6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved IAC--C.5.9 INERTIA-FREE ATTITUDE CONTROL OF SPACECRAFT WITH UNKNOWN TIME-VARYING MASS DISTRIBUTION Avishai Weiss University of Michigan, United States, avishai@umich.edu Ilya Kolmanovsky University of Michigan, United States, ilya@umich.edu Dennis S. Bernstein University of Michigan, United States, dsbaero@umich.edu We derive a continuous, inertia-free control law for spacecraft attitude tracking that is applicable to non-rigid spacecraft with translating on-board components. This control law is simulated for slew spin maneuvers. I. INTRODUCTION Attitude control of spacecraft remains a challenging nonlinear control problem of intense practical intellectual importance. Since rotational motion evolves on the set of proper orthogonal matrices, continuous control must account for the presence of multiple equilibria, whereas discontinuous control laws based on quaternions alternative parameterizations of the rotation matrices lead to additional complications. Challenges also arise depending on the properties of the actuation hardware, for example, thrusters, reaction wheels, control-moment gyros, magnetic torquers, as well as sensing hardware, for example, gyros, magnetometers, star trackers. Finally, this problem is exacerbated by uncertainty involving the mass distribution of the spacecraft. The present paper addresses an additional complication in spacecraft control, namely, the situation in which the mass distribution of the spacecraft is not only uncertain but also time-varying. Many spacecraft are built to deploy on orbit, for example, by exping solar panels or a magnetometer boom. Furthermore, a spacecraft may have moving components, such as a reflector or antenna that rotates relative to the spacecraft bus in order to track a ground station. In these cases, the mass distribution changes as a function of time, which, in turn, gives rise to a time-varying inertia matrix. In the present paper we address the problem of spacecraft attitude control with time-varying inertia. The approach that we take is an extension of the approach of ref., where continuous control laws are developed based on rotation matrices. For motion-to-rest (that is, slew) maneuvers in the absence of disturbance torques, no knowledge of the inertia matrix is needed, no estimates of the inertia matrix are constructed. For motion-to-specified-motion (for example, spin) maneuvers in the presence of harmonic (possibly constant) disturbances with known spectral content, the control law is based on an estimate of the inertia matrix; however, this estimate need not converge to the actual inertia matrix. The contribution of the present paper is an extension of the results of ref. to the case in which the mass distribution of the spacecraft is both uncertain time-varying. For motion-to-rest maneuvers, we show that the corresponding control law of ref. is effective under a special choice of the control-law parameters. This requirement can be ignored when the inertia matrix is increasing, for example, during deployment. For motion-to-specified-motion maneuvers, we make the additional assumption that the time-variation of the timevarying component of the inertia is known, whereas its spatial distribution is unknown. Under these assumptions, we extend the motion-to-specified-motion control law of ref. to the case of time-varying inertia. The contents of the paper are as follows. In Section II we develop a model of a spacecraft with time-varying inertia. In Section III we describe the attitude control objectives. Section IV deals with motion-to-rest maneuvers, while Section V treats motion-to-specified-motion maneuvers. II. SPACECRAFT MODEL Let the spacecraft be denoted by sc, let c denote its center of mass. We assume that the spacecraft is composed of a rigid bus additional moving components. These components are assumed to not rotate relative to the spacecraft; for example, they may move linearly in a body-fixed direction. We assume a bus-fixed frame F B IAC--C.5.9 Page of 9
