RECEDING HORIZON DRIFT COUNTERACTION AND ITS APPLICATION TO SPACECRAFT ATTITUDE CONTROL

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1 AAS 7-46 RECEDING HORIZON DRIFT COUNTERACTION AND ITS APPLICATION TO SPACECRAFT ATTITUDE CONTROL Robert A. E. Zidek, Christopher D. Petersen, Alberto Bemporad, and Ilya V. Kolmanovsky In this paper, a recently developed model predictive control (MPC) approach for drift counteraction optimal control (DCOC) is applied to attitude control of fully actuated and underactuated spacecraft with reaction wheels (RWs). The objective is to maximize the time that prescribed constraints on spacecraft orientation and RW spin rates are satisfied given disturbance torques due to solar radiation pressure. While the MPC/DCOC approach is based on linear programming, all closedloop simulations are performed using the nonlinear model. In case constraints are violated, a control strategy is presented that recovers constraint satisfaction (if possible). We consider the cases where either one, two, or three RWs are operable. The numerical results show that the proposed controller successfully counteracts drift in order to satisfy constraints for as long as possible. INTRODUCTION The objective of drift counteraction optimal control (DCOC) is to satisfy prescribed state and control constraints for as long as possible. Such problems arise in many space flight applications, in particular, when finite resources (fuel/energy) are of concern. For example, control strategies have been developed to extend the operational time of a satellite subject to higher-order orbital perturbations. 4 Similar problems may be formulated for spacecraft attitude control, where tight constraints on spacecraft orientation and/or angular velocity need to be satisfied for as long as possible when the controllability of the spacecraft is limited. This may be the case for underactuated spacecraft,, 6 examples of which are the Kepler spacecraft, which lost two out of its four reaction wheels (RWs), 7 and two out of three RWs of the Hayabusa spacecraft failed. 8 Furthermore, reduced controllability may result from RW saturation. In this paper, a DCOC problem of attitude control of a spacecraft with RWs is considered. The spacecraft orientation is subject to drift caused by solar radiation pressure (SRP). We use a recently developed DCOC approach 9 to maximize the time that prescribed constraints on spacecraft orientation and RW spin rates are satisfied. The approach assumes linear dynamics and solutions to the linear DCOC problem are obtained by either employing linear programming (LP) or mixed integer linear programming (MILP). While the MILP solution is guaranteed to be optimal with respect to the linear DCOC problem, LP efficiently obtains good-quality suboptimal solutions. 9 We present closed-loop simulations based on nonlinear dynamics, where a receding horizon / model predictive control (MPC) implementation of the LP formulation is used to provide state feedback control in Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 489, USA. IMT Institute for Advanced Studies, Lucca, Italy.

2 order to compensate for unmodeled effects. Furthermore, an additional control strategy is introduced that tries to recover constraint satisfaction in case constraints are violated. We also show that the proposed approach is able to satisfy constraints for an arbitrarily long time when no solution to the DCOC problem exists, i.e., when the spacecraft has enough control authority to satisfy the constraints indefinitely. The paper is structured as follows. First, the spacecraft model is described, including the nonlinear spacecraft kinematics and dynamics, the SRP torque, and the linearized spacecraft model which is used to obtain the DCOC/MPC controller. Then we state the DCOC problem (for linear dynamics) and formulate the LP and MILP to solve it. Moreover, the MPC implementation is described and a control strategy to recover constraint satisfaction is introduced. Numerical results are shown in the subsequent section, where the cases of either one, two, or three operable RWs are treated. For the case of three operable RWs, we also show that the proposed approach may satisfy constraints indefinitely depending on the initial condition and constraints. A conclusion is given in the last section. SPACECRAFT MODEL The spacecraft model, adopted from Petersen et al., 6 is summarized in this section. Notation The following notation is used for the derivation of the spacecraft model. A physical vector is denoted by r and a physical unit vector is denoted by ˆr. Resolving r or ˆr in a given frame F results in a mathematical vector denoted by r F or ˆr F, respectively. Two frames are considered, a given inertial reference frame denoted by I and the body-fixed frame B, which is assumed to be a principal frame. The notation for r or ˆr resolved in the body-fixed frame is r = r B or r = ˆr B, respectively. The skew-symmetric matrix of a mathematical vector r = [r, r, r 3 ] T is given by Kinematics r 3 r S[ r] = r 3 r. () r r The orientation of the spacecraft s body-fixed frame B relative to I is described by Θ = [φ, θ, ψ] T, where ψ (yaw), θ (pitch), and φ (roll) are the 3-- Euler angles. Then the kinematic equations read Θ = cos(θ) cos(θ) sin(φ) sin(θ) cos(φ) sin(θ) cos(φ) cos(θ) sin(φ) cos(θ) sin(φ) cos(θ) where ω is the angular velocity of B relative to I and ω = [ω, ω, ω 3 ] T. Dynamics ω, () The spacecraft is equipped with up to three RWs, where ḡ i denotes the unit vector of the ith RW spin axis resolved in the B frame. Moreover, let W = [ḡ, ḡ, ḡ 3 ] in case of three RWs and W = ḡ or W = [ḡ, ḡ ] in case of one or two operable RWs, respectively. The spin rate of the ith RW is ν i and ν = [ν, ν, ν 3 ] T, [ν, ν ] T, or ν, respectively. We assume that all RWs are identical and thin

