Inertia-Free Spacecraft Attitude Control Using. Reaction Wheels

Transcription

1 Inertia-Free Spacecraft Attitude Control Using Reaction Wheels Avishai Weiss, Ilya Kolmanovsky, Dennis S. Bernstein, Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI and Amit Sanyal, Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM This paper extends the continuous inertia-free control law for spacecraft attitude tracking derived in prior work to the case of three axisymmetric reaction wheels. The wheels are assumed to be mounted in a known and linearly independent, but not necessarily orthogonal, configuration with an arbitrary and unknown orientation relative to the unknown spacecraft principal axes. Simulation results for slew and spin maneuvers are presented with torque and momentum saturation. I. Introduction In spacecraft applications it is often expensive to determine the mass properties with a high degree of accuracy. To alleviate this requirement, the control algorithms given in refs. 3 are inertia-free in the sense that they require no prior modeling of the mass distribution. The control algorithms in refs., 3 incorporate internal states that can be viewed as estimates of the moments and products of inertia; however, these estimates need not converge to the true values, and in fact do not converge to the true values except under sufficiently persistent motion. The results of ref. 3 are based on rotation matrices 4, 5, 6 as an alternative to quaternions as used in refs. Quaternions provide a double cover of the rotation group SO(3), and thus, when used as the basis of a continuous control algorithm, cause unwinding, that is, unnecessary rotation away from and then back Graduate student. Professor, AIAA Member. Professor, AIAA Member. Assistant Professor, AIAA Member. of 3

2 to the desired physical attitude. 7 To avoid unwinding while using quaternions, it is thus necessary to resort to discontinuous control algorithms, which introduce the possibility of chatter due to noise as well as mathematical complications. 8 On the other hand, rotation matrices allow for continuous control laws but introduce multiple equilibria. Since the spurious equilibria of the closed-loop system are saddle points, the attitude of the spacecraft converges almost globally (but not globally) to the desired equilibrium. Although the derivation of the inertia-free controller in ref. 3 and the present paper is based on rotation matrices, the relevant attitude error given by the S-parameter (see (4)) can be computed from any attitude parameterization, such as quaternions or modified rodrigues parameters, and thus the continuous inertia-free controllers presented in ref. 3 are not confined to rotation matrices. The inertia-free control laws in refs. 3 assume that 3-axis input torques can be specified without onboard momentum storage, which implies implementation in terms of thrusters. However, attitude control laws are typically implemented with wheels, and thus the on-board stored momentum varies with time. To account for this effect, the contribution of this paper is the derivation of an inertia-free control law based on reaction-wheel actuation. Like the inertia-free control laws in refs., 3 the tuning of this control law requires no knowledge of the mass properties of the spacecraft, and this paper specifies the assumptions and modeling information concerning the reaction wheels and their placement relative to the bus. The paper is organized as follows. In Section II coordinate-free equations of motion for the spacecraft are derived, where, unlike ref., 5 the present paper does not assume that the wheels are aligned with the principal axes of the spacecraft bus, nor does it assume that the wheels are balanced with respect to the bus in order to preserve the location of its center of mass; in fact, the reaction wheels may be mounted at any location and in any linearly independent configuration. In Section III the control objectives are formulated, and in Section IV the controller is developed. Simulation results are reported in Section V, where robustness to variations in the spacecraft inertia is demonstrated and controller performance is examined under both torque and momentum saturation. Finally, concluding remarks are made in Section VI. II. Spacecraft Model with Reaction Wheels This section derives the equations of motion for a spacecraft with reaction wheels, while highlighting the underlying assumptions on wheel geometry, inertia, and attachment to the bus. Throughout the paper, the vector r q/p denotes the position of point q relative to point p, the vector v q/p/x = r q/p denotes the velocity of point q relative to point p with respect to frame F X, and the vector ω Y/X denotes the angular velocity of frame F Y relative to frame F X. Note that ( ) denotes a coordinate-free (unresolved) vector. All frames are orthogonal and right handed. X of 3

3 Def.. Let F X be a frame, let B be a collection of rigid bodies B,..., B l, and let p be a point. Then, the angular momentum of B relative to p with respect to F X is defined by H B/p/X = l H Bi/p/X, () where, for i =,..., l, the angular momentum H Bi/p/X of B i relative to p with respect to F X is defined by H Bi/p/X = r dm/p v dm/p/x dm. () B i The following properties of angular momentum are standard. Lemma. Let B be a rigid body, let F X and F Y be frames, and let p be a point. Then, H B/p/X = I B/p ω Y/X + H B/p/Y, (3) where the positive-definite, coordinate-free inertia tensor I B/p is defined by I B/p = B r dm/p U r dm/p r dm/p dm, (4) and where U denotes the second-order identity tensor. Lemma. Let B be a rigid body, let F X and F Y be frames, let F Y be a body-fixed frame, and let p be a point that is fixed in B. Then, H B/p/Y = (5) and H B/p/X = I B/p ω Y/X. (6) Lemma 3. Let F X be a frame, let p be a point, let B be a rigid body with mass m B, and let c be the center of mass of B. Then, H B/p/X = H B/c/X + r c/p m B v c/p/x. (7) Consider a spacecraft sc actuated by three axisymmetric wheels w, w, w 3 attached to a rigid bus b in a known and linearly independent, but not necessarily orthogonal, configuration. Although the spacecraft 3 of 3