6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved an Earth-centered inertial frame F E. We begin with Newton s second law for rotation, which states that the derivative of the angular momentum of a body relative to its center of mass with respect to an inertial frame is equal to the sum of the moments applied to that body about its center of mass. We thus have M sc/c = E H sc/c/e E {}}{ = I sc/cω B/E B {}}{ = I sc/cω B/E + ω B/E I sc/cω B/E B = I sc/c ω B/E + B I sc/c ω B/E + ω B/E I sc/c ω B/E, [] where I sc/c is the positive-definite inertia tensor of the spacecraft relative to its center of mass, ω B/E is the angular velocity of F B with respect to F E. We separate the moments on the spacecraft M sc/c into disturbance moments M dist control moments M control. We now resolve [] in F B using the notation J = I b/c B, B J = ω = ω B/E B, ω = I wi/c, B B ω B/E B, τ dist = M dist B, Bu = M control B, where the components of the vector u R 3 represent three independent torque inputs, while the rows of the matrix B R 3 3 determine the applied torque about each axis of the spacecraft frame due to u as given by the product Bu. We let the vector τ dist represent disturbance torques, that is, all internal external torques applied to the spacecraft aside from control torques. Disturbance torques may be due to onboard components, gravity gradients, solar pressure, atmospheric drag, or the ambient magnetic field. Resolving [] in F B rearranging yields J ω = Jω ω + Bu Jω + τ dist. [] The kinematics of the spacecraft model are given by Poisson s equation Ṙ = Rω, [3] which complements []. In [3], ω denotes the skewsymmetric matrix of ω, R = O E/B R 3 3 is the rotation tensor that transforms F E into F B resolved in either F E or F B. Therefore, R is the proper orthogonal matrix (that is, the rotation matrix) that transforms the components of a vector resolved in the bus-fixed frame into the components of the same vector resolved in the inertial frame. Compared to the rigid body case treated in ref., the time-varying inertia complicates the dynamic equations due to the term Jω added to []. Note that this term affects only the attitude of the spacecraft when the spacecraft has nonzero angular velocity. The kinematic relation [3] remains unchanged. Both rate (inertial) attitude (noninertial) measurements are assumed to be available. Gyro measurements y rate R 3 are assumed to provide measurements of the angular velocity resolved in the spacecraft frame, that is, y rate = ω + v rate, [4] where v rate R 3 represents the presence of noise in the gyro measurements. Attitude is measured indirectly using sensors such as magnetometers or star trackers. The attitude is determined to be y attitude = R. [5] When attitude measurements are given in terms of an alternative attitude representation, such as quaternions, Rodrigues s formula can be used to determine the corresponding rotation matrix. Attitude estimation on SO(3) is considered in ref. 3. III. OBJECTIVES FOR CONTROL DESIGN The objective of the attitude control problem is to determine control inputs such that the spacecraft attitude given by R follows a commed attitude trajectory given by a possibly time-varying C rotation matrix R d (t). For t, R d (t) is given by Ṙ d (t) = R d (t)ω d (t), [6] R d () = R d, [7] where ω d is the desired, possibly time-varying angular velocity. The error between R(t) R d (t) is given in terms of the attitude-error rotation matrix R = R T d R, IAC--C.5.9 Page of 9
