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

JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS

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 diaga1;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∕It to RCt 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. 1, the controller requests the torque req, and the torque-steering shown to achieve approximate singularity passthrough and escape law synthesizes the requested gimbal-rate control input u, resulting in from a singularity. τ the actuation torque CMG applied to the spacecraft. When the control 2) The adaptive controller uses no explicit knowledge of the τ input matrix BCMG becomes rank deficient, the torque CMG along spacecraft equations of motion, including the bus inertia and one direction cannot be generated, and thus all possible torque nonlinearities. Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 vectors are coplanar [4], which indicates a singular gimbal 3) A robustness study demonstrates the performance of the configuration. In Ref. [7], geometric methods are used to determine adaptive controller over a wide range of unknown inertia matrices, the CMG singular surfaces. If a CMG singularity can be reconfigured and in the presence of sensor/actuator misalignment. These results by null motion BCMGtut0 into a nonsingular gimbal extend preliminary results given in Refs. [39,40] and complement the configuration, it is called hyperbolic or passable; otherwise, it is application of RCAC to spacecraft with reaction wheels, magnetic called elliptic or impassable [8]. torquers, and an appendage in Refs. [41–43]. Singularities are inherent in every array of constant-speed CMGs [5]. The contents of the paper are as follows. Section II enumerates At the expense of greater hardware complexity, gimbal singularities do assumptions for a spacecraft actuated by a collection of constant- not arise with variable-speed CMGs [9,10]. Alternatively, Refs. [11,12] speed single-axis CMGs, and it summarizes the equations of motion. considered CMGs under rotor offset to avoid singularities. Singularities Section III defines the performance variable for the attitude control can be mitigated using singularity-avoidance torque-steering laws [13– problem. Section IV briefly reviews RCAC. Section V describes the 23]. In particular, rapid escape from impassable internal singularities pyramidal arrangement of four CMGs and constructs the target model was discussed in Refs. [18,19,24]. needed by RCAC for an initial gimbal configuration. Section VI Torque-steering laws based on adaptive control algorithms have investigates the performance of RCAC for a spacecraft controlled by been applied to both constant-speed and variable-speed CMGs. In CMGs in the pyramidal arrangement with sensor noise, off-nominal particular, adaptive control of constant-speed single-gimbal CMGs commands, an unknown bus inertia matrix, an approximate was considered in Refs. [25–27]; adaptive control of double-gimbal singularity passthrough, escape from an initially singular gimbal CMGs was considered in Ref. [28]; and adaptive control of variable- configuration, and sensor/actuator misalignment. Finally, the speed CMGs was considered in Refs. [29,30]. Appendix provides a derivation of the equations of motion for the spacecraft controlled by CMGs. This derivation shows explicitly how § Note that eatt attitude error and ratt attitude command in Figs. 1 and 2 each assumption is used to arrive at the equations of motion used for only. nominal-model tuning and perturbed-model testing. Article in Advance / ZHAO, CRUZ, AND BERNSTEIN 3

Controller Torque-Steering Law Spacecraft

Fig. 1 The controller requests the torque τreq, and the torque-steering law requests the gimbal rates u.

Controller Spacecraft

Fig. 2 The controller directly requests the gimbal rates u without a torque-steering law.

II. Spacecraft Attitude Dynamics with Direct Gimbal- Rate Request The dynamics of a spacecraft SC consisting of a bus B and n single- gimbal constant-speed CMGs in an arbitrary arrangement are considered. For i 1; :::;n, CMGi is composed of wheel⇀ Wi mounted on gimbal Gi as shown in Fig.! 3. Vectors denoted by r and Fig. 3 Notation and conventions for CMGi. second-order tensors denoted by R are component free.¶ Differentiation of component-free vectors and tensors is performed Assumption 1: The bus and wheels are rigid bodies. with respect to a frame; differentiation of resolved vectors and tensors Assumption 2: The gimbals are massless. does not require a frame. Assumption 3: For all i 1; :::;n, the center of mass ci of Wi is fixed in the bus. A. Kinematics and Frames Assumption 4: For all i 1; :::;n, the direction of the axis i^W is ^ ^ i Let FI be an inertial frame, and let FB be a bus frame defined by the fixed in FGi . In particular, iWi iGi . pyramidal CMG arrangement as discussed in Sec. V. The rotational Assumption 5: For all i 1; :::;n, the wheel Wi is inertially attitude kinematics of the bus are described by Poisson’s equation symmetric around i^W . i ^ Assumption 6: For all i 1;:::;n, the frame FWi spins around iWi B• at the constant rate νi > 0 relative to FG . ! ! ω⇀× i R B∕I R B∕I B∕I (1) Assumption 7: For all i 1;:::;n, gimbal Gi is actuated by requesting its rate u around j^ relative to F . ! i Gi B • Figure 3 and Assumptions 4 and 5 imply that the inertia matrix of where B denotes the frame derivative with respect to FB, ⇀R B∕I is the ω W relative to c resolved in both F and F is given by rotation tensor that rotates FI into FB (see footnote ¶), B∕I is the i i Gi Wi “×” angular velocity of FB relative to FI, and the superscript denotes 2 3 the skew-symmetric cross-product matrix. For all i 1; :::;n, the α 00 ! ! 4 i 5 frame FG is fixed to Gi, and the frame FW is fixed to Wi. J ≜ J ∕ j J ∕ j 0 β 0 (6) i ! i i Wi ci Gi Wi ci Wi i Resolving the second-order tensor R ∕ yields the rotation matrix B I 00βi R ≜ ! ! α B∕I R B∕I R B∕I (2) where i is the moment of inertia of Wi around the spin axis B I ^ ^ β iWi iGi , and i is the moment of inertia around the remaining axes

of FWi and FGi . Therefore, Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 which is a 3 × 3 direction cosine matrix. The second equality is due to the fact that the rotation tensor represents a rotation about an ! ! T T J ∕ j O J ∕ j O O J O (7) O Wi ci B i Wi ci Gi i i i i eigenaxis that is fixed in both frames. The orientation matrix B∕I is defined by and the 3 × 3 orientation matrix Oi is defined by O RT R B∕I B∕I I∕B (3) ! ! O ≜ O ∕ R ∕ j R ∕ j (8) i B Gi Gi B Gi Gi B B × ⇀ which is also a 3 3 direction cosine matrix. If x is a coordinate-free ! vector, then where R G ∕B is the rotation tensor that rotates FB into FG . For a i ⇀ ⇀ O ⇀ i coordinate-free vector x, xjB i xjGi . ⇀ O ⇀ Next, define xjB B∕I xjI (4) ⇀ ⇀ ! T ×2 J ≜ J ∕ j O J O − m r ;r≜ r ∕ j (9) and, if J is a second-order tensor, then i;c Wi c B i i i i i i ci c B

! O ! O where m is the mass of W . Note that r is the position of c relative to J jB B∕I J jI I∕B (5) i i i i c resolved in FB. Define the bus inertia matrix The following assumptions are made. ≜ ! JB J B∕cjB ¶Kovecses, J., “Dynamics of Mechanical Systems Part 1: Concepts, Geometry and Kinematic Considerations,” 2017, https://hal.archives- Then, the inertia tensor of the spacecraft relative to its center of mass ouvertes.fr/hal-01534010 [retrieved 01 June 2019]. resolved in FB is given by 4 Article in Advance / ZHAO, CRUZ, AND BERNSTEIN