4 is not a rigid body, the axial symmetry of the wheels implies that the center of mass c of the spacecraft is fixed in both the bus and the spacecraft. Since the inertia properties of the bus are assumed to be unknown, the principal axes of the bus are unknown and thus the wheel configuration has an arbitrary and unknown orientation relative to the principal axes of the bus. Each wheel is mounted so that it rotates about one of its own principal axes passing through its own center of mass. It is not assumed that the axis of rotation of each wheel passes through the center of mass of the bus, nor is it assumed that the wheels are balanced with respect to the bus in order to preserve the location of its center of mass. Thus the center of mass of the spacecraft and the center of mass of the bus may be distinct points, both of which are unknown. Assume a bus-fixed frame F B, three wheel-fixed frames F W, F W, F W3, whose x-axes are aligned with the rotation axes of w, w, w 3, respectively, and an Earth-centered inertial frame F E. The angular momentum of the spacecraft relative to its center of mass with respect to the inertial frame is given by H sc/c/e = H b/c/e + H wi/c/e, (Def. ) (8) where the angular momentum H b/c/e of the bus relative to c with respect to F E is given by H b/c/e = I b/c ω B/E, (Lemma ) (9) where I b/c is the positive-definite inertia tensor of the bus relative to the center of mass of the spacecraft, and ω B/E is the angular velocity of F B with respect to F E. The angular momentum H wi/c/e of wheel i relative to the center of mass of the spacecraft with respect to the inertial frame is given by H wi/c/e = I wi/c ω B/E + H wi/c/b (Lemma ) = I wi/c ω B/E + H wi/c i/b + r ci/c m wi v ci/c/b (Lemma 3) = I wi/c ω B/E + I wi/c i ω Wi/B, (Lemma ) () where I wi/c is the inertia tensor of wheel i relative to the center of mass of the spacecraft, I wi/c i is the inertia tensor of wheel i relative to the center of mass c i of the ith wheel, and ω Wi/B is the angular velocity of wheel i relative to the bus. Thus (8) is given by ( H sc/c/e = I b/c + I wi/c ) ω B/E + I wi/c i ω Wi/B. () 4 of 3

5 Resolving ω Wi/B in F Wi yields ω Wi/B = ψ i e, () Wi where e = [ ] T and ψ i is the angular rate relative to F B of the ith wheel about the x-axis of F Wi. Since F Wi is aligned with the principal axes of wheel i, it follows that I wi/c i Wi = diag(α i, β i, β i ). (3) Note that ω Wi/B is an eigenvector of I wi/c i with eigenvalue α i, that is, I wi/c i ω Wi/B = α i ω Wi/B. II.A. Spacecraft Equations of Motion The equations of motion for a spacecraft with reaction wheels as described above are now derived. It follows from Newton s second law for rotation that M sc/c = E H sc/c/e E E ({}} ) {{}}{ = I b/c + I wi/c ω B/E + I wi/c i ω Wi/B B {( }} ) { = I b/c + ω B/E + ω B/E + I wi/c B {}}{ I wi/c i ω Wi/B + ω B/E ( I b/c + I wi/c i ω Wi/B I wi/c ) ω B/E = ( I b/c + + ω B/E I wi/c) B ω B/E + (( I b/c + I wi/c α i B ω Wi/B ) ω B/E + ) α iω Wi/B. (4) 5 of 3

6 To resolve (4) in F B, the following notation is used: B J b = I b/c, J wi = I wi/c, B J w = I wi/c, J sc = Jb + J w, B ω = ω B/E B, ω = B ω B/E B, ν i = ω Wi/B, B ν i = B ω Wi/B, B τ dist = M sc/c B. The vector τ dist represents disturbance torques, that is, all external torques applied to the spacecraft aside from control torques. Disturbance torques may be due to gravity gradients, solar pressure, atmospheric drag, or the ambient magnetic field. As in (), the angular acceleration ν i of each wheel has one degree of freedom. In F Wi, B ω Wi/B = Wi W i ω Wi/B = ψ i e. (5) Wi Thus, ν i = B ω Wi/B = O B/Wi B B ω Wi/B = O B/Wi ψ i e, (6) Wi where the proper orthogonal matrix O B/Wi R 3 3 is the orientation matrix that transforms components of a vector resolved in F Wi into the components of the same vector resolved in F B. With the above notation, resolving (4) in F B yields ( ) τ dist = (J b + J w ) ω + α i ν i + ω (J b + J w ) ω + α i ν i 6 of 3

7 ( ) = J sc ω + α i O B/W ψ i e + ω J sc ω + α i O B/W ψ i e = J sc ω + J α ν + ω (J sc ω + J α ν), (7) where J α = [ α O B/W e α O B/W e α 3 O B/W3 e ], (8) ν = [ψ ψ ψ 3 ] T, and ν = [ ψ ψ ψ 3 ] T. Rearranging (7) and choosing the control input u to be ν yields the equations of motion for a spacecraft with reaction wheels, which have the form J sc ω = (J sc ω + J α ν) ω J α u + τ dist, (9) ν = u. () In practice, a servo loop is closed around each reaction wheel in order to produce the desired wheel angular accelerations given in (). Instead of commanding wheel angular accelerations by implementing servo loops, motor torque commands can be used. To determine the relationship between the desired angular acceleration and the required motor torque, the dynamic equations for each wheel must be derived. It follows that M wi/c i = E H wi/c i/e = E {}}{ I wi/c i ω Wi/E = I wi/c i W ω Wi/E + ω Wi/E I wi/c i ω Wi/E ( ) = B B I wi/c i ω B/E + ω Wi/B ω Wi/B ω B/E ( ω ) + B/E + ω Wi/B ( ω ) I wi/c i B/E + ω Wi/B. () 7 of 3