6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved which satisfies the differential equation R = R ω, [8] where the angular velocity error ω is defined by ω = ω R T ω d. We rewrite [] in terms of the angular-velocity error as J ω = [J( ω + R T ω d )] ( ω + R T ω d ) + Bu + τ dist + J( ω R T ω d R T ω d ) J( ω + R T ω d ). [9] A scalar measure of attitude error is given by the rotation angle θ(t) about an eigenaxis needed to rotate the spacecraft from its attitude R(t) to the desired attitude R d (t), which is given by 4 θ(t) = cos ( [tr R(t) ]). [] IV. MOTION-TO-REST CONTROL Two controllers are presented in ref.. When no disturbances are present, the inertia-free control law given by (38) of ref. achieves almost global stabilization of a constant desired configuration R d, that is, a slew maneuver that brings the spacecraft to rest. As in ref., define the Lyapunov cidate V (ω, R) = ωt Jω + K p tr(a A R), [] where K p is a positive number A R 3 3 is a diagonal positive-definite matrix given by A = diag(a, a, a 3 ). Let u be given by (38) of ref., that is, u = B (K p S + K v ω), [] where K v R 3 3 is positive definite, S is defined as S = a i ( R T e i ) e i, [3] where, for i =,, 3, e i denotes the ith column of the 3 3 identity matrix. Taking the derivative of () along the trajectories of () yields V (ω, R) = ω T J ω + ωt Jω + K p ω T S = ω T ( Jω ω + Bu Jω + Jω + K p ω T ( S = ω T K p S K v ω ) Jω + K p ω T S ( = ω T K v + ) J ω, [4] ) where the derivative of K p tr(a A R) is given by K p ω T S as shown in Section V. Selecting K v > J + εi for some ε > ensures that () decays as in ref. but otherwise, the controller requires no modification for the case of time-varying inertia. This condition is automatically satisfied when the inertia matrix is increasing, that is, J(t ) J(t ), for all t t, which implies that J. Thus, for every positive-definite choice of K v, it follows that K v > J + εi for some ε >. This is the case, for example, during solar panel or magnetometer boom deployment. During retraction, a bound on J must be known in order to properly select K v. For simulation, we assume that the inertia of the spacecraft takes the form J = J(t) = J + J (t), where J is constant represents the rigid part of the spacecraft, J (t) is time-varying represents a moving part of the spacecraft. We simulate a point mass moving linearly in time outward along the spacecraft s y-axis, representing solar panel deployment. The inertia matrix J (t) is thus given by J (t) = min(t, t d ) min(t, t d ) kg-m, where t d is the time it takes to deploy. The following parameters are used. The inertia matrix J is given by J = 5..5..5 3.5 kg-m, with principal moments of inertia.4947, 3.7997, 5.56 kg-m. Let t d =, set K p = 5 K v = 5 I 3. Since the inertia is increasing, any positive definite K v is acceptable. We use controller () for an aggressive slew maneuver, where the objective is to bring the spacecraft from the initial attitude R = I 3 initial angular velocity ω() = [.5 ] T rad/sec to rest (ω d = ) at the desired final orientation R d = diag(,, ), which represents a rotation of 8 degrees about the x-axis. Figures -3 show, respectively, the attitude error, angular velocity components, control torque components. The spacecraft attitude angular velocity components are brought close to the desired values in about 5 sec, before the solar panel deployment is complete, are maintained throughout the remainder of the deployment. IAC--C.5.9 Page 3 of 9
6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved 3.5 3 3 Eigenaxis Attitude Error, rad.5.5 Torque Inputs, N m 3.5 4 4 6 8 4 6 8 5 4 6 8 4 6 8 Fig. : Eigenaxis attitude error using the control law [] for a slew maneuver during translational motion of an internal mass. Fig. 3: Control torque components using the control law [] for a slew maneuver during translational motion of an internal mass. 3.5 3.5 3 3 Angular Velocity Components, rad/sec.5.5.5 Eigenaxis Attitude Error, rad.5.5.5.5 4 6 8 4 6 8 5 5 Fig. : Spacecraft angular-velocity components using the control law [] for a slew maneuver during translational motion of an internal mass. Fig. 4: Eigenaxis attitude error using the control law [] of ref. for a slew maneuver during translational motion of an internal mass. V. MOTION-TO-SPECIFIED-MOTION CONTROL A more general control law that tracks a desired attitude trajectory for rigid spacecraft in the presence of disturbances is given by [] of ref.. We apply this controller to the non-rigid spacecraft presented in the previous section. Additionally, we assume a