Xn Xn ! ! ⇀ ! ! ! R ∕ ≜ R ∕ j R ∕ j ; ω ≜ ω ∕ j (17) J ≜ J ∕ j J ∕ j J ∕ j J J (10) C I C I C C I I C C I C SC c B B c B Wi c B B i;c i1 i1 R R The error between B∕I and C∕I is given in terms of the attitude- B. Dynamics error rotation matrix In equation (4.81) of Ref. [4], setting the inertial derivative of the ! ! ! ! T ! T! R~ ≜ R ∕ j R ∕ j R ∕ j R ∕ R ∕ j R ∕ j R ∕ j spacecraft angular momentum B C B B I B I C B B I B I B C I I B I B RT R ∕ B∕I ⇀ C I NdH (18) dt which satisfies τ to be CMG and setting the wheel angular accelerations to be zero yield B• ! ! ⇀ × R_~ ω R~ ω× Xn R B∕CjB R B∕CjB B∕CjB ~ B (19) Jω_ ω × Jω α ν O e τ τ (11) B B B i i i 1 CMG dist ! ω⇀ i1 where R B∕C is the rotation tensor that rotates FC into FB, B∕C is the angular velocity of FB relative to FC, and the angular-velocity error where ω~ B is defined as ⇀ ⇀ ⇀ ⇀ ~ T ω ≜ ω ω~ B ≜ ωB∕Cj ωB∕I − ωC∕Ij ωB − R ωC (20) B B∕IjB B B τ Note that dist is the torque disturbance applied to the spacecraft, and B. Performance Variable the torque τ applied to the spacecraft by the CMGs is given by CMG The objective of the attitude control problem is to align FB with the attitude-command frame F . To follow attitude commands, a τ ≜ C CMG BCMGu B1u_ (12) performance variable z is defined and used for adaptive control. A vector representation of the attitude error R~ in Eq. (18) is given by ≜ ω× − ∈ R3×n 2 3 BCMG BB1 B2 (13) R~ − R~ X3 a3 32 a2 23 ≜ R~ T × 4 R~ − R~ 5 ∈ R3 ∈ R3×n S ai ei ei a1 13 a3 31 (21) where the ith column of B1 is given by i1 ~ ~ a2R21 − a1R12 −β O ∈ R3 B1i i ie2 (14) where a1, a2, and a3 are real numbers. Note that S depends on the off- ∈ R3×n R~ and the ith column of B2 is given by diagonal entries of the error matrix in Eq. (19). Lemma 1: Assume that a1, a2, and a3 are distinct, positive O × − × OTω − α ν O ∈ R3 numbers. Then, S 0 if, and only if, B2i ie2 Ji Jie2 i B i i ie3 (15) R~ ∈ − − − where ei denotes the ith column of the 3 × 3 identity matrix I3.Asin fI3; diag1; 1; 1; diag 1; 1; 1; diag1; 1; 1g Ref. [4] (p. 80), the term B u_ is not considered in the subsequent 1 □ development. Proof: See lemma 3 of Ref. [38]. In Ref. [44], the equations of motion for a spacecraft controlled by Lemma 1 shows that using S as the sole metric of attitude error constant-speed single-gimbal CMGs are derived where, in the gives rise to three spurious attitude equilibria. These equilibria can be σ removed by considering the additional attitude-error metric notation of Ref. [44], B1u_ in Eq. (12) is expressed as B and BCMGu is σ expressed as D1 D2 D3_. Various torque-steering laws ~ ~ ~ ~ s ≜ traceA − ARa11 − R11a21 − R22a31 − R33 involving Bσ, D2, and D3 were considered in Ref. [44]. In equation (4.81) in Ref. [4], the term (22)

Xn where the attitude parameter matrix A ≜ diaga1;a2;a3 is positive R~ R~ R~ R~ Jg;iδig^ i definite; and 11, 22, and 33 are the diagonal entries of .In R~ Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 i1 contrast to S, which depends on the off-diagonal entries of , the error metric s depends on the diagonal entries of R~ . The following is equivalent to B1u_ in Eq. (12). Related developments appeared in result is needed. Refs. [29,45]. Lemma 2: For all i; j ∈ f1; 2; 3g, the i; j entry Rij of a rotation matrix R satisfies jRijj ≤ 1. III. Performance Variable for the Attitude Control Proof: See fact 4.13.14 of Ref. [46]. □ Problem The following result shows that S and s can be used together to ensure R~ I . A. Commanded Attitude and Attitude Error 3 Proposition 1: Assume that a1, a2, and a3 are distinct, positive Let FC be a frame that represents the commanded attitude for FB. R~ numbers. Then, S 0 and s 0 if, and only if, I3. The control objective is to have the spacecraft attitude Proof: Sufficiency is immediate. To prove necessity, note that s 0 implies that R ≜ ! B R B∕IjB − R~ − R~ − R~ − R~ a11 11a21 22a31 33traceA A 0 follow the commanded attitude maneuver given by ~ Lemma 2 implies that, for i 1; 2; 3, ai1 − Rii ≥ 0, and C• − R~ − R~ − R~ _ d ! ! ! ⇀ × × thus a11 11a21 22a31 330. RC∕I R C∕Ij R C∕Ij R C∕Ij ωC∕Ij RC∕Iω (16) dt C C C C C Because a1, a2, and a3 are positive, it follows that R~ R~ R~ R~ 11 22 33 1. Because, by Lemma 1, is diagonal, it ! ω⇀ R~ □ where R C∕I is the rotation tensor that rotates FI into FC, C∕I is the follows that I3. • angular velocity of FC relative to FI, C indicates the frame derivative To apply RCAC to attitude control, Lemma 1 and Proposition 1 are with respect to FC, and used to define the performance variable z by Article in Advance / ZHAO, CRUZ, AND BERNSTEIN 5

ω~ B about the leading sign of the numerator, the relative degree, and z ≜ S ∈ R7 (23) nonminimum-phase zeros [36]. For linear multi-input multioutput s systems, information about the transmission zeros is captured by using where ω~ B is the angular-velocity error defined by Eq. (20). Markov parameters of the plant. For nonlinear systems, Gf can be Proposition 2: Assume that a1, a2, and a3 are distinct, positive constructed by using Markov parameters from of the linearized ω ω R R numbers. Then, z 0 if, and only if, B C and B∕I C∕I. dynamics [47]. In the present paper, Gf is constructed by analyzing the Proof: For necessity, assume that z 0. Then, S 0, s 0, and effect of the control input on the performance variable z for the ~ Proposition 1 imply that R I3. Therefore, Eq. (18) yields RB∕I initial gimbal configuration. Additional parameters and weights needed R R~ R ω ω R~ Tω C∕I C∕I. If, in addition, ~ B 0, then B C by RCAC are determined from the closed-loop response of a fully ω ω R R ω ω ~ B C. Thus, B∕I C∕I and B C. nonlinear nominal simulation model; RCAC uses no explicit knowledge R R To prove sufficiency, assume that C∕I B∕I. Then, Eq. (18) of the spacecraft equations of motion. No linearized model of the ~ implies R I3. Thus, Lemma 1 implies that S 0, and Proposition spacecraft dynamics is needed, created, or used for any purpose in ω ω ω ω − 1 implies that s 0. Next, let B C. Then, ~ B B this paper. R~ Tω ω − ω R R ω ω C B C 0. Thus, B∕I C∕I and B C imply ω □ that s 0 and S ~ B 0. Hence, z 0. C. Retrospective Cost

To update the controller parameter Θk, the retrospective cost function is defined by IV. RCAC Algorithm Xk This section summarizes the RCAC algorithm as used in the ^ ^ T ^ ^ T ^ J^kΘ ≜ z^iΘ Rzz^iΘΦiΘ RuΦiΘ present paper. Because RCAC is a discrete-time control algorithm, it i1 is used to update the coefficients of a discrete-time controller that Θ^ − Θ T Θ^ − Θ operates on sampled data. The sampled values of the performance 0 RΘ 0 (29) variable zt are given by z ≜ zkT . Likewise, the continuous-time ≥ η η η η k s where, for all i 1, Rz zIl , Ru uIl , and RΘ ΘIlΘ ; and z, u t z u control input , which is the vector of gimbal rates, is given by a ηu, and ηΘ are positive numbers. As in Ref. [36], the updated zero-order-hold device. In particular, for all t ∈ kT ; k 1T ,it Θ s s controller coefficient matrix k1 that minimizes Eq. (29) is given by follows that utu , where u is given by the feedback controller. −1 k k recursive least squares. Let P0 RΘ and, for all k ≥ 1, let Θk1 be the unique global minimizer of the retrospective cost function A. Controller Structure [Eq. (29)]. Then, Θk1 is given by Define the strictly proper discrete-time dynamic compensator of Θ Θ − ΦT Γ−1 Φ Θ −1 − order nc given by k1 k Pk f;k k f;k k Rz Ru Rzzk uf;k