8 Resolving () in F B and projecting it along each motor axis yields τ motor,i = e T i [ Jwi/c i ( ω + ν i ν i ω) + (ω + ν i ) J wi/c i (ω + ν i ) ], () where J wi/c i = I wi/c i B. Although () can be used for torque control, the measurements of ω, ω, ν i, and ν i needed to implement it demonstrate why reaction wheels are typically angular-acceleration commanded and feedback-controlled rather than torque-commanded. II.B. Specialization: principal-axis alignment As in ref., 5 the equations of motion (9), () are now specialized by assuming that the principal axes of the bus are aligned with the rotational axes of the wheels, that the wheels are mass-balanced relative to the center of mass of the bus so that the center of mass of the spacecraft coincides with the center of mass of the bus, and, finally, that the moments of inertia β, β, β 3 of the wheels are lumped into the bus inertia J b = diag(j b, J b, J b3 ), where J b = Jb + β + β 3, J b = Jb + β + β 3, and J b3 = Jb3 + β + β. In this configuration, O B/W e =, O B/W e =, O B/W3 e =. (3) Therefore, J α = J w = diag(α, α, α 3 ). Rewriting the equations of motion (9), () as J b ω = ((J b + J α )ω + J α ν) ω + u + τ dist, (4) u = J α ( ω + ν), (5) and simplifying, yields J b ω = (J b J b3 ) ω ω 3 + α ω 3 (ω + ν ) α 3 ω (ω 3 + ν 3 ) + u + τ dist, (6) J b ω = (J b3 J b ) ω 3 ω + α 3 ω (ω 3 + ν 3 ) α ω 3 (ω + ν ) + u + τ dist, (7) J b3 ω 3 = (J b J b ) ω ω + α ω (ω + ν ) α ω (ω + ν ) + u 3 + τ dist3, (8) u = α ( ω + ν ), (9) u = α ( ω + ν ), (3) u 3 = α 3 ( ω 3 + ν 3 ), (3) 8 of 3

9 which are equations (7.59) and (7.6) of ref. 5 III. Spacecraft Model, Assumptions, and Control Objectives For the control laws (44) and (53) given below, the assumptions presented in Section II.B are not invoked. The kinematics of the spacecraft are given by Poisson s equation Ṙ = Rω, (3) which complements (9) and (). In (3), ω denotes the skew-symmetric matrix of ω, and R = O E/B R 3 3. Both rate (inertial) and attitude (noninertial) measurements are assumed to be available. Compared to the case of thrusters treated in ref., 3 reaction-wheel actuation complicates the dynamic equations due to the term J α ν in (9), as well as the integrators () augmented to the system. The kinematic relation (3) remains unchanged. The torque inputs applied to each reaction wheel are constrained by current limitations on the electric motors and amplifiers as well as angular-velocity constraints on the wheels. These constraints are addressed indirectly in Section V. The objective of the attitude control problem is to determine control inputs such that the spacecraft attitude given by R follows a commanded 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), (33) R d () = R d, (34) where ω d is the desired, possibly time-varying angular velocity. The error between R(t) and R d (t) is given in terms of the attitude-error rotation matrix R = R T d R, (35) which satisfies the differential equation R = R ω, (36) where the angular velocity error ω is defined by ω = ω R T ω d. (37) 9 of 3

10 Rewrite (9) in terms of ω as J sc ω = [J sc ( ω + R T ω d ) + J α ν] ( ω + R T ω d ) + J sc ( ω R T ω d R T ω d ) J α u + τ dist. (38) III.A. Attitude Error A scalar measure of attitude error is given by the eigenaxis attitude error, which is the rotation angle θ(t) about the eigenaxis needed to rotate the spacecraft from its attitude R(t) to the desired attitude R d (t). This angle is given by 7 θ(t) = cos ( [tr R(t) ]). (39) III.B. Spacecraft Inertia Since the control laws in this paper require no inertia modeling, examples that span a range of possible inertia matrices are considered. The inertia of a rigid body is determined by its principal moments of inertia, that is, the diagonal entries of the inertia tensor resolved in a principal body-fixed frame, in which case the inertia matrix is a diagonal matrix. If the inertia tensor is resolved in a non-principal body-fixed frame, then the diagonal entries are the moments of inertia and the off-diagonal entries are the products of inertia. The off-diagonal entries of the inertia matrix are thus a consequence of an unknown rotation between a principal body-fixed frame and the chosen body-fixed frame. Figure shows the triangular region of feasible principal moments of inertia of a rigid body. There are five cases that are highlighted for the principal moments of inertia λ λ λ 3 >, where λ, λ, λ 3 satisfy the triangle inequality λ < λ + λ 3. Let m denote the mass of the rigid body. The point λ = λ = λ 3 5λ corresponds to a sphere of radius r = m, a cube whose sides have length l = 6λ m, and a cylinder of length l and radius r, where l/r = 3 and r = length l and radius r, where l/r = 3 and r = λ m. The point λ = λ = λ 3 corresponds to a cylinder of λ triangular region, corresponds to a solid rectangular body with sides l = m. The point λ = 6 5 λ = λ 3, located at the centroid of the 8λ m > l 4λ = m > l λ 3 = m. The remaining cases in Figure are nonphysical, limiting cases. The point λ = λ = λ 3 corresponds to a thin disk of radius r = λ m cylinder of radius r = and length l = and length l =. The point λ = λ and λ 3 = corresponds to a thin λ m. Finally, each point along the line segment λ = λ + λ 3, λ m > l λ = 3 m. where λ > λ 3, corresponds to a thin rectangular plate with sides of length l = For all simulations of the inertia-free control laws, the principal axes are viewed as the nominal bodyfixed axes, and thus the nominal inertia matrix is a diagonal matrix whose diagonal entries are the principal moments of inertia. To demonstrate robustness, the principal moments as well as the orientation of the of 3