constant nonzero disturbance torque, τ dist = [.7.3 ] T. The parameters of the controller are chosen to be K = D = I 3, A = diag(,, 3), K p = 6, K v = 6 I 3, Q = I 6. We first consider the slew maneuver. Figures 4-6 show, respectively, the attitude error, angular velocity components, control torque components. The spacecraft attitude angular velocity components are brought close to the desired values in under 5 sec. The persistent nonzero control torque seen in Figure 6 is due to the constant nonzero disturbance torque. While the controller [] of ref. assumes that the spacecraft is rigid, it successfully completes the slew maneuver by treating the term Jω as a disturbance that gradually disappears as the spacecraft is brought to rest (ω = ). This suggests that it might not succeed at spin maneuvers where ω as t. Before simulating a spin maneuver, we modify J (t) so that it is persistent throughout the simulation, rather IAC--C.5.9 Page 4 of 9
6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved 5..9 4.8 Angular Velocity Components, rad/sec 3 Eigenaxis Attitude Error, rad.7.6.5.4.3.. 5 5 Fig. 5: Spacecraft angular-velocity components using the control law [] of ref. for a slew maneuver during translational motion of an internal mass. 3 4 5 6 7 8 9 Fig. 7: Eigenaxis attitude error using the control law [] of ref. for a spin maneuver during accordion-like motion. 6.8 4.6 Torque Inputs, N m 4 Angular Velocity Components, rad/sec.4...4 6.6 8 5 5 Fig. 6: Control torque components using the control law [] of ref. for a slew maneuver during translational motion of an internal mass..8 3 4 5 6 7 8 9 Fig. 8: Spacecraft angular-velocity components using the control law [] of ref. for a spin maneuver during accordion-like motion. than coming to rest at sec., as before. We let J (t) = sin (πt)j, which represents an accordion like motion, while still preserving the required inertia inequalities J a J b + J c, J b J a + J c, J c J a + J b, where J a, J b, J c are the principal moments of inertia. We now consider a spin maneuver with the spacecraft initially at rest R() = I 3. The specified attitude is given by R d () = I 3 with the desired constant angular velocity ω d = [.5.5.3 ] T rad/sec. Figures 7-9 show, respectively, the attitude error, angular velocity components, control torque components. The spacecraft attitude angular velocity components are brought close to the desired values in about sec. but do not settle, as was expected. The non-rigidity of the spacecraft acts as a disturbance torque that cannot be rejected by the control. V.I. Extended Control Law We now extend controller [] of ref. to the case of a non-rigid spacecraft whose inertia matrix has the form J(t) = J + f(t)j, IAC--C.5.9 Page 5 of 9
6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved 8 6 Similarly, we let where ζ R 6 is defined by J ω = L(ω)ζ [5] Torque Inputs, N m 4 4 6 ζ = [ J J J 33 J 3 J 3 J ] T. Next, let Ĵ R 3 3 denote an estimate of J, Ĵ R 3 3 denote an estimate of J define the inertiaestimation errors J = J Ĵ, 8 3 4 5 6 7 8 9 Fig. 9: Control torque components using the control law [] of ref. for a spin maneuver during accordionlike motion. where f(t) is known but J J are unknown. The following preliminary results are needed. Let I denote the identity matrix, whose dimensions are determined by context, let M ij denote the i, j entry of the matrix M. The following result is given in ref.. Lemma. Let A R 3 3 be a diagonal positivedefinite matrix. Then the following statements hold for a proper orthogonal matrix R: i) For all i, j =,, 3, R ij [, ]. ii) tr (A AR). iii) tr (A AR) = if only if R = I. For convenience we note that, if R is a rotation matrix x, y R 3, then, therefore, (Rx) = Rx R T, R(x y) = (Rx) Ry. Next we introduce the notation where γ R 6 is defined by J ω = L(ω)γ, γ = [ J J J 33 J 3 J 3 J ] T L(ω) = ω ω 3 ω ω ω 3 ω ω 3 ω ω. J = J Ĵ. Letting ˆγ, γ R 6 represent Ĵ, J, respectively, ˆζ, ζ R 6 represent Ĵ, J, respectively, it follows that γ = γ ˆγ, ζ = ζ ˆζ. Likewise, let ˆτ dist R 3 