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Adaptive Spacecraft Attitude Control Using Single-Gimbal Control Moment Gyroscopes Without Singularity Avoidance

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Adaptive Spacecraft Attitude Control Using Single-Gimbal Control Moment Gyroscopes Without Singularity Avoidance JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Adaptive Spacecraft Attitude Control Using Single-Gimbal Control Moment Gyroscopes Without Singularity Avoidance Zhanzhan Zhao,∗ Gerardo Cruz,† and Dennis S. Bernstein‡ University of Michigan, Ann Arbor, Michigan 48109 DOI: 10.2514/1.G003926 This paper applies retrospective cost adaptive control (RCAC) to spacecraft attitude control with a rotation-matrix attitude parameterization using constant-speed single-gimbal control moment gyroscopes. Unlike control laws that use torque-steering laws to synthesize torque requests while avoiding gimbal singularities, the adaptive control law requests the rate of each gimbal. Because no torque request is provided, there is no need to invert the mapping from the gimbal rates to the control torque, and thus no attempt is made to avoid gimbal singularities. The RCAC implementation is based on a target model involving a single Markov parameter corresponding to the initial gimbal configuration. The parameters and weights for RCAC are based on nominal-model tuning, which uses no explicit knowledge of the nonlinear equations of motion. This paper investigates the robustness of RCAC to off-nominal conditions of commands, noise, and unknown bus inertia; and it examines the performance of RCAC in the case where an approximately singular gimbal configuration occurs during attitude command, following as well as the case of an initial gimbal-lock singularity. The performance of the adaptive controller in the presence of sensor/actuator misalignment is also investigated. Nomenclature k = discrete-time step kw = initial waiting period A = attitude parameter matrix diag a1;a2;a3 lz = dimension of z a1;a2;a3 = attitude parameters lu = dimension of u BCMG;B1;B2 = control input matrices Θ c = center of mass of the spacecraft lΘ = dimension of mi = mass of Wi ci = center of mass of the ith wheel e = rotation eigenaxis for the rotated bus inertia n = number of the gimbals ⇀ nc = controller order O matrix J B;rot B∕G = direction cosine matrix obtained by resolving × i ! ei = ith column of the 3 3 identity matrix I3 R G ∕B in FB 0 i e = misalignment eigenaxis from F to F 0 O 0 B B B ∕B =! direction cosine matrix obtained by resolving FB = bus-fixed frame R B∕B 0 in FB 0 FB 0 = sensor-fixed frame O 0∕ = direction cosine matrix obtained by resolving B Gi ! F = frame representing the attitude command for F 0 0 C B R Gi∕B in FB F = frame fixed to gimbal G O O Gi i i = B∕Gi F = inertial frame O 0 O 0 I i = B ∕Gi FWi = frame fixed to wheel Wi Pi;k = controller coefficient matrix Gi = ith gimbal Qi;k = controller coefficient matrix Gf q = finite impulse response filter used by q = forward shift operator R retrospective cost adaptive control B∕I = direction cosine matrix obtained by resolving H = matrix that models the initial gimbal configu- ! R B∕I in FB ration used in Gf R C∕I = direction cosine matrix obtained by resolving I3 = 3 × 3 identity matrix ! ! R C∕I in FC J = inertia tensor J SC∕c of the spacecraft relative η Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 Ru = control weighting matrix uIlu to c resolved in FB η ! Rz = performance weighting matrix zIlz J = inertia tensor J ∕ of the bus relative to c η B B c RΘ = regularization weighting matrix ΘIlΘ resolved in F ~ B R = attitude-error rotation matrix between RB∕I and J = rotated bus inertia matrix R B;rot ! ! C∕I J = inertia tensor J ∕ of W relative to c i Wi ci i i R B∕B 0 = rotation tensor that rotates FB 0 into FB resolved in FW ! !i R B∕I = rotation tensor that rotates FI into FB Ji;c = inertia tensor J W ∕c of Wi relative to c resolved ! i R C∕I = rotation tensor that rotates FI into FC in FB ! ^ Θ^ R ∕ = rotation tensor that rotates F into F Jk = retrospective cost function !Gi B B Gi 0 0 R Gi∕B = rotation tensor that rotates FB into FGi ri = position of ci relative to c resolved in FB Received 22 June 2018; revision received 8 June 2019; accepted for S = vector representation of the attitude error R~ publication 1 July 2019; published online 19 August 2019. Copyright © 2019 s = scalar attitude-error metric by Dennis S. Bernstein. Published by the American Institute of Aeronautics ϕ and Astronautics, Inc., with permission. All requests for copying and Ts = settling time to bring and maintain eig within permission to reprint should be submitted to CCC at www.copyright.com; ϕ∞ employ the eISSN 1533-3884 to initiate your request. See also AIAA Rights t = continuous-time variable and Permissions www.aiaa.org/randp. u = vector of gimbal rates *Graduate Student, Department