11 λ 3 Sphere λ = λ = λ 3 λ Thin disk λ = λ = λ 3 λ = λ λ = 6 5 λ = λ 3 λ = λ = λ 3 λ λ Thin cylinder λ = λ, λ 3 = Figure : Feasible region of the principal moments of inertia λ, λ, λ 3 of a rigid body satisfying < λ 3 λ λ, where λ < λ + λ 3. The shaded region shows all feasible values of λ and λ 3 in terms of the largest principal moment of inertia λ. The open dots and dashed line segment indicate nonphysical, limiting cases. body-fixed frame relative to the principal axes is varied. For convenience, λ is normalized to kg-m, and the inertia matrices J, J, J 3, J 4, J 5 are chosen to correspond to the points noted in Figure. These matrices, which correspond to the sphere, cylinder with l/r = 3, centroid, thin disk, and thin cylinder, respectively, are defined as J = diag(,, ), J = diag(,, 5), J 3 = diag(, 5/3, 5), J 4 = diag(, 5, 5), J 5 = diag(,,.). (4) The inertia matrix J 3 corresponding to the centroid of the inertia region serves as the nominal inertia matrix, while the inertia matrices J, J, J 4, J 5 are used as perturbations to demonstrate robustness of the control laws. A perturbation J(λ) of J i in the direction of J j thus has the form J(λ) = ( λ)j i + λj j, (4) where λ [, ]. Finally, in order to facilitate numerical integration of Euler s equation, note that J 5 is of 3

12 chosen to be a nonsingular approximation of the limiting inertia of a thin cylinder. IV. Controller Design Let I denote the identity matrix, whose dimensions are determined by context, and let M ij denote the i, j entry of the matrix M. The following result is given in ref. 3 Lemma. Let A R 3 3 be a diagonal positive-definite matrix, and let R R 3 3 be a rotation matrix. Then the following statements hold: i) For all i, j =,, 3, R ij [, ]. ii) tr (A AR). iii) tr (A AR) = if and only if R = I. For convenience note that, if R is a rotation matrix and x, y R 3, then (Rx) = Rx R T, and, therefore, R(x y) = (Rx) Ry. Next, introduce the notation J sc ω = L(ω)γ, where γ R 6 is defined by [ γ = J J J 33 J 3 J 3 J ] T and L(ω) = ω ω 3 ω ω ω 3 ω ω 3 ω ω. Next, let Ĵsc R 3 3 denote an estimate of J sc, and define the inertia-estimation error J sc = Jsc Ĵsc. of 3

13 Letting ˆγ, γ R 6 represent Ĵsc, J sc, respectively, it follows that γ = γ ˆγ. Likewise, let ˆτ dist R 3 denote an estimate of τ dist, and define the disturbance-estimation error τ dist = τdist ˆτ dist. The assumptions upon which the following development is based are now stated. Assumption. J sc is constant but unknown. Assumption. J α defined by (8) is constant, nonsingular, and known. That is, the spacecraft has three linearly independent, axisymmetric wheels, with known moments of inertia about their spin axes and known configuration relative to the bus. The controllers presented in ref. 3 are now extended to the case of reaction-wheel actuation. IV.A. Control Law for Slew Maneuvers When no disturbances are present, the inertia-free control law given by (38) of ref. 3 achieves almost global stabilization of a constant desired attitude R d, that is, a slew maneuver that brings the spacecraft to rest. The initial conditions of the slew maneuver may be arbitrary, that is, the spacecraft may have nonzero initial velocity. Given a, a, a 3 R, define the vector measure of attitude error S = a i ( R T e i ) e i, (4) where, for i =,, 3, e i denotes the ith column of the 3 3 identity matrix. When attitude measurements are given in terms of an alternative representation, such as quaternions, the corresponding attitude-error R defined by (35) can be computed, and thus (4) can be evaluated and used by the controller given in Theorem below. Consequently, S can be computed from any attitude parameterization. Theorem. Let K p be a positive number and let A = diag(a, a, a 3 ) be a diagonal positive-definite matrix. Then the function V (ω, R) = ωt J sc ω + K p tr(a A R), (43) is positive definite, that is, V is nonnegative, and V = if and only if ω = and R = I. 3 of 3

14 Proof. It follows from statement of Lemma that tr (A A R) is nonnegative. Hence V is nonnegative. Now suppose that V =. Then, ω =, and it follows from iii) of Lemma that R = I. Theorem. Let K p be a positive number, let K v R 3 3 be a positive-definite matrix, let A = diag(a, a, a 3 ) be a diagonal positive-definite matrix with distinct diagonal entries, let R d be constant, define S as in (4), and define V as in Theorem. Consider the control law u = J α (K p S + K v ω). (44) Then, V (ω, R) = ω T K v ω (45) is negative semidefinite. Furthermore, the closed-loop system consisting of (9), (), (36), and (44) is almost globally asymptotically stable, 8 and for all initial conditions not in an embedded submanifold of R 3 SO(3) R 6 R 3 (see ref. 3 ), ω and R I as t. Proof. Noting that d dt tr (A A R) = tr A R = tr A( Rω ω d R) = = = = = 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 [ ] T a i e i R T e i ω [ ] T = a i ( R T e i ) e i ω = ω T S, 4 of 3

15 then V (ω, R) = ω T J sc ω + K p ω T S = ω T [(J sc ω + J α ν) ω J α u] + K p ω T S = ω T ( K p S K v ω) + K p ω T S = ω T K v ω. The proof of the final statement follows from invariant set arguments that are similar to those used in ref. 3 Note that J α is substituted for the input matrix B used in the inertia-free control law (38) of ref., 3 but otherwise the controller requires no modification for the case of reaction-wheel actuation in order to achieve almost global stabilization of a constant desired attitude R d. IV.B. Control Law for Attitude Tracking A control law that tracks a desired attitude trajectory in the presence of disturbances is given by () of ref. 3 This controller is based on an additional assumption. Assumption 3. Each component of τ dist is a linear combination of constant and harmonic signals, whose frequencies are known but whose amplitudes and phases are unknown. Assumption 3 implies that τ dist can be modeled as the output of the autonomous system d = A d d, (46) τ dist = C d d, (47) where d is the disturbance state, A d R n d n d and C d R 3 n d are known matrices, and the eigenvalues of A d are nonrepeated on the imaginary axis. In this model, d() is unknown, which is equivalent to the assumption that the amplitude and phase of each harmonic component of the disturbance is unknown. The eigenvalues of A d are chosen to represent all frequency components that may be present in the disturbance signal, where the zero eigenvalue corresponds to a constant disturbance. By providing infinite gain at the disturbance frequencies, the controller asymptotically rejects the harmonic disturbances. In particular, an integral controller provides infinite gain at DC in order to reject constant disturbances. In the case of orbitdependent 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. 5 of 3

