JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 25, No. 1, January– February 2002 Adaptive Control of Double-Gimbal Control-Moment Gyro with Unbalanced Rotor Jasim Ahmed ¤ Robert Bosch Corporation, Palo Alto, California 94303 and Dennis S. Bernstein † University of Michigan, Ann Arbor, Michigan 48109-2140 A double-gimbalcontrol-moment gyro (CMG) is modeled using Lagrange’s equations, and an adaptive feedback control law is developed to follow a commanded CMG motion. The control law does not require knowledge of the mass properties of the CMG. A Lyapunov argument is used to prove that command following is achieved globally with asymptotic convergence. Numerical simulationsare performed to illustrate the command following algorithm. A CMG testbed was designed and constructed to implement and demonstrate the adaptive algorithm. This testbed is described, and experimental results are given. I. Introduction A CTUATORS for spacecraft fall into two distinct categories, namely, inertial and noninertial. Inertial actuators provide torques to a spacecraft by reacting against inertial space and, thus, change the angular momentum of the spacecraft. Thrusters, which require fuel, are the principal type of inertial actuators. Magnetic actuators, which react to the Earth’s magnetic eld and which do not require fuel, also serve as inertial actuators. Noninertial actuators include reaction wheels, momentum wheels, and single-gimbal and double-gimbal control moment gy- ros (CMGs). Whereas these actuators require electric power, they do not requirefuel, and they do not changethe totalangularmomen- tum of the spacecraft. 1 Among noninertial actuators, the dual-axis CMG is the most sophisticated because it has the ability to apply control torques around arbitrary axes with the exception of singular orientationscorrespondingto gimbal lock. In applications,multiple CMGs are used for reliability, to avoid gimbal lock, and to avoid large angle motion. Becausea CMG is a multibodysystem, its dynamicsare complex, especially if large angle motion is allowed. Theoretical analysis of CMG dynamics is given in Ref. 2, whereas engineering aspects of CMG controlareconsideredin Refs.3– 5. In thepresentpaperwe are concerned with the problem of wheel imbalance in CMGs. At high rotational speeds (5,000– 10,000 rpm is typical for CMGs), small mass imbalances in the CMG wheel or due to asymmetric lubricant buildup can produce signicant disturbance forces at the frequency of the wheel angular velocity. 6 Mass imbalance effects also arise in momentum wheels where active isolation stages are used for distur- bancesuppression. 7 These studiessuggestthat noninertialactuators are often the source of the largest component of disturbance forces on spacecraft. In the present paper we model a double-gimbal CMG with un- known mass imbalance,mounted on a supportstructurexed to the Earth. Using Lagrange’s equations(see Ref. 8), we accountfor large anglemotionand the presenceof imbalance,which to the bestof our knowledge has not been done. A double-gimbalCMG testbed was designed and constructed to permit various control experiments to Received 26 June 2000; revision received 10 April 2001; accepted for publication 13 April 2001. Copyright c ° 2001 by the American Institute of Aeronautics and Astronautics, 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 Rose- wood Drive, Danvers, MA 01923; include the code 0731-5090/02 $10.00 in correspondence with the CCC. ¤ Systems Engineer, Research and Technology Center, 4009 Miranda Avenue; jasim.ahmed@rtc.bosch.com. † Professor, Department of Aerospace Engineering; dsbaero@engin. umich.edu. be performed. The control objective is to develop a control law that counteractsthe effects of mass imbalance without knowledgeof the mass distribution of the CMG while allowing the CMG to follow a commanded motion that includes unbounded rotational motion of the CMG gimbals and wheel. Adaptive control laws for mechanical systems with linear depen- dence on parameters that are able to follow a commanded motion have been developed. 9¡11 However, in Ref. 9 and 10, the class of commanded motions do not include those that permit unbounded rotational motion of the mechanical systems because the rotational motionof the systemisdescribedin terms ofanglesand theirderiva- tives. In Ref. 11, differentialgeometric techniquesare used to guar- antee convergence to a set consisting of four states, one of which is the desired state, but requiresknowledgeof boundsof the unknown parameters. In this paper, the rotational motion of the CMG is described by using trigonometric functions of the half-anglesof the gimbals and wheel, which transform unboundedCMG rotational motion to mo- tion on a compact set. This formulation permits the development of a control algorithm, which is presented in Sec. IV, that allows unbounded rotational motion of the CMG gimbals and wheel. A proof of the control law is based on a variant of standard Lyapunov arguments found in Ref. 12 to account for the motion on closed sets. The control law is a direct adaptive control law 13¡16 and has the form of a dynamic compensator whose order depends on the numberof uncertainparametersand whose statesprovideestimates of the wheel and gimbal moments of inertiaand centersof mass.Our controller can be viewed as an extension of the control law derived in Ref. 17 for spacecraft tracking with unknown mass distribution. The contents of the paper are as follows. In Sec. II, we describe the equations that govern the CMG motion; in Sec. III, we dene the command following problem; and in Sec. IV, we formulate a controlalgorithmthat permits the CMG to tracka desiredtrajectory. In Sec. V, we illustrate the adaptive control law using a numerical example, in Sec. VI, we describe the experimental setup, and in Sec. VII, we present experimentalresults. II. Equations of Motion In this section, we model the CMG as a system of interconnected rigid bodies and apply Lagrange’s equations for a system of rigid bodies to obtain the equations that govern the CMG’s motion. For a system of rigid bodies, Lagrange’s equations are given by d dt  @ L @ P q i ´ ¡ @ L @ q i D Q 0 i ; i D 1;:::; n (1) where n is the number of degrees of freedom, q 1 ;:::; q n 2< are the independentgeneralizedcoordinates, P q 1 ;:::; P q n 2< are the deriva- tives of q 1 ;:::; q n , and L D L .q ; P q / 2< is the Lagrangian of the 105