Xnc Xnc (30) uk Pi;kuk−i Qi;kzk−i (24) − ΦT Γ−1Φ i1 i1 Pk1 Pk Pk f;k k f;kPk (31) × ≥ ∈ Rlu lu ∈ where, for all i 1; :::;nc and all k 1, Pi;k and Qi;k −1 × where Φf;k ≜ GfqΦk, uf;k ≜ Gfquk, and Γk ≜ Rz Ru Rlu lz are controller coefficient matrices. The control law [Eq. (24)] Φ P ΦT . can be rewritten in regressor form as f;k k f;k 0;k< kw; uk (25) G ΦkΘk;k≥ kw V. Construction of f A. Pyramidal Arrangement of Four CMGs Φ where the regressor matrix k, which contains past performance and Consider a spacecraft actuated by n 4 CMGs mounted in the control data, is defined by pyramidal arrangement shown in Fig. 4. Because the CMGs apply a 2 3 pure torque to the bus, the location of the pyramidal arrangement on T uk−1 the bus is immaterial. The pyramid has a square base, for which the 6 . 7 ^ ^ 6 . 7 sides define the axes iB and jB of the bus. Consequently, the sides of 6 . 7 ^ ^ β 6 7 the pyramidal base are aligned with iB and jB, and is the angle uk−n × Φ ≜ 6 c 7 ⊗ ∈ Rlu lΘ between a vector orthogonal to each face and the apex direction k^ . k 6 7 Ilu (26) B Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 z − 6 k 1 7 The CMGs are arranged such that, for i 1; 2; 3; 4, each axis j^ of 6 7 Gi 4 . 5 . rotation of gimbal Gi relative to the bus is perpendicular to the corresponding face; each axis i^ of rotation of wheel W relative to zk−n Gi i c the gimbal is parallel with the corresponding edge of the base. Therefore, the angular momentum of W , which aligns with i^ , lies where lΘ ≜ l n l l , and k is an initial waiting period during i Gi u c u z w in the plane spanned by the corresponding face. A counterclockwise which Φk is populated with data. The controller coefficient vector Θk gimbal angular velocity ui around j^G is defined to be positive, and is defined as _ i thus ui θi. The initial orientation matrices corresponding to this Θ ≜ T ∈ RlΘ k vecP1;k ··· Pnc;kQ1;k ··· Qnc;k (27) CMG arrangement are given by π T B. Retrospective Performance Variable O θ 0 R β − ;e Rθ 0;e ; 1 1 2 1 1 2 The retrospective performance variable is defined by π T π T O θ 0 R β − ;e R − ;e Rθ 0;e ; Θ ≜ Φ Θ^ − 2 2 2 2 2 3 2 2 z^k zk Gf q k uk (28) π T T Θ^ ∈ RlΘ O θ 0 R − β;e R−π;e Rθ 0;e ; where is the retrospectively optimized controller coefficient 3 3 2 1 3 3 2 vector obtained from the optimization in the following. The updated × T T Θ Θ^ ∈ Rlz lu π π controller is given by k1 . As explained in Ref. [36], Gf O θ 0 R − β;e R ;e Rθ 0;e (32) is a finite-impulse response filter that contains the essential plant 4 4 2 2 2 3 4 2 modeling information, and q is the forward shift operator. For linear single-input single-output systems, Gf contains modeling information where Rϕ; ξ is given by Rodrigues’s formula 6 Article in Advance / ZHAO, CRUZ, AND BERNSTEIN

where the leading minus sign is due to the error junction in Fig. 2. For all numerical examples, Gf is fixed and given by Eq. (34), with H given by Eq. (35).

VI. Numerical Examples A. Numerical Integration of the Sampled-Data Dynamics RCAC updates the coefficients of the discrete-time controller [Eq. (24)] for which the output uk is applied to the continuous-time spacecraft dynamics using a zero-order-hold device. The closed-loop system thus has sampled-data dynamics. To capture the intersample dynamics, the MATLABfunction ODE45 is used with a variable step size in order to ensure integration accuracy. In addition, ODE45 is Fig. 4 Pyramidal arrangement of four CMGs. applied in each sampling interval t ∈ kTs; k 1Ts with a fixed integration interval of length Ts in order to ensure that the integration R ϕ ξ ϕ − ϕ ξξT ϕ ξ× is synchronous with updates of the zero-order-hold control signal. ; cos I3 1 cos sin (33)

B. Simulation Performance Metric 3 ξ ∈ R is a unit vector, and ϕ ∈ −π; π. The accuracy of RCAC is assessed by using the eigenangle ϕeig of R~ as the error metric, where

B. Construction of Gf −1 1 ~ For the pyramidal arrangement of four CMGs and with the initial ϕeig ≜ cos trR − 1 (36) θ θ θ θ 2 gimbal configuration 10 20 30 400, note that this initial gimbal configuration is nonsingular in the sense that The eigenangle ϕ is the rotation angle around the eigenaxis that there is no constraint on the achievable torque vectors. Also, u eig i rotates R ∕ t to the attitude command R ∕ t. Using Rodrigues’s changes the direction of the angular momentum of W , which B I C I i formula [Eq. (33)], R ∕ can be represented by the commanded produces the gimbal torque τ . For gimbal G , τ for the initial C I Gi i Gi eigenangle ϕC and the commanded eigenaxis ξC resolved in FI. The gimbal configuration lies in a face of the pyramid, orthogonal to i^ , Gi transient performance is given by the settling time Ts to be the time as shown in Fig. 5. The vector sum of τ , τ , τ , and τ is τ ϕ ϕ G1 G2 G3 G4 CMG needed to bring and maintain eig within a specified bound ∞ for all in Eq. (12). ≥ t Ts, For the initial gimbal configuration θ10θ20θ30 θ τ 400, Fig. 5 shows that G can be decomposed in the directions ϕ t ≤ ϕ∞ (37) ^ − ^ τ 1 −^ − ^ eig jB and kB; G2 can be decomposed in the directions iB and kB; τ −j^ −k^ τ Δ G3 can be decomposed in the directions B and B; and G4 can be where ϕ∞3 deg. The final error metric is the value of ϕ t ^ − ^ eig decomposed in the directions iB and kB. Based on these averaged over the last 1 s of simulation. observations, Gf is chosen to be For all examples, the controller coefficients are initialized at Θ00. This choice reflects the absence of prior modeling 1 information. If additional modeling information is available, this G q H (34) f q information may be advantageous for initializing Θ0.

where H captures the effect of the gimbal control input on the C. Nominal-Model Tuning performance variable z for the given initial gimbal configuration. H is Nominal-simulation tuning refers to the tuning of RCAC based on thus given by simulation of a nominal model. Instead of attempting to identify the system, the simulation is run a small number of times to tune the 2 3 0 −10 1 parameters of RCAC to the nominal simulation. No attempt is made 6 7 to optimize the response. 6 10−107 6 7 To demonstrate this approach, consider a rest-to-rest maneuver, Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 − − − − 6 1 1 1 1 7 where the spacecraft is assumed to be initially at rest with H −6 0 −10 17 (35) 6 7 R ∕ 0I , ω 00, and u00; and the attitude and 6 − 7 B I 3 B 6 10 107 angular-velocity commands are given by R ∕ R ∕ and ω 0. 4 − − − − 5 B I C I C 1 1 1 1 For nominal-model tuning, the commanded eigenangle is ϕC T 0000 −150 deg around the commanded eigenaxis ξC 111 . The nominal simulation model assumes the following parameter 2 values. The bus inertia matrix is JB diag10; 25∕3; 5 kg ⋅ m , which implies that FB is a principal-axis frame for the bus inertia of the nominal simulation model. For i 1; 2; 3; 4, the wheel inertia is 2 Ji diag0.02; 0.012; 0.012 kg ⋅ m , the wheel mass is mi 0.001 kg, and the wheel spin speed is νi 600 rad∕s. T The gimbal positions are r1 0 0.1 0 m, r2 − T − T 0.1 0 0 m, r3 0 0.1 0 m, and r4 0.1 0 0 T m. The pyramidal face angle is β 54 deg, and the initial gimbal angles are θ10θ20θ30θ400 deg. The attitude parameter matrix is chosen to be A diag1; 2; 3. Nominal-model tuning with the sample time of 0.1 s yields the parameter choices nc 2, ηz ηu 1, ηΘ 0.01, and kw 5 steps. These values reflect the fact that the nominal-model tuning is limited to a small number of runs with no attempt to optimize the response. With these tuning parameters, the settling time is Ts Fig. 5 Gimbal torque directions for the initial gimbal configuration. 12.3 s with the final error of 1.1 × 10−7 deg. Article in Advance / ZHAO, CRUZ, AND BERNSTEIN 7