16 Assumption 3 implies that A d can be chosen to be skew symmetric, which is henceforth done. Let ˆd R n d denote an estimate of d, and define the disturbance-state estimation error d = d ˆd. The attitude tracking controller in the presence of disturbances given in ref. 3 is modified below for reaction-wheel actuators. Theorem 3. Let K p be a positive number, let K R 3 3, let Q R 6 6 and D R n d n d be positivedefinite matrices, let A = diag(a, a, a 3 ) be a diagonal positive-definite matrix, and define S as in (4). Then the function V ( ω, R, γ, d) = ( ω + K S) T J sc ( ω + K S) + K p tr (A A R) + γt Q γ + d T D d (48) is positive definite, that is, V is nonnegative, and V = if and only if ω =, R = I, γ =, and 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 =, and d =, and it follows from iii) of Lemma that R = I, and thus S =. Therefore, ω =. The following result concerns attitude tracking without knowledge of the spacecraft inertia. This control law does not regulate the speed of the wheels. Consequently, the function V defined by (48), which is used as a Lyapunov function in the proof of Theorem 4 below, is not a positive-definite function of the angular rates of the wheels relative to the bus. Theorem 4. Let K p be a positive number, let K v R 3 3, K R 3 3, Q R 6 6, and 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 with distinct diagonal entries, define S and V as in Theorem 3, and let ˆγ and ˆd satisfy ˆγ = Q [L T (ω)ω + L T (K Ṡ + ω ω R T ω d )]( ω + K S), (49) where Ṡ = a i [( R T e i ) ω] e i (5) 6 of 3

17 and ˆd = A d ˆd + D C T d ( ω + K S), (5) ˆτ dist = C d ˆd, (5) so that ˆτ dist is the disturbance-torque estimate. Consider the control law u = J α (v + v + v 3 ), (53) where v = ( Ĵ sc ω + J α ν) ω Ĵsc(K Ṡ + ω ω R T ω d ), (54) v = ˆτdist, (55) and v 3 = Kv ( ω + K S) K p S. (56) Then, V ( ω, R, γ, 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 (57) is negative semidefinite. Furthermore, the closed-loop system consisting of Eqs. (), (36), (38), and (53) is almost globally asymptotically stable, and for all initial conditions not in an embedded submanifold of R 3 SO(3) R 6 R 3 (see ref. 3 ), ω and R I as t. 7 of 3

18 Proof. V ( ω, R, γ, d) = ( ω + K S) T (J sc ω + J sc K Ṡ) K p tr A R γ T Q ˆγ + d T D d = ( ω + K S) T [(J sc ω + J α ν) ω + J sc ( ω ω R T ω d ) J α u + τ dist + J sc K Ṡ] + K p ω T S γ T Q ˆγ + d T D d = ( ω + K S) T [(J sc ω + J α ν) ω + J sc (K Ṡ + ω ω R T ω d ) + v + v + v 3 + τ dist ] + K p ω T S γ T Q ˆγ + d T D d = ( ω + K S) T [( J sc ω) ω + J sc (K Ṡ + ω ω R T ω d )] + ( 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 + d T D d = ( ω + K S) T [L(ω) γ ω + L(K Ṡ + ω ω R T ω d ) γ] ( ω + K S) T K v ( ω + K S) K p S T K S γ T Q ˆγ + d T Cd T ( ω + K S) + d T D[A d d D Cd T ( 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 + 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)] + 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. The closed-loop spacecraft attitude dynamics (38) and the control law (53)-(56) can be expressed as J ω = [L(ω) γ] ω + L( ω R T ω d R T ω d ) γ L(K Ṡ)ˆγ + z d K v ( ω + K S) K p S. (58) From Lemma 3 and Lemma 4 of ref., 3 the closed-loop system consisting of (49)-(5) and (58) has four disjoint equilibrium manifolds. These equilibrium manifolds in R 3 SO(3) R 6 R 3 are given by E i = {( ω, R, γ, d) R 3 SO(3) R 6 R 3 : R = R i, ω, ( γ, d) Q i }, (59) 8 of 3

19 where, for all i {,,, 3}, Q i is the closed subset of R 6 R 3 defined by Q i = {( γ, d) R 6 R 3 : [L(R T i ω d ) γ] (R T i ω d ) L(R T i ω d ) γ + C d d =, γ =, d = A d d}. Furthermore, the equilibrium manifold ( ω, R, ( γ, d)) = (, I, Q ) of the closed-loop system given by (49)-(5) and (58) is locally asymptotically stable, and the remaining equilibrium manifolds given by (, R i, Q i ) for i {,, 3} are unstable. Finally, the set of all initial conditions converging to these equilibrium manifolds forms a lower dimensional submanifold of R 3 SO(3) R 6 R 3. V. Examples Simulations are now provided to illustrate the inertia-free control laws (44) and (53). To simulate slew and spin maneuvers, the following spacecraft parameters are assumed. The bus inertia matrix J b is nominally given by J 3, which corresponds to the centroid of the inertia region shown in Figure with the body-fixed frame assumed to be a principal body-fixed frame. The quantity J b is unknown to the controller. The axes of rotation of the reaction wheels are aligned with the spacecraft body-fixed frame unit vectors, and the wheel inertias are given by J w = diag(α, β, β ) kg-m, J w = diag(β, α, β ) kg-m, and J w3 = diag(β 3, β 3, α 3 ) kg-m, where α = α = α 3 =.5 and β = β = β 3 =.375. The values of β i are unknown to the controller. Let K p be given by K p = γ tr A, (6) and, as in ref., 3 let K v = K v (ω) be given by K v = η + ω + ω + ω 3. (6) Alternative choices of K v are given in ref. 9 V.A. Slew Maneuver Using Control Law (44) with No Disturbance Controller (44) is used for an aggressive slew maneuver, where the objective is to bring the spacecraft from the initial attitude R = I 3 and initial angular velocity [ ω() =.5 ] T rad/sec 9 of 3

