentum wheel has a reference frame F Wk (k = 1...M) that is fixed in the body. The vector components of the unit axes ( c sk, c tk, c gk ) are assumed to be given in F b. We define a matrix C s = [ c s1, c s2,..., c sk ] (10) to describe the orientation of the spin axis in the body. We form an additional inertia like matrix for the momentum wheels formed from the flywheels spin inertia as, I swmw = diag(i swmw 1, I swmw 2,..., I swmw M ) (11) The spin axis angular momentum of the k-th MW is not contained in the body angular momentum terms, so we form an M 1 matrix h swmw a = I swmw C T s ω + I swmw Ω MW (12) where Ω MW is an M 1 matrix formed with the relative angular velocities of the flywheels about their spin axis. We define the momentum wheel s change in absolute angular momentum about its spin axes as ḣ swmw a = g MW (13) The equations that describe the motion of each CMG are similar to the equations for a GMW. We exclude the time rate of change of absolute angular momentum of the CMG spin axis because an independent controller ensures that the angular velocity of the flywheel about its spin axis is constant relative to the CMG frame. Kinematics For this research we choose to use Modified Rodrigues Parameters (MRPs) because singularities can be avoided with use of the shadow set [9]. Euler s Theorem states that any rotation about a single fixed point can be described by a single rotation about a fixed axis through that point. The axis and angle are called Euler axis and Euler angle, respectively. It is common to define ê as the Euler axis and Φ as the Euler angle. The three element set of MRPs is defined as As presented in Ref. [9] we use a shadow set of MRPs σ = ê tan(φ/4) (14) σ S = 1 σ T σ σ (15) to avoid singularities. The shadow set replaces the original MRPs when σ T σ > 1. The magnitude of the MRP that switches to the shadow set is user defined. The conversion from MRPs to a direction cosine matrix is R = 1 + 4(1 σt σ) (1 + σ T σ) 2 σ + 3 8 (1 + σ T σ) 2 (σ ) 2 (16)

Knowing the angular velocity as a function of time we can determine the attitude by integrating σ = G(σ)ω (17) where ω is the angular velocity between the two frames and G(σ) = 1 ( 1 + σ + σσ T 1 + ) σt σ 1 2 2 (18) Summary The equations that describe the motion of a rigid body with N CMGs and M momentum wheels are where ḣ = ω h + g e (19) ḣ ga = diag((i t (I sg + I sw ))A T s ω I sw Ω)(A T t ω) + g g (20) ḣ swmw a = g MW (21) δ = I 1 g h ga A T g ω (22) σ = G(σ)ω (23) h = (I B + A t I t A T t + A s I sg A T s )ω + A s h swa + A g h ga + C s h swmw a (24) h swa = I sw A T s ω + A s Ω (25) h ga = I g A T g ω + I g δ (26) h swmw a = I swmw C T s ω + I swmw Ω MW (27) CONTROL LAWS We wish to define control laws that use the torque amplification of CMGs for large rapid slew and the fine tuning abilities of momentum wheels. In this section we present the reference body dynamics and develop the state equations for this problem. We present a CMG feedback control law developed by Oh and Vadali [5]. This control law allows us to find the open loop CMG commands for the tracking maneuver. Derivation of Lyapunov and feedback linearization control laws conclude this section. Reference Body The use of a reference body for tracking follows that of Ref. [8]. We start by defining a reference trajectory. Consider a rigid reference body that has a body-fixed reference frame F R. The center of mass of the reference body is coincident with the origin of F R and is fixed in inertial space. The orientation of F R with respect to the inertial reference frame F I is expressed by the direction cosines matrix R RI. The MRPs that describe R RI are R RI = 1 + 4(1 σt R σ R) (1 + σ T R σ R) 2 σ R + 8 (1 + σ T R σ R) 2 (σ R )2 (28) The equations that describe the motion of the reference body are ḣ R = ω R I Bω R + g R (29) σ R = G(σ R )ω R (30) Note that the reference body has the same inertia of the controlled body without the actuator inertias. 4