D. Perturbed-Model Testing is assumed to be diagonal and is given by the 422 points of the In perturbed-model testing, the parameters and weights obtained triangular regions in Fig. 8, which correspond to the shaded triangular under nominal-model tuning are fixed, and the robustness and region in figure 1 in Ref. [48]. Figure 8 shows the settling time for ω performance of RCAC are evaluated through simulations based on each bus inertia matrix. For all simulations, the measurements of B perturbations of the nominal model. These perturbations are chosen are corrupted by zero-mean Gaussian white sensor noise with to reflect off-nominal conditions of the model, commands, sensor covariance of 0.01I3. noise, actuator misalignment, and the singularities that can potentially occur in real-world operations. 4. Robustness to Off-Diagonal Perturbations of the Bus Inertia Matrix

As shown in Fig. 9, the rotated bus inertia matrix JB;rot is defined as 1. Rest-to-Rest Rotation Maneuvers with Sensor Noise ϕ − T The commanded eigenangle C varies from 180 to 180 deg JB;rot ≜ Rθ;e JBRθ;e (38) around the commanded eigenaxes 111T, 100T, T T 010 , and 001 . Zero-mean Gaussian white noise where Rθ;e is given by Rodrigues’s formula [Eq. (33)] with with a covariance of 0.01I3 is assumed to corrupt the sensors that e 111T. Consequently, F is no longer a principal-axis ω B measure the angular velocity B of the bus. Figure 6 shows the frame of the bus. For the simulation, zero-mean Gaussian white noise settling time and error at 50 s. with covariance of 0.01I3 is assumed to corrupt the measurements of ω B. The end time for all simulations in Fig. 9 is 300 s. The 2. Rest-to-Rest Rotation Maneuver nonmonotonic trend suggests a complex relationship between the The commanded attitude maneuver is a rest-to-rest rotation around initial gimbal configuration and the rotated inertia matrix. T the eigenaxis ξC 010 , beginning at 0.5 s, ending at 15 s, and with the constant rate of 0.18 rad∕s. Figure 7a compares the 5. Approximate Singularity Passthrough actual attitude to the attitude command; and Fig. 7b shows the Rerunning the 422 maneuvers tested in Fig. 8 in the absence of corresponding eigenangle error. sensor noise, there are 32 times at which the minimum singular value of BCMG is less than 0.1. Of these 32 cases, the smallest minimum 3. Robustness to Diagonal Perturbations of the Bus Inertia Matrix singular value of BCMG occurs for the inertia matrix The robustness of RCAC to uncertainty in the principal moments diag30; 22; 15 kg ⋅ m2 shown in Fig. 10. In particular, Fig. 10b of inertia is assessed by considering a collection of diagonal bus shows that, at the time resolution of 0.1 s of the numerical integration, λ λ λ inertia matrices JB diag 1; 2; 3. Figure 1 in Ref. [48] provided an approximate singularity passthrough occurs at a time of 31.2 s, a convenient classification of all possible principal moments of when the minimum singular value of BCMG reaches its smallest value, inertia in relation to the underlying geometry. In particular, the namely, 0.0313. A closer approach to a singularity occurs in Fig. 11 λ λ triangular shaded region shows all feasible values of 2 and 3 in where, at a time of 27.5 s, the minimum singular value reaches 0.0038 λ terms of the largest principal moment of inertia 1. The vertices of the without any adverse effect. Because the control input matrix BCMG is region correspond to a sphere, a thin disk, and a thin pencil. For time dependent, rank-deficient conditions tend to hold at only additional details, see Ref. [48]. isolated points, and thus the use of requested gimbal rates is able to The attitude command is to rotate −150 deg around overcome singularities encountered instantaneously during attitude ξ T λ λ λ C 111 . The unknown bus inertia matrix diag 1; 2; 3 command following.

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Fig. 6 Plots of a) settling time Ts and b) error at 50 s.

Fig. 7 Plots of a) commanded and actual eigenangles, and b) eigenangle error. 8 Article in Advance / ZHAO, CRUZ, AND BERNSTEIN

2 Fig. 8 Settling time Ts as a function of the bus inertia matrix diag λ1; λ2; λ3 for λ2 in the ranges [4, 8], [5, 10], [10, 20], and 15; 30 kg ⋅ m .

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Fig. 9 Plots of a) settling time Ts, and b) error at 300 s.

6. Escape from an Initial Gimbal-Lock Singularity simulated with the same bus inertia, initial conditions, and attitude To consider the case of an exact, impassable gimbal singularity, the command as the spacecraft with CMGs. Figure 11b shows that, under initial gimbal configuration is chosen to be different from the initial thruster control, the attitude of the spacecraft generally decreases, gimbal configuration considered in the previous subsections; where the oscillations indicate that the attitude overshoots the however, no change is made to H defined by Eq. (35) for setpoint but eventually damps out. In contrast, due to the initial θ00000 deg. The attitude command is to rotate the gimbal-lock singularity, RCAC drives the attitude in the opposite T Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 spacecraft −150 deg around the eigenaxis 001 . Letting direction, as evidenced by the increasing eigenangle. At a time of θ0−90 90 −90 90 deg, no torque can be generated in 330 s, however, the eigenangle decreases monotonically and rapidly ^ the direction of kB. Therefore, the initial torque required for the to the attitude command, which it reaches at a time of 357.8 s with a × −5 commanded attitude maneuver is not in the range of BCMG, and thus it final error of 3.88 10 deg. is a singular direction of the initial gimbal configuration. Consequently, it is impossible for the spacecraft to initially rotate 7. Robustness to Sensor/Actuator Misalignment in the commanded direction; this case represents an impassable In all previous simulations, the attitude sensors are assumed to be singularity corresponding to gimbal lock. perfectly aligned with the pyramidal CMGs. In particular, the sensing The minimum singular value of BCMG during the first five steps is axes are assumed to be exactly aligned with the axes of the bus frame × −15 1.4 10 (due to numerical roundoff) because RCAC must wait FB, which also defines the directions of the CMGs. However, sensor/ five steps due to Eq. (25), where kW 5. The minimum singular actuator misalignment can occur in practice; in which case, the value of BCMG at t 0.6 s is 0.24. As shown in Fig. 11b, RCAC sensing axes may be different from the axes that define the gimbal initially takes an indirect route in following the attitude command in directions [49,50]. This misalignment is assumed to be unknown to order to escape from the gimbal lock. For the same initially singular RCAC, and it is of interest to determine the effect of the sensor/ gimbal configuration, a nonsingular torque request is used in actuator misalignment on the attitude error. Ref. [24] to escape the singularity. To determine the performance degradation due to sensor/actuator To assess the performance degradation due to the initial gimbal misalignment, it is assumed, as in previous sections, that the base of singularity, RCAC is applied to the spacecraft with thruster actuation the CMG pyramid is aligned with the axes of the bus frame FB. only: that is, without CMGs. Based on the response of the nominal However, the sensors, which are misaligned with the CMGs, are − − T 0 ! model, H is redefined to be H I3 I3 03×1 ; all of the aligned with a rotated frame FB. The rotation tensor R B 0∕B thus remaining parameters and weights of RCAC are the same as in the determines the alignment of the CMGs relative to the attitude case of CMGs. Using RCAC, the spacecraft with thrusters is sensors; this rotation tensor is assumed to be unknown to RCAC. In Article in Advance / ZHAO, CRUZ, AND BERNSTEIN 9

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c) d) Fig. 11 Plots of a) minimum singular value, b) eigenangle error, c) gimbal-rate control input u, and d) CMG and thruster torques.

! the absence of misalignment, R B 0∕B is the identity second-order ! 0 0 T O 0 ≜ O 0∕ R ∕ 0 j O 0∕ O ∕ Rθ ;e O ∕ tensor. i B Gi Gi B B 0 B B B Gi B Gi Because the attitude error is determined by the sensor (39) measurements, the goal is to determine the performance degradation 0 0 0 due to the misalignment between FB and FB. Note that, due to the where θ and e are the misalignment eigenangle and misalignment misalignment, Eq. (8) is replaced by from FB to FB 0 . The commanded attitude maneuver is to rotate 10 Article in Advance / ZHAO, CRUZ, AND BERNSTEIN

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Fig. 12 Plots of a) settling time Ts, and b) error at 50 s.