20 to rest (ω d = ) at the desired final orientation R d = diag(,, ), which represents a rotation of 8 deg about the x-axis. The reaction wheels are initially not spinning relative to the spacecraft, that is, [ ν() = ] T rad/sec. No disturbance is present, and the parameters γ = η = 5 are used in (6) and (6). Figures (a)-(d) show, respectively, the attitude error, angular-velocity components, angular rates of the wheels, and the control inputs, which are the angular accelerations of the wheels. The spacecraft attitude and angular-velocity components reach the commanded values in about sec. The angular rates of the wheels approach constant values that are consistent with the initial, nonzero angular momentum. Eigenaxis Attitude Error (rad) S Angular Velocity Components (rad/sec) ω ω ω (a) Top: Eigenaxis attitude error. Bottom: Norm of the S parameter (b) Spacecraft angular-velocity components 4 ν u 3 ν ν 3 u u 3 Wheel Angular Velocity Components (rad/sec) Angular Acceleration Components (rad/sec ) (c) Angular rates of the reaction wheels (d) Angular accelerations of the reaction wheels Figure : Slew maneuver using the control law (44) with no disturbance. In practice, reaction wheels have a maximum instantaneous acceleration. Angular-acceleration saturation is enforced in Figures 3 and 4, where convergence is slower than in Figure, although stability is maintained. Additionally, reaction wheels have a maximum rotational rate. Figure 5 shows the effect of wheel-rate saturation at 5 rad/sec, corresponding to about 4 rpm. The reaction-wheel rates are saturated for up of 3

21 to 5 seconds, although this does not impact the control objective. Figure 6 shows plots for wheel-rate saturation at rad/sec, or about 9 rpm. Although this constraint on the rotation rate is too stringent to obtain zero steady-state error for the desired maneuver, the performance of the controller degrades gracefully by achieving zero spacecraft angular velocity at an offset attitude ω 3 ω ω 3 Eigenaxis Attitude Error (rad).5.5 Angular Velocity Components (rad/sec) (a) Eigenaxis attitude error (b) Spacecraft angular-velocity components 4 4 ν u 3 ν ν 3 3 u u 3 Wheel Angular Velocity Components (rad/sec) Angular Acceleration Components (rad/sec ) (c) Angular rates of the reaction wheels (d) Angular accelerations of the reaction wheels Figure 3: Slew maneuver using the control law (44) with no disturbance. The acceleration of the reaction wheels is saturated at 4 rad/sec. V.A.. Inertia Robustness To evaluate performance for slew maneuvers, define the settling-time metric k = min k> {k : for all i {,..., }, θ((k i)t s) <.5 rad}, (6) where k is the simulation step, T s is the integration step size, and θ(kt s ) is the eigenaxis attitude error (39) at the kth simulation step. The metric k is thus the minimum time such that the eigenaxis attitude error is less than.5 rad during the most recent simulation steps. To illustrate the inertia-free property of the control laws (44) and (53), the inertia of the spacecraft is of 3

22 3.5.5 ω 3 ω ω 3 Eigenaxis Attitude Error (rad).5.5 Angular Velocity Components (rad/sec) (a) Eigenaxis attitude error (b) Spacecraft angular-velocity components 4 ν u ν u 3 ν 3.5 u 3 Wheel Angular Velocity Components (rad/sec) Angular Acceleration Components (rad/sec ) (c) Angular rates of the reaction wheels 5 5 (d) Angular accelerations of the reaction wheels Figure 4: Slew maneuver using the control law (44) with no disturbance. The acceleration of the reaction wheels is saturated at rad/sec. varied using J b (λ) = ( λ)j 3 + λj i, (63) where λ [, ] and i =, 4, 5. Figure 7 shows how the settling time depends on λ. Next, the robustness to misalignment of the reaction wheels relative to the principal axes is investigated. Here, the inertia matrix is rotated by an angle φ about one of the axes of frame F b. For a rotation about the x axis of F b, the inertia of the spacecraft is varied using J b (φ) = O (φ)j 3 O T (φ), (64) where the proper orthogonal matrix O (φ) rotates vectors about the x axis by the angle φ. Similar relations exist for rotations about the y and z axes. Figure 8 shows how a thruster misalignment angle φ affects the settling time, where φ is varied from 8 deg to +8 deg. of 3

