JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 24, No. 1, January– February 2001 Adaptive Control of Nonlinear Attitude Motions Realizing Linear Closed Loop Dynamics Hanspeter Schaub ¤ Sandia National Laboratories, Albuquerque, New Mexico 87185 Maruthi R. Akella † University of Texas at Austin, Austin, Texas 78712 and John L. Junkins ‡ Texas A&M University, College Station, Texas 77843 An adaptive attitude control law is presented to realize linear closed loop dynamics in the attitude error vector. The Modied Rodrigues Parameters (MRPs) are used along with their associated shadow set as the kinematic variables since they form a nonsingular set for all possible rotations. The desired linear closed loop dynamics can be of either PD or PID form. Only a crude estimate of the moment of inertia matrix is assumed to be known. A nonlinear control law is developed which yields linear closed loop dynamics in terms of the MRPs. An adaptive control law is then developed that enforces these desired linear closed loop dynamics in the presence of large inertia and external disturbance model errors. Because the unforced closed loop dynamics are nominally linear, standard linear control methodologies such as pole placement can be employed to satisfy design requirements such as control bandwidth. The adaptive control law is shown to track the desired linear performance asymptotically without requiring apriori knowledge of either the inertia matrix or external disturbance. Introduction W HILE the traditional approach to attitude control is based on linear control theory, recent efforts by several authors indicate a shift toward nonlinear control methods. For example, Wie and Barba 1 and Wei et al. 2 develop the rotational equations of motion using the redundant set of Euler parameters. In contrast, Dwyer 3, 4 outlinesan approachbased on a minimal set of three Euler parameters wherein a nonlinear transformation maps the complete equations of motion into a locally valid linear model that may en- counter singular attitudes. It is a well known fact that every three- parameter attitude representation has the problem of singularities. The work of Slotine and Li based on Euler angles also has the same limitation. 5 To address the problem of singular orientations while using a minimal set of three rigid body attitude coordinates, more recently the Modied Rodrigues Parameters (MRPs) have been proposed. Any rigid body orientation can be described through two numer- ically distinct sets of MRPs which abide by the same differential kinematic equation. By switching between the original and alter- nate sets of MRPs (also referred to as the shadow set), it is possible to achieve a globally nonsingular attitude parameterization for all possible §360 deg rotations. 6 ¡9 Given this advantage, there have been several recent attitude control applications employing MRPs as rotational kinematic variables. 10¡ 13 A common feature within all these efforts and other developments by Wen and Kruetz-Delgado 14 , Wen et al., 15 Meyer, 16,17 Reyhanoglu et al., 18 and Slotine and Li 5 is the control law that is based on a stability analysis driven by an associatedLya- punov analysis. Although such attitude feedback control laws can be found by rst dening a candidate Lyapunov function and then extracting the corresponding stabilizing nonlinear control, certain Received 29 October 1999; revision received 28 February 2000; accepted for publication 17 April 2000. Copyright c ° 2000 by the authors. Pub- lished by the American Institute of Aeronautics and Astronautics, Inc., with permission. ¤ Research Engineer, Sandia National Laboratories, Mail Stop-1003; hschaub@sandia.gov. Member AIAA. † Assistant Professor, Department of Aerospace Engineering and Engi- neering Mechanics; makella@mail.utexas.edu. Member AIAA. ‡ George J. Eppright Professor, Aerospace Engineering Department; junkins@tamu.edu. Fellow AIAA. very important concepts from linear control theory, such as closed loop damping and bandwidth, are not very well dened because the correspondingclosed loop dynamicsare generallynonlinear.To achieve a desired closed loop behavior, the closed loop dynamics are linearized about a reference motion in order to use linear con- trol theory techniques to pick the feedback gains. Depending on the nonlinearity of the exact closed loop equations of motion, the desired closed loop performance will be achieved only in a local neighborhoodand not globally. Instead of rst nding a feedback control law and then analyzing the closed loop dynamics stability, it is possible to start out instead with a desired(or prescribed) setof stableclosedloopdynamicsand then extract the correspondingnonlinear control law using a feed- back linearization approach 4, 14 common in robotics path planning problems. For example, the closed loop dynamics could be a stable linear differential equation. This technique is very general and can be applied to a multitude of systems. However, depending on the nonlinearity of the dynamical system, the nonlinear control laws extracted from such a feedback linearization approach can be po- tentially very complex. Paielli and Bach 19 present such an attitude control law derived in terms of the Euler parameter components, and that law is remarkably simple. Compared to standard Lyapunov functionderivedattitudecontrollaws,theircontrollawexpressionis only slightly more complex. Further, Paielli and Bach illustratethat thistypeof controllaw is ratherrobustfor attitudecontrolproblems. However, this controllaw feeds back the Gibbs vector 7 as an attitude measurethat is singularat §180deg (error)rotationsaboutany axis. As an important contribution of this paper, we develop a feedback linearizing control law based on the MRP vector that achieves the desired set of stable closed loop trajectories without encountering singular orientations. This paper also addresses the issue of uncer- tainty in the moment of inertia matrix. Even if the attitude control law (based on some nominal value of inertia) is robust with respect to inertia uncertainties,the closed loop dynamics will no longer ex- hibit the desired performanceif an incorrectinertia matrix is used in the feedback control law. While the inertia matrix is assumed to be essentially unknown in this development, the goal is to ensure that the feedbackcontrollaw would still producethe desiredclosedloop dynamics.To accomplishthistask, time-varyingupdatelaws for the feedback gain matrices are developed that ensure stability for the dynamics of the system adaptively. While classical adaptive con- trol theory due to Narendra 20 and Sastry 21 has also been employed 95