T ⇀ ⇀ ⇀ ⇀ ⇀ −150 deg around ξC 111 . The misalignment eigenangle H ∕ ∕ H ∕ ∕ m r ∕ × ω ∕ × r ∕ θ 0 varies from −40 to 25 deg around four choices of the eigenaxis e 0, Wi c I Wi ci I i ci c B I ci c T T T T ⇀ namely, 111 , 100 , 010 , and 001 . ⇀×2 ⇀ H ∕ ∕ − m r ∕ ω ∕ (A4) Figure 12 shows the settling time and error at 50 s. Wi ci I i ci c B I ⇀ Because, by Assumption 3, c is fixed in W , H ∕ ∕ is given by VII. Conclusions i i Wi ci I Adaptive control of a rigid spacecraft controlled by a pyramidal ⇀ ! ⇀ H ∕ ∕ J ∕ ω ∕ (A5) arrangement of four single-gimbal constant-speed CMGs was Wi ci I Wi ci Wi I considered. Retrospective cost adaptive control was used with ! ω⇀ where J Wi∕ci is the inertia tensor of Wi relative to ci, and Wi∕I is the gimbal-rate-requested CMGs; no torque-steering law was used. The ⇀ RCAC implementation was based on a target model involving a ω angular velocity of FWi relative to FI. Expanding Wi∕I in Eq. (A5) single Markov parameter corresponding to the initial gimbal yields configuration. The parameters and weights for RCAC were based on a nominal simulation; no explicit knowledge of the nonlinear ⇀ ! ⇀ ⇀ H ∕ ∕ J ∕ ω ∕ ω ∕ (A6) equations of motion was used by the adaptive controller. Wi ci I Wi ci Wi B B I Numerical simulations with a perturbed model showed that the adaptive controller is robust to large variations in the spacecraft inertia ! ω⇀ ⇀ ω⇀ in the presence of gyroscope noise. The performance across these J Wi∕ci Wi∕Gi ui B∕I (A7) variations shows the robustness of RCAC to the choice of the target ⇀ model. The controller was able to escape from an initial gimbal-lock where, by Assumption 7, the control vector u for CMG is the singularity and not suffer from close approach to a gimbal singularity. i i angular velocity of F relative to F ; that is, An open problem is to consider the effect of the gimbal angular Gi B acceleration u_ on the closed-loop performance. Although this term is ⇀ ≜ ω⇀ typically neglected in the analysis of torque-steering laws ([4] p. 80), ui Gi∕B (A8) its effect on the performance of CMG-controlled spacecraft remains to be investigated. Substituting Eq. (A7) into Eq. (A4) yields

⇀ ! ω⇀ ⇀ ! ω⇀ Appendix: Derivation of the Equations of Motion HWi∕c∕I J Wi∕ci Wi∕Gi ui J Wi∕c B∕I (A9) Assumptions 1, 2, and 3 imply that the center of mass c of the ⇀ where, by the parallel axis theorem, spacecraft is fixed in B. Therefore, the angular momentum H ∕ ∕ of Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 B c I B relative to c with respect to FI is given by ! ! − ⇀×2 J Wi∕c J Wi∕ci mi r ci∕c (A10) ⇀ ! ⇀ ω ⇀ HB∕c∕I J B∕c B∕I (A1) By Assumption 2, the angular momentum HGi∕c∕I of Gi relative⇀ to c ! with respect to FI is zero. Therefore, the angular momentum HSC∕c∕I where J B∕c is the inertia tensor of the bus relative to c. The angular ⇀ of the spacecraft relative to c with respect to FI is given by

momentum HWi∕c∕I of Wi relative to c with respect to FI is given by Xn I• ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ HSC∕c∕I HB∕c∕I HW ∕c∕I (A11) H ∕ ∕ H ∕ ∕ r ∕ × m r ∕ (A2) i Wi c I Wi ci I ci c i ci c i1 ⇀

where HWi∕ci∕I is the angular momentum of Wi relative to its center of ⇀ Using Eqs. (A1) and (A9), Eq. (A11) can be rewritten as

mass ci with respect to FI, r ci∕c is the position of ci relative to c, mi is • Xn the mass of Wi, and I denotes the derivative with respect to FI. ⇀ ! ⇀ ! ⇀ ⇀ I• H ∕ ∕ J ∕ ω ∕ J ∕ ω ∕ u (A12) ⇀ SC c I SC c B I Wi ci Wi Gi i i1 Applying the transport theorem to r ci∕c in Eq. (A2) yields B• ⇀ ⇀ ⇀ ⇀ ⇀ ⇀ where the inertia tensor of the spacecraft relative to its center of mass H ∕ ∕ H ∕ ∕ r ∕ × m r ∕ ω ∕ × r ∕ (A3) Wi c I Wi ci I ci c i ci c B I ci c is given by • where B denotes the derivative with respect to FB. Because c and ci Xn B• ! ! ! ⇀ J SC∕c J B∕c J Wi∕c (A13) are fixed in the bus, it follows that r ci∕c 0. Therefore, i1 Article in Advance / ZHAO, CRUZ, AND BERNSTEIN 11

• A. Dynamics Gi ω⇀ A.1. Euler’s Equation for the Angular Momentum of the Spacecraft Wi∕Gi 0 Applying the angular momentum theorem to the spacecraft yields

Thus, using Eq. (A8), the angular acceleration of FWi relative to FG with respect to FB is given by I• i ⇀ ⇀ • G • HSC∕c∕I MSC (A14) B i ω⇀ ω⇀ ω⇀ × ω⇀ ω⇀ × ω⇀ Wi∕Gi Wi∕Gi Gi∕B Wi∕Gi Gi∕B Wi∕Gi ⇀ ⇀×⇀ where M is the sum of all external torques acting on the spacecraft. ω SC ui Wi∕Gi (A21) Applying the transport theorem to the left side of Eq. (A14) yields Because • • B• I B ⇀ ⇀ ⇀ ⇀ ⇀ r c ∕c 0 HSC∕c∕I HSC∕c∕I ωB∕I × HSC∕c∕I (A15) i

the derivative of the cross term in Eq. (A19) with respect to FB is given by Hence, Eqs. (A14) and (A15) imply B• B• B• z}|{ z}|{ z}|{ B• B• • ⇀×2 ⇀× ⇀× ⇀× ⇀× ⇀× ⇀ × ⇀ ×⇀× B r ∕ r ∕ r ∕ r ∕ r ∕ r ∕ r ∕ r ∕ r ∕ ⇀ ⇀ ⇀ ⇀ ci c ci c ci c ci c ci c ci c ci c ci c ci c H ∕ ∕ ω ∕ × H ∕ ∕ M (A16) SC c I B I SC c I SC 0(A22)

which is Euler’s equation for the angular momentum of the Substituting Eqs. (A20–A22) into Eq. (A19) yields the derivative of spacecraft. Eq. (A9) with respect to FB;thatis,

• B B• B• ⇀ ⇀×! ! ⇀× ⇀ ⇀ ⇀ ! ⇀×⇀ ⇀ ! ⇀ H ∕ ∕ ui J ∕ − J ∕ ui ω ∕ ω ∕ ui J ∕ ui ω ∕ u i J ∕ ω ∕ Wi c I Wi ci Wi ci B I Wi Gi Wi ci Wi Gi Wi c B I (A23) B• B• ⇀×! − ! ⇀× ω⇀ ⇀×! ω⇀ ⇀ ! ⇀ ! ω⇀ ui J Wi∕ci J Wi∕ci ui B∕I ui J Wi∕ci Wi∕Gi ui J Wi∕ci u i J Wi∕c B∕I

A.2. Derivatives of the Wheel Angular Momenta with Respect to FB B. Resolving the Equations of Motion in FB

Differentiating Eq. (A11) with respect to FB yields Resolving Poisson’s equation [Eq. (1)] in FB yields B• B• B• B• d ⇀ × ⇀ ⇀ Xn ⇀ R_ ! ! ! ω R ω× B∕I R B∕IjB R B∕IjB R B∕IjB B∕IjB B∕I B (B1) HSC∕c∕I HB∕c∕I HWi∕c∕I (A17) dt i1 where ! Because, by Assumption 1, the inertia J ∕ is constant with B c R ≜ ! ! ω ≜ ω⇀ respect to FB, differentiating Eq. (A1) with respect to FB yields B∕I R B∕IjB R B∕IjI; B B∕IjB (B2)