denote an estimate of τ dist, define the disturbance-estimation error τ dist = τdist ˆτ dist. We now state the assumptions upon which the following development is based: Assumption. J J are constant unknown. Assumption. f(t) is time-varying known. Assumption 3. Each component of τ dist is a linear combination of constant harmonic signals, whose frequencies are known but whose amplitudes phases are unknown. Assumption 3 implies that τ dist can be modeled as the output of an autonomous system of the form d = A d d, [6] τ dist = C d d, [7] where A d R n d n d C d R 3 n d are known matrices A d is a Lyapunov-stable matrix. In this model, d() is unknown, which is equivalent to the assumption that the amplitude phase of each harmonic component of the disturbance is unknown. The matrix A d is chosen to include eigenvalues of all frequency components that may be present in the disturbance signal, where the zero eigenvalue corresponds to a constant disturbance. In effect, the controller provides infinite gain at the disturbance frequency, which results in asymptotic rejection of harmonic disturbance components. In particular, an integral controller provides infinite gain at DC in order to reject constant disturbances. In the case IAC--C.5.9 Page 6 of 9
6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved of orbit-dependent disturbances, the frequencies can be estimated from the orbital parameters. Likewise, in the case of disturbances originating from on-board devices, the spectral content of the disturbances may be known. In other cases, it may be possible to estimate the spectrum of the disturbances through signal processing. Assumption 3 implies that A d can be chosen to be skew symmetric, which we do henceforth. Let ˆd R n d denote an estimate of d, define the disturbance-state estimation error d = d ˆd. Theorem. Let K p be a positive number, let K R 3 3, let Q, Z R 6 6 D R n d n d be positive definite matrices, let A = diag(a, a, a 3 ) be a diagonal positive-definite matrix, define Then the function S = a i ( R T e i ) e i. V ( ω, R, γ, d) = ( ω + K S) T J( ω + K S) + K p tr (A A R) + γt Q γ + ζ T Z ζ + d T D d [8] is positive definite, that is, V is nonnegative, V = if only if ω =, R = I, γ =, d =. Proof. It follows from statement of Lemma that tr (A A R) is nonnegative. Hence V is nonnegative. Now suppose that V =. Then, ω + K S =, γ =, d =, it follows from statement 3 of Lemma that R = I, thus S =. Therefore, ω =. Theorem. Let K p be a positive number, let K v R 3 3, K R 3 3, Q, Z R 6 6, D R n d n d be positive definite matrices, assume that A T d D + DA d is negative semidefinite, let A = diag(a, a, a 3 ) be a diagonal positive-definite matrix, define S V as in Theorem, let ˆγ, ˆζ, ˆd satisfy where ˆγ = Q [L T (ω)ω + L T (K Ṡ + ω ω R T ω d )]( ω + K S), [9] ˆζ = Z [f(t)l T (ω)ω + f(t)l T (K Ṡ + ω ω R T ω d ) + f(t)l T ( ( ω + KṠ) ω)]( ω + K S), [] Ṡ = a i [( R T e i ) ω] e i [] ˆd = A d ˆd + D C T d ( ω + K S), [] ˆτ dist = C d ˆd, [3] so that ˆτ dist is the disturbance torque estimate. Furthermore, consider the control law where u = B (v + v + v 3 ), [4] v = ( Ĵ + f(t)ĵ) ω ω Then, f(t)ĵ( ( ω + KṠ) ω) (Ĵ + f(t)ĵ)(k Ṡ + ω ω R T ω d ), [5] v = ˆτdist, [6] v 3 = Kv ( ω + K S) K p S. [7] V ( ω, R, γ, d) = ( ω + K S) T K v ( ω + K S) is negative semidefinite. Proof. Noting that K p S T K S d dt tr (A A R) = tr A R + d T (A T d D + DA d ) d [8] = tr A( Rω ω R) d = a i e T i ( Rω ω R)e d i = = = = [ = [ a i e T R(ω i R T ω R)e d i a i e T R(ω i R T ω d ) e i a i e T Re i i ω a i e i R T e i ] T ω a i ( R T e i ) e i ] T ω = ω T S, IAC--C.5.9 Page 7 of 9