23 3.5.5 ω 3 ω ω 3 Eigenaxis Attitude Error (rad).5.5 Angular Velocity Components (rad/sec) (a) Eigenaxis attitude error (b) Spacecraft angular-velocity components 3 ν u 5 ν ν 3 u u 3 Wheel Angular Velocity Components (rad/sec) 5 5 Angular Acceleration Components (rad/sec ) (c) Angular rates of the reaction wheels (d) Angular accelerations of the reaction wheels Figure 5: Slew maneuver using the control law (44) with no disturbance. The maximum rotation rate of each wheel is saturated at 5 rad/sec. V.B. Slew Maneuver Using Control Law (53) under Constant Disturbance The unknown constant disturbance torque τ dist = [.7.3 ] T is now considered. Note that the controller (53) is used in place of the controller (44), which lacks an integrator, and thus has a constant steady-state error bias due to the persistent disturbance. The parameters of the controller (53) are chosen to be K = I 3, A = diag(,, 3), γ = η =, D = I 3, and Q = I 6. Figures 9(a)-(f) show, respectively, the attitude error, angular velocity components, angular rates of the wheels, angular momentum, disturbance-estimate errors, and inertia-estimate errors. The spacecraft attitude and angular velocity components reach the commanded values in about 8 sec. Figure 9c indicates that the reaction-wheel rotational speed grows unbounded. Figure 9d shows that the total angular momentum of the spacecraft increases, which is consistent with the constant disturbance torque acting on the spacecraft. In practice, the spacecraft needs a method to dump the stored angular momentum so that the reaction wheel rates do not grow unbounded. Figure repeats the maneuver with maximum wheel saturation at rad/sec, corresponding to roughly 3 of 3

24 3.5.5 ω 3 ω ω 3 Eigenaxis Attitude Error (rad).5.5 Angular Velocity Components (rad/sec) (a) Eigenaxis attitude error (b) Spacecraft angular-velocity components 5 ν u ν ν 3 u u 3 Wheel Angular Velocity Components (rad/sec) 5 5 Angular Acceleration Components (rad/sec ) (c) Angular rates of the reaction wheels (d) Angular accelerations of the reaction wheels Figure 6: Slew maneuver using the control law (44) with no disturbance. The maximum rotation rate of each wheel is saturated at rad/sec. rpm. The controller brings the spacecraft to the desired orientation in about 6 sec at which time one of the angular rates of the reaction wheels reaches rad/sec, disturbance and inertia estimates diverge, and the system is destabilized. V.C. Spin Maneuver Using Using Control Law (53) Consider a spin maneuver with the spacecraft initially at rest and R() = I 3. The desired attitude is determined by R d () = I 3, and the commanded constant angular velocity is [ ω d = ] T rad/sec. Assume no disturbance. Figures (a)-(f) show, respectively, the attitude errors, angular-velocity components, angular rates of the wheels, the control inputs, which are the angular accelerations of the wheels, angular momentum, and inertia-estimate errors. For this maneuver, the spin command consists of a specified time history of rotation about a specified body axis aligned in a specified inertial direction. The controller 4 of 3

25 Centroid to Thin Disk Centroid to Sphere Centroid to Thin Cylinder 65 Centroid to Thin Disk Centroid to Sphere Centroid to Thin Cylinder Settling 7 65 Settling λ (a) λ Figure 7: Settling time as a function of λ for various combinations (4) of inertia matrices resolved in principal frames. Convergence is achieved for (a) control law (44), and (b) control law (53). Each controller is implemented in all cases with a single tuning. In all cases, the bus inertia J 3 is unknown. (b) 85 8 x y z 65 x y z 6 75 Settling 7 65 Settling φ (deg) (a) φ (deg) Figure 8: Settling time as a function of principal-frame/body-frame rotation angle φ for rotations about each of the three principal axes of J 3. Convergence is achieved for (a) control law (44), and (b) control law (53). (b) achieves the commanded spin within about sec. VI. Conclusion Almost global stabilizability (that is, Lyapunov stability with almost global convergence) of spacecraft tracking is feasible without inertia information and with continuous feedback using three linearly independent reaction wheels, whose axes of rotation are not necessarily aligned with the principal axes of the spacecraft bus, do not necessarily pass through the spacecraft s center of mass, and are not necessarily mass balanced in order to preserve the location of the spacecraft s center of mass. In addition, asymptotic rejection of harmonic disturbances (including constant disturbances as a special case) is possible with knowledge of the disturbance spectrum but without knowledge of either the amplitude or phase. 5 of 3

26 Under these assumptions, the adaptive control laws presented in this paper provide an alternative to previous controllers that ) require exact or approximate inertia information or ) are based on attitude parameterizations such as quaternions that require discontinuous control laws or fail to be physically consistent (that is, specify different control torques for the same physical orientation). A future extension will address spacecraft actuation using control moment gyroscopes (CMGs). Acknowledgments The authors wish to thank Marc Camblor for generating Figure, and the reviewers for their helpful comments. References Junkins, J. L., Akella, M. R., and Robinett, R. D., Nonlinear Adaptive Control of Spacecraft Maneuvers, AIAA Journal of Guidance, Control, and Dynamics, Vol., pp. 4, 997. Ahmed, J., Coppola, V. T., and Bernstein, D. S., Asymptotic Tracking of Spacecraft Attitude Motion with Inertia Matrix Identification, AIAA Journal of Guidance, Control, and Dynamics, Vol., pp , Sanyal A., Fosbury A., Chaturvedi N., and Bernstein D. S., Inertia-Free Spacecraft Attitude Tracking with Disturbance Rejection and Almost Global Stabilization, AIAA J. Guid. Contr. Dyn., Vol. 3, pp , 9. 4 Chaturvedi N., Sanyal A., and 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,. 5 Wie, B., and Barba, P. M., Quaternion Feedback for Spacecraft Large Angle Maneuvers, AIAA Journal of Guidance, Control, and Dynamics, Vol. 8, pp , Joshi, S. M., Kelkar, A. G., and Wen, J. T., Robust Attitude Stabilization using Nonlinear Quaternion Feedback, IEEE Transactions on Automatic Control, Vol. 4, pp. 48 6, Bhat, S. P., and Bernstein, D. S., A Topological Obstruction to Continuous Global Stabilization of Rotational Motion and the Unwinding Phenomenon, Systems and Control Letters, Vol. 39, pp. 63 7,. 8 Crassidis, J. L., Vadali, S. R., and Markley, F. L., Optimal Variable-Structure Control Tracking of Spacecraft Maneuvers, AIAA Journal of Guidance, Control, and Dynamics, Vol. 3, pp ,. 9 Mayhew, C. G., Sanfelice, R. G., and Teel, A. R., Quaternion-Based Hybrid Control for Robust Global Attitude Tracking, IEEE Transactions on Automatic Control, Vol. 56, No., pp ,. Cortes, J., Discontinuous Dynamic Systems, IEEE Control Systems Magazine, Vol. 8, pp. 36-7, June 8. Kasdin, N. J. and Paley, D.A., Engineering Dynamics, Princeton University Press, Princeton, NJ, USA,. Arambel, P. O., Manikonda, V., and Mehra, R. K., Spacecraft Attitude Tracking in the Presence of Input Magnitude Constraints, Proceedings of the American Control Conference, pp , June. 3 Boskovic, J. D., Li, S.-M., and Mehra, R. K., Robust Adaptive Variable Structure Control of Spacecraft Under Control Input Saturation, AIAA Journal of Guidance, Control, and Dynamics, Vol. 4, pp. 4,. 4 Banga, H., Tahka, M.-J., and Choib, H.-D., Large Angle Attitude Control of Spacecraft with Actuator Saturation, Control Engineering Practice, Vol., pp , 3. 5 Levine, W. S., Control Systems Applications, CRC Press, Inc., Boca Raton, FL, USA, 999, page 3. 6 of 3