B• • ⇀ B Resolving Eq. (A1) in FB yields ! ω⇀ HB∕c∕I J B∕c B∕I (A18) ⇀ ω HB∕c∕IjB JB B (B3) Furthermore, differentiating Eq. (A9) with respect to FB and using Eq. (A10) yields where Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 ! • ≜ B B• B• B• JB J B∕cjB (B4) ⇀ ! ⇀ ⇀ ! ⇀ ⇀ HW ∕c∕I J W ∕c ωW ∕G ui J W ∕c ωW ∕G u i i i i i i i i i i Furthermore, resolving Eq. (A18) in FB yields B• • • z}|{ B B B• ⇀ d ⇀ ! ⇀×2 ⇀ ! ⇀ ω J ∕ − m r ∕ ω ∕ J ∕ ω ∕ (A19) HB∕c∕IjB HB∕c∕IjBJB _ B (B5) Wi ci i ci c B I Wi c B I dt ! where Next, using Eq. (A8), the derivative of J Wi∕ci with respect to FB is given by B• ω ω⇀ _ B B∕IjB (B6) B• Gi• ! ! ω⇀× ! − ! ω⇀× Using Fig. 3 and Assumptions 4, 5, and 6 to resolve the angular J Wi∕ci J Wi∕ci Gi∕B J Wi∕ci J Wi∕ci Gi∕B velocity of FWi relative to FGi in FGi yields ⇀×! − ! ⇀× ui J Wi∕ci J Wi∕ci ui (A20) ⇀ ω ∕ j ν e (B7) Wi Gi Gi i 1 where • ν ^ Gi where i > 0 is the angular rate of Wi around iWi relative to FGi . ! ⇀ J Wi∕ci 0 Similarly, the gimbal angular velocity ui defined by Eq. (A8) and the Gi• ⇀

due to Assumptions 1, 5, and 6. Furthermore, Assumptions 4 and 6 gimbal angular acceleration ui of FGi relative to FB resolved in FGi imply that are given by 12 Article in Advance / ZHAO, CRUZ, AND BERNSTEIN

⋅ Gi Using Eqs. (1) and (A8), it follows that ⇀ ⇀ d ⇀ uijGi uie2;uijGi uijGi u_ie2 (B8) dt Gi• O_ ! ! ⇀× O × i R G ∕Bj R G ∕B ui j iuie2 (B20) where ui is the gimbal-rate request for CMGi, and u_i is its derivative. i Gi i Gi Because ui is the spin rate of gi, the corresponding gimbal angle θi is defined such that Therefore,

_ × θi ui (B9) _ −β O_ −β O B1i i ie2 i iuie2 e2 0 (B21) It thus follows from Eqs. (A8) and (B8) that which implies that B1 in Eq. (B18) is constant. ⇀ Resolving Eq. (A23) in FB yields the derivative of Eq. (B16), ω ∕ j θ_ e (B10) Gi B Gi i 2 which is given by

Δ θ θ θ T θ_ B• Defining the gimbal angle vector 1 ··· n yields u, ⇀ ⇀ Δ d × × T T ∈ Rn H ∕ ∕ j H ∕ ∕ j u O e J − J e O ω where uu1 ··· un is the control input. Furthermore, Fig. 3 dt Wi c I B Wi c I B i i 2 i i 2 i B and Assumptions 4 and 5 imply that the inertia matrix of Wi relative O ×OTO ν O ω iuie2 i iJiuie2 ie1u_i iJie2 Ji;c _ B to ci resolved in both FGi and FWi is given by × × T × × " # uiOie Ji − Jie O ωB uiOie βiuie2 e αiνie1 α 00 2 2 i 2 2 ! ! i O β ω J ≜ J ∕ j J ∕ j 0 β 0 (B11) u_i i ie2 Ji;c _ B i Wi ci Gi Wi ci Wi i × × 00βi O − OTω − α ν O β O ω ie2 Ji Jie2 i B i i ie3ui iu_i ie2 Ji;c _ B α ^ ^ (B22) where i is the moment of inertia of Wi around the spin axis iWi iGi β and i is the moment of inertia around the remaining axes of FWi Using Eqs. (B19) and (B22) to resolve the sum in Eq. (A17) in F and FGi . B Next, note that yields B• Xn ⇀ Xn • • B• Gi Gi ω − ⇀ ⇀ ⇀ ⇀ ⇀ HWi∕c∕IjB Ji;c _ B B1iu_i B2iui u u ω ∕ × u u (B12) i1 i1 i i Gi B i i Xn ω − Resolving Eqs. (B7), (B8), and (6) in FB and using Eq. (B12) yield Ji;c _ B B1u_ B2u (B23) i1 ⇀ ⇀ ⇀ ω ∕ j ν O e ;uj ω ∕ j u O e ; × Wi Gi B i i 1 i B Gi B B i i 2 T ∈ R3 n where u_ u_1 ··· u_n , and the ith column of B2 is given by B• G • (B13) ⇀ ⇀i u j u j u_ O e O × − × OTω − α ν O ∈ R3 i B i B i i 2 B2i ie2 Ji Jie2 i B i i ie3 (B24)

where Using Eqs. (B3) and (B18) to resolve Eq. (A11) in FB yields

! ! T T ⇀ Xn J W ∕c j Oi J W ∕c j O OiJiO (B14) i i B i i Gi i i ≜ ω α ν O − HSC HSC∕c∕IjB J B i i ie1 B1u (B25) i1 and the orientation matrix Oi is defined by ! ! where, using Eq. (A13), Oi ≜ R G ∕Bj R G ∕Bj (B15) i Gi i B Xn ≜ ! ! J J SC∕cjB JB Ji;c (B26) where R Gi∕B is the rotation tensor that transforms FB into FGi . i1

Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 OTO Resolving Eq. (A9) in FB and using the fact that i i I3 yields ⇀ Using Eqs. (B5) and (B23) to resolve Eq. (A17) in FB yields T T ×2 H ∕ ∕ j O J O ν O e u O e O J O − m r ω Wi c I B i i i i i 1 i i 2 i i i i i B B• O α ν β ω ⇀ ⇀ i i ie1 iuie2Ji;c B _ d ω − HSC HSC∕c∕IjBHSC∕c∕IjB J _ B B1u_ B2u (B27) (B16) dt Furthermore, define Resolving Eq. (A16) in FB using Eqs. (B25) and (B27) yields ⇀ ! T ×2 Xn J ≜ J ∕ j O J O − m r ;r≜ r ∕ j (B17) i;c Wi c B i i i i i i ci c B ω − −ω × ω α ν O − τ J _ B B1u_ B2u B J B i i ie1 B1u dist i Using Eq. (B16) to resolve the sum in Eq. (A11) in F yields 1 B (B28) Xn ⇀ Xn H ∕ ∕ j O α ν e β u e J ω Wi c I B i i i 1 i i 2 i;c B where the torque applied to the spacecraft by the disturbances is given i1 i1 by Xn α ν O e J ω − B u (B18) ⇀ i i i 1 i;c B 1 τ ≜ i1 dist MdistjB (B29) ∈ R3×n where the ith column of B1 is given by Rearranging Eq. (B28) yields the dynamics for a spacecraft actuated by n angular-velocity-requested single-gimbal constant- −β O ∈ R3 B1i i ie2 (B19) angular-rate CMGs subject to external torque disturbances; that is, Article in Advance / ZHAO, CRUZ, AND BERNSTEIN 13