State Equations With the reference body and the actual body fully defined we can form the terms δω and δσ which are the difference between the body and reference angular velocity and attitude, respectively: From these equations we define δσ = (1 σt R σ R)σ (1 σ T σ)σ R + 2σ σ R 1 + (σ T R σ R)(σ T σ) + 2σ R σ (31) δω = ω R BR ω R (32) δh = h I B R BR ω R (33) the difference between the angular momentum of the bodies, where R BR = R(δσ). Substituting equation 24 into δh and simplifying gives δh = I B ω + (A t I t A T t + A s I sg A T s )ω + A s h swa + A g h ga (34) + C s h swmw a I B R BR ω R = I B δω + (A t I t A T t + A s I sg A T s )ω + A s h swa + A g h ga (35) + C s h swmw a Taking the time derivative of equation 33 yields Hall et.al. [8] showed that δḣ = ḣ I B dr BR dt ω R I B R BR ω R (36) dr BR I B ω R = I B ω δω (37) dt Taking the time derivative of Equation 35 gives δḣ = I Bδ ω + Ȧsh swa + A s ḣ swa + A g ḣ ga + C s ḣ swmw a (38) +(ȦtI t A T t + A t I t Ȧ T t + ȦsI sg A T s + A s I sg Ȧ T s )ω + (A t I t A T t + A s I sg A T s ) ω The state equations are the attitude and angular velocity error rates. Equation 17 to express the states as We use Equation 38 and δ σ = G(δσ)δω (39) δ ω = I 1 B [δḣ Ȧsh swa A s ḣ swa A g ḣ ga (A t I t A T t + A s I sg A T s ) ω (40) (ȦtI t A T t + A t I t Ȧ T t + ȦsI sg A T s + A s I sg Ȧ T s )ω C s g MW ] Open Loop CMG Control We use a gimbal velocity law first presented by Oh and Vadali [5] to find the CMG control maneuver for a large rapid slew. Presented in this notation the gimbal torques are where g g = I g δ + I g A T g ω diag((i t I s )A T s ω I sw Ω)A T t ω (41) δ = k δ ( δ des δ) (42) δ des = D (k 3 δω + k 4 δσ JR BR ω R J 1 ω (Jω + A s h swa + A g h ga )) (43) D = A t diag(i sw Ω) (44) +1/2[(a s1 a T t1 + a t1 a T s1)(ω + R BR ω R ) (a s2 a T t2 + a t2 a T s2)(ω + R BR ω R )......(a sn a T tn + a tn a T sn )(ω + R BR ω R )](I t I s ) The coefficients k δ, k 3, and k 4 are positive constant scalars. 5

Lyapunov Controller We have developed the equations that we need to derive the control law. candidate function Consider a Lyapunov where K 1 = K T 1 > 0 and k 2 > 0. The derivative of V is where Equation 47 can be expanded to V = 1 2 δωt K 1 δω + 2k 2 ln(1 + δσ T δσ) (45) V = ḢT I 1 B K 1δω + k 2 δσ T δω (46) Ḣ = δḣ Ȧsh swa A s ḣ swa C s ḣ swmw a A g ḣ ga (47) (ȦtI t A T t + A t I t Ȧ T t + ȦsI sg A T s + A s I sg Ȧ T s )ω (A t I t A T t + A s I sg A T s ) ω Ḣ = ω ((I B + A t I t A T t + A s I sg A T s )ω + A s h swa + A g h ga (48) +C s h swmw a ) + g e I B ω δω I B R BR I 1 B [ ω R Iω R + g R ] Ȧsh swa A s g w C s g MW A g (diag((i t I s )A T s ω I sw Ω)(A T t ω) + g g ) (ȦtI t A T t + A t I t Ȧ T t + ȦsI sg A T s + A s I sg Ȧ T s )ω (A t I t A T t + A s I sg A T s ) ω With Ḣ in expanded form we identify the control torques for the body g e, g w, g g, and g MW. For any CMG to maintain a constant velocity of the flywheel about its spin axis relative to the body the spin torque is g w = I sw (ȦT s ω + A T s ω) (49) For this problem we consider g e, g w, and g g as known parameters. With the g MW insure that V < 0 when δω 0 and δσ 0. By choosing K 1 = I B equation 46 becomes input we can We choose the control torque g MW to be which yields V = δω T (Ḣ + k 2δσ) (50) C s g MW = δḣ Ȧsh swa A s g w A g ḣ ga (A t I t A T t + A s I sg A T s ) ω (51) since k 1 > 0. From this result we see that With Equations 35, 38, and 51, we have (ȦtI t A T t + A t I t Ȧ T t + ȦsI sg A T s + A s I sg Ȧ T s )ω +k 2 δσ + k 1 δω V = k 1 δω T δω 0 (52) lim δω(t) = 0 (53) t I B δ ω = k 2 δσ k 1 δω (54) 6