of Aerospace Engineering; zhanzhao@ ui = gimbal rate of Gi around j^G relative to FB umich.edu. i † Wi = ith wheel Staff Member, NASA Goddard Space Flight Center; gerardo.e.cruz- z = performance variable [email protected]. ‡ z^ = retrospective performance variable Professor, Department of Aerospace Engineering; [email protected]. k Article in Advance / 1 2 Article in Advance / ZHAO, CRUZ, AND BERNSTEIN αi = moment of inertia of Wi around the spin axis In contrast to CMG-driven attitude control laws based on torque- ^ iWi steering laws, the present paper focuses on direct gimbal-rate-requested β = pyramidal face angle adaptive control of single-gimbal constant-speed CMGs without a β ^ ^ i = moment of inertia of Wi around jWi and kWi torque-steering law. In Fig. 2, the controller directly requests the gimbal Θk = controller coefficient vector rates u without a torque-steering law, and thus without specifying a Θ^ = retrospectively optimized controller coefficient requested torque. The gimbal motors implement the requested gimbal vector rates u that, as a result of the CMG dynamics, produce the actuation θ = rotation eigenangle for the rotated bus inertia torque τCMG applied to the spacecraft. Because no torque request is matrix JB;rot specified, there is no need to invert the mapping from the gimbal rates to θi = rotation angle of Gi the control torque, and thus no attempt is made to avoid gimbal θ 0 = misalignment eigenangle from FB to FB 0 . singularities. The direct gimbal-rate approach was used in Refs. [31– ν i = spin rate of the frame FW relative to FG around 33]. In particular, the approach of Ref. [31] was confined to open-loop ^ i i iWi control, Ref. [32] linearized the nonlinear system and applies an ξC = commanded eigenaxis resolved in FI optimal control law, and Ref. [33] applied sliding mode control. τ CMG = torque applied to the spacecraft by the control Nevertheless, approximate gimbal singularities can occur using moment gyroscopes the direct gimbal-rate approach. In these cases, the impact of τdist = torque applied to the spacecraft by the singularities encountered during a maneuver is limited due to the fact disturbances that the control input matrix B is time dependent, and thus rank- τ CMG G = torque applied to the spacecraft by Gi deficient conditions tend to hold at only isolated points. This property Φ i k = regressor matrix thus facilitates a simplified approach to the use of constant-speed ϕ C = commanded eigenangle CMGs for spacecraft attitude control. ϕ eig = rotation angle around the eigenaxis that rotates The adaptive control law adopted in the present paper is R B∕I t to RC t retrospective cost adaptive control (RCAC). RCAC was developed ϕ ϕ ∞ = bound on eig for linear systems in Refs. [34–36]. As shown in the present paper, ω B = angular velocity of FB relative to FI resolved in RCAC requires extremely limited modeling information for a FB spacecraft with CMG actuation, namely, geometric information ω C = angular velocity of FC relative to FI resolved in concerning the initial gimbal configuration. The ability of RCAC to FC control the spacecraft with limited modeling information is ω ~ B = angular velocity of FB relative to FC resolved in demonstrated by tuning RCAC in numerical simulations based on a FB nominal model and by subsequently evaluating its performance using ω⇀ B∕I = angular velocity of FB relative to FI a perturbed model. In this approach, the spacecraft equations of ω⇀ C∕I = angular velocity of FC relative to FI motion are used only for numerical simulation; no explicit knowledge of these equations is used by RCAC. Closed-loop stability for linear systems using RCAC was addressed in Ref. [35]. I. Introduction Proof of closed-loop stability for nonlinear systems in general and CMG control of spacecraft is an open problem. CONTROL moment gyroscope (CMG) consists of a spinning A pyramidal CMG arrangement consisting of four CMGs is A wheel mounted on one or two gimbals; the wheel may considered in this paper. The RCAC law is based on a rotation-matrix spin at either a constant or variable rate. The gimbal or gimbals attitude parameterization, which avoids singularities arising with ’ rotate the wheel s spin axis relative to the spacecraft bus, thereby Euler angles and Rodrigues parameters as well as the double covering applying a reaction torque to the spacecraft. The high torque levels and arising from quaternions. These issues and their ramifications were accuracy provided by CMGs make them well suited for attitude control discussed in Refs. [37,38]. – of large space stations, orbiting telescopes, and agile spacecraft [1 3]. The specific contributions of this paper are as follows: – CMGs are typically operated using torque-steering laws [2 6]. In 1) In contrast with torque-steering laws, a gimbal-rate controller is § τ Fig.
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