Xn on Robotics, Vol. 25, No. 6, 2009, pp. 1262–1270. ω ω × ω α ν O τ τ J _ B B J B i i ie1 CMG dist (B30) doi:10.1109/TRO.2009.2032953 i1 [2] Schaub, H., and Junkins, J. L., Analytical Mechanics of Space Systems, 2nd ed., AIAA, Reston, VA, 2003, p. 264. τ where the torque CMG applied to the spacecraft by the CMGs is given doi:10.2514/4.867231 by [3] Wie, B., Space Vehicle Dynamics and Control, 2nd ed., AIAA, Reston, VA, 2008, p. 461. τ ≜ doi:10.2514/4.860119 CMG BCMGu B1u_ (B31) [4] Leve, F., Hamilton, B., and Peck, M., Spacecraft Momentum Control Systems, Springer, New York, 2015, p. 157. where doi:10.1007/978-3-319-22563-0 [5] Wie, B., “Singularity Analysis and Visualization for Single-Gimbal ≜ ω× − ∈ R3×n ” Journal of Guidance, Control, and BCMG BB1 B2 (B32) Control Moment Gyro Systems, Dynamics, Vol. 27, No. 2, 2004, pp. 271–282. If the wheel angular rate ν is much larger than the maximum doi:10.2514/1.9167 i [6] Paradiso, J. A., “Global Steering of Single Gimbaled Control Moment component of the bus angular velocity (that is, ” ν ≫ ω ω ω Gyroscopes Using a Directed Search, Journal of Guidance, Control, j ij j maxf 1; 2; 3gj), then it follows from Eqs. (B19), (B24), and Dynamics, Vol. 15, No. 5, 1992, pp. 1236–1244. and (B32) that doi:10.2514/3.20974 [7] Kurokawa, H., “A Geometric Study of Single Gimbal Control Moment ≈ α ν O BCMG;i i i ie3 (B33) Gyros,” Report of Mechanical Engineering Laboratory, Vol. 175, Jan. 1998, pp. 135–138. [8] Bedrossian, N. S., Paradiso, J., Bergmann, E. V., and Rowell, D., which is not necessarily constant due to Oi. The simplified matrix [Eq. (B33)] is the Jacobian used in Refs. [5,13] to formulate torque- “Redundant Single Gimbal Control Moment Gyroscope Singularity ” steering laws and analyze gimbal singularities. Analysis, Journal of Guidance, Control, and Dynamics, Vol.13, No. 6, 1990, pp. 1096–1101. doi:10.2514/3.20584 [9] Schaub, H., Vadali, S., and Junkins, J., “Feedback Control Law for C. Time Dependence of J and BCMG Variable Speed Control Moment Gyros,” Journal of the Astronautical The rotation rate of the gimbals due to the requested gimbal rates u Sciences, Vol. 46, No. 3, 1998, pp. 307–328. “ changes the gimbal configuration, and thus the inertia of the [10] Yoon, H., and Tsiotras, P., Singularity Analysis of Variable-Speed Control Moment Gyros,” Journal of Guidance, Control, and Dynamics, spacecraft. Therefore, J may be time varying. Using Eqs. (B17) and Vol. 27, No. 3, 2004, pp. 374–386. (B26), the spacecraft inertia is expressed as doi:10.2514/1.2946 [11] Viswanathan, S. P., Sanyal, A. K., Leve, F., and McClamroch, N. H., Xn “Dynamics and Control of Spacecraft with a Generalized Model of O OT − ×2 J JB iJi i miri (C1) Variable Speed Control Moment Gyroscopes,” Journal of Dynamic i1 Systems, Measurement and Control, Vol. 137, No. 7, 2007, Paper 071003. where Assumption 1 implies that JB and Ji are constant. Therefore, [12] Prabhakaran, V. S., and Sanyal, A., “Adaptive Singularity-Free Control Moment Gyroscopes,” Journal of Guidance, Control, and Dynamics, Xn Xn Vol. 41, No. 11, 2018, pp. 2416–2424. _ O_ OT O O_ T O × − × OT J iJi i iJi i ui ie2 Ji Jie2 i (C2) doi:10.2514/1.G003545 i1 i1 [13] Leve, F. A., “Evaluation of Steering Algorithm Optimality for Single- Gimbal Control Moment Gyroscopes,” IEEE Transactions on Control – The control input matrix BCMG defined by Eq. (B32) has a similar Systems Technology, Vol. 22, No. 3, 2014, pp. 1130 1134. dependence on u. Using Eqs. (B21) and (B30) to rewrite the doi:10.1109/TCST.2013.2259829 derivative of Eq. (B32) yields [14] Takada, K., Kojima, H., and Matsuda, N., “Control Moment Gyro Singularity-Avoidance Steering Control Based on Singular-Surface ” _ ω× − _ Cost Function, Journal of Guidance, Control, and Dynamics, Vol. 33, BCMG _ BB1 B2 (C3) No. 5, 2010, pp. 1442–1450. doi:10.2514/1.48381 ω Solving Eq. (B30) for _ B yields [15] Wang, L., Guo, Y., Wu, L., and Chen, Q., “Improved Optimal Steering Law for SGCMG and Adaptive Attitude Control of Flexible Spacecraft,” Xn Journal of Systems Engineering and Electronics, Vol. 26, No. 6, 2015, Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 ω −1 −ω× ω α ν O τ _ B J B J B i i ie1 BCMGu B1u_ dist pp. 1268–1276. i1 doi:10.1109/JSEE.2015.00139 (C4) [16] Elliott, D. S., Peck, M. A., and Nesnas, I., “Novel Method for Control Moment Gyroscope Singularity Avoidance Using Constraints,” AIAA ω× τ Guidance, Navigation, and Control Conference, AIAA SciTech Forum, Note that the term _ BB1 in Eq. (C3) depends on u and dist due to Eq. (C4). The derivative of the ith column B of B , defined by AIAA Paper 2018-1327, 2018. 2i 2 doi:10.2514/6.2018-1327 Eq. (B24), is given by [17] Geng, Y., Hou, Z., and Huang, S., “Global Singularity Avoidance Steering Law for Single-Gimbal Control Moment Gyroscopes,” Journal _ O × × − × − × − × × OTω – B2i ui ie2 e2 Ji Jie2 e2 Ji Jie2 e2 i B of Guidance, Control, and Dynamics, Vol. 40, No. 11, 2017, pp. 3027 3036. O × − × OTω − α ν O ie2 Ji Jie2 i _ B i i iuie1 (C5) doi:10.2514/1.G002331 [18] Geng, Y., Hou, Z., Guo, J., and Huang, S., “Rapid Singularity-Escape _ ” Therefore, the term B2 in Eq. (C3) depends on u directly as well as Steering Strategy for Single-Gimbal Control Moment Gyroscopes, ω indirectly through its dependence on _ B given by Eq. (C4). Journal of Guidance, Control, and Dynamics, Vol. 40, No. 12, 2017, pp. 3199–3210. Furthermore, Eq. (C5) also depends on τdist due to Eq. (C4). Hence, doi:10.2514/1.G002686 both terms in Eq. (C3) depend on the control input u and the external “ torque τ . Consequently, B may be time varying. [19] Valk, L., Berry, A., and Vallery, H., Directional Singularity Escape and dist CMG Avoidance for Single-Gimbal Control Moment Gyroscopes,” Journal of Guidance, Control, and Dynamics, Vol. 41, No. 5, 2018, pp. 1095– 1107. References doi:10.2514/1.G003132 “ [1] Carpenter, M. D., and Peck, M. A., “Reducing Base Reactions with [20] Jones, L. L., Zeledon, R. A., and Peck, M. A., Generalized Framework ” Gyroscopic Actuation of Space-Robotic Systems,” IEEE Transactions for Linearly Constrained Control Moment Gyro Steering, Journal of 14 Article in Advance / ZHAO, CRUZ, AND BERNSTEIN