Due to Equation 53 we have lim Jδ ω = lim k 2δσ (55) t t so that δσ 0. From Equation 39 we know that δ σ 0, so that the limit of δσ is a constant, therefore lim δσ = 0 (56) t 0 With LaSalle s Theorem [10], the attitude and angular velocity error due to tracking with this control law is globally asymptotically stable. Input-State Feedback Linearization Controller We now take a different approach to controlling this problem. For a certain class of nonlinear systems we can use a state feedback control and a change of variables that transforms the nonlinear system into an equivalent linear system [10]. We investigate input-state linearization of this nonlinear system. We begin with the state equations 39 and 40 which we transform with The transformed state equations are z 1 = δσ (57) z 2 = G(δσ)δω = δ σ (58) ż 1 = z 2 (59) ż 2 = Ġδω + Gδ ω (60) where Ġ = G(δ σ). Substitution of Equation 40 allows the state equations to be written as (61) ż 1 = z 2 (62) ż 2 = Ġδω + G(I 1 B (δḣ Ȧsh swa A s ḣ swa A g ḣ ga (63) (ȦtI t A T t + A t I t Ȧ T t + ȦsI sg A T s + A s I sg Ȧ T s )ω (A t I t A T t + A s I sg A T s ) ω C s g MW ) In this form we can choose C s g MW to cancel the nonlinear terms. We choose the momentum wheel control torques to be where (64) C s g MW = δḣ Ȧsh swa A s ḣ swa A g ḣ ga (65) (ȦtI t A T t + A t I t Ȧ T t + ȦsI sg A T s + A s I sg Ȧ T s )ω (A t I t A T t + A s I sg A T s ) ω + I B ν The constants k 5 and k 6 are positive. The system with control is ([ ] [ 0 1 k6 G 0 ż = + 0 Ġ 0 k 5 G ν = k 5 δω + k 6 δσ (66) ]) z (67) With the system represented linearly we can use linear stability analysis. From linear theory we can guarantee asymptotic stability if the coefficient matrix is Hurwitz [10]. 7