6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved we have V ( ω, R, γ, d) = ( ω + K S) T (J ω + JK Ṡ) K p tr A R γ T Q ˆγ + ( ω + K S) T J( ω + K S) ζ T T ˆζ + dt D d = ( ω + K S) T [Jω ω + J( ω ω R T ω d ) Bu Jω + τ dist + JK Ṡ + J( ω + K Ṡ)] + K p ω T S γ T Q ˆγ ζ T T ˆζ + dt D d = ( ω + K S) T [Jω ω + J(K Ṡ + ω ω R T ω d ) + v + v + v 3 + τ dist + J( ( ω + KṠ) ω)] + K p ω T S γ T Q ˆγ ζ T T ˆζ + dt D d = ( ω + K S) T [( J + f(t) J )ω ω + ( J + f(t) J )(K Ṡ + ω ω R T ω d ) + f(t) J ( ( ω + K Ṡ) ω)] + ( ω + K S) T τ dist ( ω + K S) T K v ( ω + K S) K p ( ω + K S) T S + K p ω T S γ T Q ˆγ ζ T T ˆζ + dt D d = ( ω + K S) T [L(ω) γ ω + L(K Ṡ + ω ω R T ω d ) γ] + ( ω + K S) T [f(t)l(ω) ζ ω + f(t)l(k Ṡ + ω ω R T ω d ) ζ + f(t)l( ( ω + K S) ω) ζ] ( ω + K S) T K v ( ω + K S) K p S T K S γ T Q ˆγ ζ T T ˆζ + dt C T d ( ω + K S) + d T D[A d d D C T d ( ω + K S)] = ( ω + K S) T K v ( ω + K S) K p S T K S γ T Q ˆγ + ( ω + K S) T [ ω L(ω) + L(K Ṡ + ω ω R T ω d )] γ ζ T Z ˆζ + ( ω + K S) T [ f(t)ω L(ω) + f(t)l(k Ṡ + ω ω R T ω d ) + f(t)l( ( ω + K S)] ζ + d T (A T d D + DA d ) d = ( ω + K S) T K v ( ω + K S) K p S T K S + γ T [ Q ˆγ + (L T (ω)ω + L T (K Ṡ + ω ω R T ω d ))( ω + K S)] + ζ T [ Z ˆζ + (f(t)l T (ω)ω + f(t)l T (K Ṡ + ω ω R T ω d ))( ω + K S) + f(t)l T ( ( ω + K S)] + d T (A T d D + DA d ) d = ( ω + K S) T K v ( ω + K S) K p S T K S + d T (A T d D + DA d ) d. Note that a bound on J need not be known as in the case of for the motion-to-rest controller. This is due to the J estimator [] accounting for the extra term that appears in the derivative of [8] due to the spacecraft having a time-varying inertia matrix. We simulate the spin maneuver using controller [4] on the non-rigid spacecraft with J (t) = sin (πt)j, as before. We assume a constant nonzero disturbance torque, τ dist = [.7.3 ] T. The parameters of the controller are chosen to be K = I 3, A = diag(,, 3), K p = 5, K v = I 3, D = I 3, Q = I 6. Figures - show, respectively, the attitude error, angular velocity components, control torque components. The spacecraft attitude angular velocity components are brought close to the desired values in under sec. The modified controller [4] is able to reject the persistent disturbance caused by the non-rigidity of the spacecraft. Eigenaxis Attitude Error, rad.4...8.6.4. 3 4 5 6 7 8 9 Fig. : Eigenaxis attitude error using the control law [4] for a spin maneuver during accordion-like motion. VI. CONCLUSIONS We extended the control laws of ref. to the case of non-rigid spacecraft, without requiring knowledge of the spacecraft s time-varying inertia. These results have practical advantages relative to previous controllers that ) require exact or approximate inertia information or ) are based on attitude parameterizations such as quaternions that require discontinuous control laws. We simulated these controllers for various slew spin maneuvers, demonstrating their effectiveness. Future work will complete the proof for almost global stabilization (that is, Lyapunov stability with almost global convergence) of non-rigid spacecraft atti- IAC--C.5.9 Page 8 of 9
6nd International Astronautical Congress, Cape Town, SA. Copyright by the International Astronautical Federation. All rights reserved Angular Velocity Components, rad/sec.5.5.5 with Disturbance Rejection Almost Global Stabilization, AIAA J. Guid. Contr. Dyn., Vol. 3, pp. 67-78, 9. 3 Sanyal, A. K., Lee, T., Leok, M., McClamroch, N. H., Global Optimal Attitude Estimation Using Uncertainty Ellipsoids, Systems Control Letters, Vol. 57, pp. 36 45, 8. 4 Hughes, P. C., Spacecraft Attitude Dynamics, Wiley, 986; reprinted by Dover, 8, page 7..5 3 4 5 6 7 8 9 Fig. : Spacecraft angular-velocity components using the control law [4] for a spin maneuver during accordion-like motion. 8 6 4 Torque Inputs, N m 4 6 8 3 4 5 6 7 8 9 Fig. : Control torque components using the control law [4] for a spin maneuver during accordion-like motion. tude tracking using controller [4]. Additionally, extensions of this method may be applicable to non-rigid spacecraft with moving components that are not necessarily translating relative to the spacecraft frame. REFERENCES Chaturvedi N., Sanyal A., McClamroch, N. H., Rigid Body Attitude Control: Using rotation matrices for continuous, singularity-free control laws, IEEE Control Systems Magazine, Vol. 3(3), pp. 3-5,. Sanyal A., Fosbury A., Chaturvedi N., Bernstein D. S., Inertia-Free Spacecraft Attitude Tracking IAC--C.5.9 Page 9 of 9