Guidance, Control, and Dynamics, Vol. 35, No. 4, 2012, pp. 1094– [35] Hoagg, J. B., and Bernstein, D., “Retrospective Cost Model Reference 1103. Adaptive Control for Nonminimum-Phase Systems,” Journal of doi:10.2514/1.56207 Guidance, Control, and Dynamics, Vol. 35, No. 6, 2012, pp. 1767– [21] Gersh, J., and Peck, M., “Violet: A High-Agility Nanosatellite for 1786. Demonstrating Small Control-Moment Gyroscope Prototypes and doi:10.2514/1.57001 Steering Laws,” Guidance, Navigation, and Control Conference, AIAA [36] Rahman, Y., Xie, A., and Bernstein, D. S., “Retrospective Cost Adaptive Paper 2009-5900, 2009. Control: Pole Placement, Frequency Response, and Connections doi:10.2514/6.2009-5900 with LQG Control,” IEEE Control Systems, Vol. 37, No. 5, 2017, [22] Zhang, J., Ma, K., Meng, G., and Tian, S., “Spacecraft Maneuvers via pp. 28–69. Singularity-Avoidance of Control Moment Gyros Based on Dual-Mode doi:10.1109/MCS.2017.2718825 Model Predictive Control,” IEEE Transactions on Aerospace and [37] Chaturvedi, N., Sanyal, A. K., and McClamroch, N. H., “Rigid Body Electronic Systems, Vol. 51, No. 4, 2015, pp. 2546–2559. Attitude Control,” IEEE Control Systems Magazine, Vol. 31, No. 3, doi:10.1109/TAES.2015.130715 2011, pp. 30–51. [23] Takada, K., and Kojima, H., “Receding Horizon Control on Steering of doi:10.1109/MCS.2011.940459 Control Moment Gyro for Fast Attitude Maneuver,” Transactions of the [38] Sanyal, A., Fosbury, A., Chaturvedi, N., and Bernstein, D. S., “Inertia- Japan Society for Aeronautical and Space Sciences, Vol. 52, No. 175, Free Spacecraft Attitude Tracking with Disturbance Rejection and 2009, pp. 1–10. Almost Global Stabilization,” Journal of Guidance, Control, and doi:10.2322/tjsass.52.1 Dynamics, Vol. 32, No. 4, 2009, pp. 1167–1178. [24] Guo, Y., Cui, H., Ma, G., and Li, C., “Singular Direction Escape Steering doi:10.2514/1.41565 Law for Control Moment Gyros,” Journal of Aerospace Engineering, [39] Agarwal, K., Weiss, A., Kolmanovsky, I., and Bernstein, D. S., “Inertia- Vol. 30, No. 5, 2017, pp. 648–656. Free Spacecraft Attitude Control with Control-Moment-Gyro doi:10.2514/2.4610 Actuation,” AIAA Guidance, Navigation, and Control Conference, [25] MacKunis, W., Dupree, K., FitzCoy, N., and Dixon, W. E., “Adaptive AIAA Paper 2012-5003, 2012. Satellite Attitude Control in the Presence of Inertia and CMG Gimbal doi:10.2514/6.2012-5003 Friction Uncertainties,” Journal of Astronautical Sciences, Vol. 56, [40] Camblor, M., Cruz, G., Esteban, S., Leve, F. A., and Bernstein, D. S., No. 1, 2008, pp. 121–134. “Retrospective Cost Adaptive Spacecraft Attitude Control Using doi:10.1007/BF03256544 Control Moment Gyros,” Proceedings of the American Control [26] Wang, L., Zhong, C., Guo, Y., Wu, Y., and Guo, J., “Robust Adaptive Conference, IEEE Publ., Piscataway, NJ, 2014, pp. 2492–2497. Attitude Control for Flexible Spacecraft in the Presence of SGCMG doi:10.1109/ACC.2014.6859136 Friction Nonlinearity,” International Journal of Robust and Nonlinear [41] Cruz, G., and Bernstein, D. S., “Adaptive Spacecraft Attitude Control Control, Vol. 28, No. 9, 2018, pp. 3324–3341. with Reaction Wheel Actuation,” Proceedings of the American Control doi:10.1002/rnc.v28.9 Conference, IEEE Publ., Piscataway, NJ, 2013, pp. 4832–4837. [27] MacKunis, W., Leve, F., Patre, P., Fitz-Coy, N., and Dixon, W., doi:10.1109/ACC.2013.6580586 “Adaptive Neural Network-Based Satellite Attitude Control in the [42] Cruz, G., and Bernstein, D. S., “Retrospective Cost Adaptive Control of Presence of CMG Uncertainty,” Aerospace Science and Technology, Spacecraft Attitude Using Magnetic Actuators,” AIAA Guidance, Vol. 54, July 2016, pp. 218–228. Navigation, and Control Conference, AIAA Paper 2013-4563, 2013. doi:10.1016/j.ast.2016.04.022 doi:10.2514/6.2013-4563 [28] Cui, P., and Liu, F., “Attitude-Tracking Control with Path Planning [43] Zhao, Z., Cruz, G., Lee, T., and Bernstein, D. S., “Adaptive Attitude for Agile Satellite Using Double-Gimbal Control Moment Control of a Dual-Rigid-Body Spacecraft with Unmodeled Non- Gyros,” Mathematical Problems in Engineering, Vol. 2012, 2012, minimum-Phase Dynamics,” 2018 Annual American Control pp. 1–19. Conference (ACC), IEEE Publ., Piscataway, NJ, 2018, pp. 2503–2508. doi:10.1155/2012/342043 doi:10.23919/ACC.2018.8431163 [29] Yoon, H., and Tsiotras, P., “Spacecraft Adaptive Attitude and Power [44] Vadali, S., “Feedback Control and Steering Laws for Spacecraft Using Tracking with VariableSpeed Control Moment Gyroscopes,” Journal of Single Gimbal Control Moment Gyros,” Guidance, Navigation and Guidance, Control, and Dynamics, Vol. 25, No. 6, 2002, pp. 1081– Control Conference, AIAA Paper 1988-3475, 1988. 1090. doi:10.2514/6.1989-3475 doi:10.2514/2.4987 [45] Asghar, S., “Exact Steering in Control Moment Gyroscope Systems,” [30] Yoon, H., and Agrawal, B. N., “Adaptive Control of Uncertain Ph.D. Thesis, Univ. of Surrey, Guildford, England, U.K., 2008. Hamiltonian Multi-Input Multi-Output Systems with Application to [46] Bernstein, D. S., Scalar, Vector, and Matrix Mathematics: Theory, Spacecraft Control,” Proceedings of the American Control Conference, Facts, and Formulas, Princeton Univ. Press, Princeton, NJ, 2018, IEEE Publ., Piscataway, NJ, 2008, pp. 2969–2974. p. 388. doi:10.1109/ACC.2008.4586947 [47] Morozov, A. V., Hoagg, J. B., and Bernstein, D. S., “Retrospective Cost [31] Sun, Z., and Ding, S., “SGCMG Non-Singularity Steering Based on Adaptive Control of a Planar Multilink Arm with Nonminimum-Phase Adaptive Gauss Pseudospectral Method,” International Conference on Zeros,” Proceedings of the IEEE Conference on Decision and Control, Informative and Cybernetics for Computational Social Systems, IEEE IEEE Publ., Piscataway, NJ, 2010, pp. 3706–3711. Downloaded by UNIV OF MICHIGAN on August 20, 2019 | http://arc.aiaa.org DOI: 10.2514/1.G003926 Publ., Piscataway, NJ, 2014, pp. 96–101. doi:10.1109/CDC.2010.5718121 doi:10.1109/ICCSS.2014.6961823 [48] Weiss, A., Kolmanovsky, I., Bernstein, D. S., and Sanyal, A., “Inertia- [32] Ozawa, R., and Takahashi, M., “Agile Attitude Control and Singularity Free Spacecraft Attitude Control Using Reaction Wheels,” Journal of Avoidance/Escape by the SDRE Method Using a Biased State- Guidance, Control, and Dynamics, Vol. 36, No. 5, 2013, pp. 1425– Dependent Weighting Matrix,” Applied Sciences, Vol. 8, No. 1, 2018, 1439. Paper 140. doi:10.2514/1.58363 doi:10.3390/app8010140 [49] Hu, Q., Li, B., and Zhang, A., “Robust Finite-Time Control Allocation [33] Zhang, F., Jin, L., and Xu, S., “Fault Tolerant Attitude Control for in Spacecraft Attitude Stabilization Under Actuator Misalignment,” Spacecraft with SGCMGs Under Actuator Partial Failure and Actuator Nonlinear Dynamics, Vol. 73, Nos. 1–2, 2013, pp. 53–71. Saturation,” Acta Astronautica, Vol. 132, March 2017, pp. 303–311. doi:10.1007/s11071-013-0766-2 doi:10.1016/j.actaastro.2016.12.033 [50] Mercker, T. H., and Akella, M. R., “Adaptive Estimation and Control [34] Santillo, M. A., and Bernstein, D. S., “Adaptive Control Based on Algorithms for Certain Independent Control Axis Misalignments,” Retrospective Cost Optimization,” Journal of Guidance, Control, and Journal of Guidance, Control, and Dynamics, Vol. 37, No. 1, 2014, Dynamics, Vol. 33, No. 2, 2010, pp. 289–304. pp. 72–85. doi:10.2514/1.46741 doi:10.2514/1.61735