NUMERICAL SIMULATION We present the results of numerical simulations of the derived control laws for illustration. We first present the feed forward control inputs as calculated by the closed loop control moment gyro control law. Then we show how slight variations in initial conditions make the system diverge (as expected) from the desired reference conditions with only CMG feed forward inputs. Implementation of the closed loop momentum wheel control laws show that initial errors are rejected with minimal torques. We define the actuator orientations in F B which is aligned with the principal inertia reference frame. The principal inertia matrix is I B = diag(6, 5, 10) kg m 2. Three CMGs are used for the feed forward control aligned such that A s0 = A g = 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 (68) (69) All three CMGs have the same inertia properties, I sw = 0.01 kg m 2, I sg = 0.01 kg m 2, I t = 0.01 kg m 2, and I g = 0.01 kg m 2. The nominal relative angular velocity of each CMG flywheel is 2000 rpm. Three momentum wheels are aligned with the principal axes with I sw = 0.04 kg m 2 for each momentum wheel. The transverse inertias of the momentum wheels along with inertias of the torque devices for the momentum wheels are included in the body inertia. The reference motion to be tracked is a 90 o slew about the 3 axis of the body. The reference motion is generated with bang-bang external torques to simulate the motion of a body with thrusters. The Modified Rodrigues Parameter representation of the reorientation of the reference body is shown in Figure 1. The gains for determining the control moment gyro maneuvering for the reorientation are k δ = 100, k 4 = 100, and k 3 = 2I ii k 4. Figure 2 shows the attitude and velocity error for the body with CMG feedback control. We now take the exact inputs to the CMGs that produced the previous result and implement them as feed forward inputs. Figure 3 shows the feed forward CMG control with an initial error of σ = [0.01, 0.02, 0.01]. Clearly without feedback control, any initial error or perturbation will result in errors in tracking and final orientation. We apply the Lyapunov momentum wheel feedback control laws. The gains are k 1 = 54 and k 2 = 47. Figure 4 shows that the momentum wheels correct for initial error while allowing the majority of the torque for the slew to come from the CMGs. Figure 5 shows the momentum wheel applied torques throughout the maneuver. Numerical simulations of the feedback linearization control law were performed. The linear gains were chosen to be k 5 = 5 and k 6 = 4 for this simulation. Figure 6 is the tracking error. The control torques that the momentum wheels apply to the system are shown in Figure 7. CONCLUSIONS The objective of this research is to derive and validate mixed CMG and momentum wheel control laws. A control moment gyro gimbal rate feed forward control law provides output torques required for large rapid angular slew maneuvers. Lyapunov and feedback linearization techniques are used to develop momentum wheel control laws for initial condition and tracking error rejection. The controllers are derived from the nonlinear state equations for attitude and angular velocity error. Both control laws guarantee asymptotic convergence. Numerical simulations are preformed to verify convergence of the system with the derived control laws. Momentum wheel control torques for each controller are small in comparison to the torques required for the angular slew maneuver. (70) 8

Figure 1: MRP to be tracked Future work will include implementation of the mixed actuator attitude control laws on the Distributed Spacecraft Attitude Control System Simulator (DSACSS) at Virginia Tech. Exploration of momentum wheel feedback laws that do not have knowledge of the open loop CMG dynamics will be preformed. Power studies of mixed CMG and momentum wheel attitude control versus momentum wheel feedback control must be conducted to verify the advantage of the mixed actuator attitude control concept. References [1] J. R. Wertz and W. J. Larson, Space Mission Analysis and Design. New York, New York: Microcosm Press, 1999. [2] S. J. P. Robert H. Bishop and J. W. Sunkel, Adaptive control of space station with control moment gyros, IEEE Control Systems, vol. 12, pp. 23 28, October 1992. [3] S. N. Singh and T. C. Bossart, Exact feedback linearization and contol of space station using CMG, IEEE Transactions on Automatic Control, vol. 38, pp. 184 187, January 1993. [4] K. Ford and C. Hall, Singular direction avoidance steering for control-moment gyros, Journal of Guidance, Control, and Dynamics, vol. 23, pp. 648 656, July 2000. [5] H. Oh and S. Vadali, Feedback control and steering laws for spacecraft using single gimbal control moment gyros, Journal of the Astronautical Sciences, vol. 38, pp. 183 203, April 1991. [6] K. A. Ford, Reorientations of Flexible Spacecraft Using Momentum Exchange Devices. PhD thesis, Air Force Institute of Technology, 1997. 9

Figure 2: Tracking Reference Body with CMG feedback [7] H. Schaub, S. R. Vadali, and J. L. Junkins, Feedback control law for variable speed control moment gyros, Journal of the Astronautical Sciences, vol. 46, pp. 307 328, July 1998. [8] C. D. Hall, P. Tsiotras, and H. Shen, Tracking rigid body motion using thrusters and momentum wheels, The Journal of the Astronautical Sciences, vol. 50, no. 3, 2002. [9] M. D. Shuster, A survey of attitude representations, The Journal of the Astronautical Sciences, vol. 41, no. 4, pp. 439 518, 1993. [10] H. K. Khalil, Nonlinear Systems. New York, New York: Macmillian, 1992. 10

Figure 3: Tracking Reference Body with CMG Feed Forward and Initial Error Figure 4: Tracking Reference Body with CMG Feed Forward,Initial Error, and MW Lyapunov feedback control 11

Figure 5: MW Lyapunov feedback control torques Figure 6: Tracking Reference Body with CMG Feed Forward, Initial Error, and MW feedback linearization 12

Figure 7: MW feedback linearization torques 13