27 6 Sanyal, A. K., Lee, T., Leok, M., and McClamroch, N. H., Global Optimal Attitude Estimation Using Uncertainty Ellipsoids, Systems and Control Letters, Vol. 57, pp , 8. 7 Hughes, P. C., Spacecraft Attitude Dynamics, Wiley, 986; reprinted by Dover, 8, page 7. 8 Koditschek, D. E., The Application of Total Energy as a Lyapunov function for Mechanical Control Systems, Proc. of AMS-IMS-SIAM Joint Summer Research Conf., Bowdin College, ME, pp. 3 57, G. Cruz, X. Yang, A. Weiss, I. Kolmanovsky, and D. S. Bernstein, Torque-saturated, Inertia-free Spacecraft Attitude Control, AIAA Guid. Nav. Contr. Conf., Portland, OR, August, AIAA of 3

28 3.5.5 ω 3 ω ω 3.5 Eigenaxis Attitude Error (rad).5.5 Angular Velocity Components (rad/sec) (a) Eigenaxis attitude error (b) Spacecraft angular-velocity components. Wheel Angular Velocity Components (rad/sec) (c) Angular rates of the reaction wheels. The spin rate grows unbounded due to the constant disturbance torque. ν ν ν 3 Angular momentum, X Angular momentum, Y Angular momentum, Z 5 Total, x Bus, x Wheels, x Total, y Bus, y Wheels, y Total, z Bus, z Wheels, z (d) Angular momentum of the spacecraft relative to its center of mass with respect to the inertial frame resolved in the inertial frame. The total angular momentum is not conserved due to the constant disturbance torque in the bus-fixed frame. Disturbance Estimate Errors (N m) (e) Disturbance-estimate errors. Inertia Estimate Errors (kg m ) Inertia Estimate Errors (kg m ) Jtilde Jtilde Jtilde Jtilde 3 Jtilde 3 Jtilde (f) Inertia-estimate errors. Figure 9: Slew maneuver using the control law (53) under a disturbance that is constant with respect to the bus-fixed frame. 8 of 3

29 ω ω ω 3 Eigenaxis Attitude Error (rad).5.5 Angular Velocity Components (rad/sec) (a) Eigenaxis attitude error (b) Spacecraft angular-velocity components. 5 ν 4 u ν ν 3 3 u u 3 Wheel Angular Velocity Components (rad/sec) 5 5 Angular Acceleration Components (rad/sec ) (c) Angular rates of the reaction wheels. The spin rate grows until reaching the saturation limit of rad/sec (d) Angular accelerations of the reaction wheels. Disturbance Estimate Errors (N m) (e) Disturbance-estimate errors. Inertia Estimate Errors (kg m ) Inertia Estimate Errors (kg m ) Jtilde Jtilde Jtilde Jtilde 3 Jtilde 3 Jtilde (f) Inertia-estimate errors. Figure : Slew maneuver using the control law (53) under a disturbance that is constant with respect to the bus-fixed frame.. The maximum rotation rate of each wheel is saturated at rad/sec. 9 of 3

30 .5 ω.9.8 ω ω 3 Eigenaxis Attitude Error, rad Angular Velocity Components, rad/sec Time, sec (a) Eigenaxis attitude error Time, sec (b) Spacecraft angular-velocity components. 5 3 ν u ν u ν 3 u 3 5 Wheel Angular Velocity Components, rad/sec Angular Acceleration Components (rad/sec ) Time, sec (c) Angular rates of the reaction wheels Time, sec (d) Angular accelerations of the reaction wheels. Angular momentum, X Angular momentum, Y Angular momentum, Z Total, x Bus, x Wheels, x Time, sec Total, y Bus, y Wheels, y Time, sec Total, z Bus, z Wheels, z Time, sec (e) Angular momentum of the spacecraft relative to its center of mass with respect to the inertial frame resolved in the inertial frame. Inertia Estimate Errors, kg m Inertia Estimate Errors, kg m Jtilde Jtilde Jtilde Time, sec Jtilde 3 Jtilde 3 Jtilde Time, sec (f) Inertia-estimate errors. Figure : Spin maneuver using the control law (53). 3 of 3