3 (moments of inertia about axes transversal to spin axis are approximately zero), where J w denotes the moment of inertia about the RW spin axis. The moment of inertia matrix of the spacecraft bus resolved in the B frame is given by J = diag(j, J, J 3 ) and J = J + J w W W T (3) is the locked inertia. The rotational dynamics of the spacecraft described in the B frame are given by 6 J ω + S[ ω]( J ω + J w W ν) + J w W ν = τ srp, (4) where τ srp is an external torque due to SRP. SRP Torque For modeling the SRP torque in (4), a cuboid spacecraft (six flat panels) is assumed. We define κ = Φ sun,tot c(d SC,sun /d ), β = 4 9 C diff, () where c is the speed of light, Φ sun,tot is the solar flux experienced by the spacecraft, d = AU is the nominal distance between Earth and Sun, d SC,sun is the distance between the spacecraft and Sun (d SC,sun =.99 AU in this paper), and C diff denotes the diffusion coefficient, which is assumed to be the same for all panels. Assuming the SRP acts identically across all points on the jth panel, the SRP acting on panel j is given by6, P j = κ(ˆq j ˆq sun )(ˆq j + β ˆq sun ), (6) where ˆq j is the normal to the surface of the jth panel pointing outward from the spacecraft and ˆq sun represents the direction of the sun. Consequently, the SRP torque due to the jth panel resolved in the B frame is τ srp,j = S[ r j/o r C/O ]A j Pj, (7) where r C/O = [l x, l y, l z ] T denotes the mathematical vector of the position of the spacecraft s center of mass, C, relative to the geometric center, O, of the cuboid. The vector from O to the geometric center of the jth panel is represented by r j/o. With j {x+, x, y+, y, z+, z }, it is r x+/o = r x /O = [L x /,, ] T, r y+/o = r y /O = [, L y /, ] T, r z+/o = r z /O = [,, L z /] T, where the surface areas of the panels are A x+ = A x = L y L z, A y+ = A y = L x L z, and A z+ = A z = L x L y. The total SRP torque is the sum of all panel contributions, τ srp = 6 τ srp,j I j, (8) j= with I j as a function indicating if the jth panel is facing the Sun or not, i.e., I j = if ˆq j ˆq sun > and I j = otherwise. 3

4 Linear Discrete-Time Model The continuous-time nonlinear spacecraft model given by () and (4) is linearized and transformed into a discrete-time model in order to obtain the MPC drift counteraction controller (see subsequent section). We define the state and control input vectors, respectively, by x = [φ, θ, ψ, ω, ω, ω 3, ν,..., ν m ] T, u = [α,..., α m ] T, (9) with m {,, 3} as the number of operable RWs and α i = ν i as the acceleration of the ith RW. The linearization point is given by x lin = [ 6, ν T]T and u lin = m. Thus, the matrices corresponding to the continuous-time state space model, ẋ = Ãx + Bu + d, are I m à = J T J J w S[W ν ] 3 m, m 3 m 3 m m B = 3 m J J w W I m m, d = 3 J τ srp Θ=3 m, () where τ srp Θ=3 is the SRP torque when Θ = 3 and T results from numerically linearizing (8) about Θ = 3, i.e., τ srp T Θ + τ srp Θ=3. The discrete-time model is obtained from the continuous-time model () using Euler s forward method, where t denotes the sampling time and t Z, x t+ = Ax t + Bu t + d, () with A = I (6+m) (6+m) + à t, B = B t, and d = d t. DRIFT COUNTERACTION OPTIMAL CONTROL In this section, a class of DCOC problems for linear discrete-time systems is described. Based on this, we obtain a controller that maximizes the time until constraint violation occurs. The approach, which uses LP and MILP formulations, was recently developed by Zidek et al. 9. Problem Formulation Let the set of admissible control sequences {u t } be defined by where the time-varying control constraints are given by Furthermore, let the first exit-time from a time-varying set, U seq = {{u t } : u t U t for all t}, () U t = {u R m : C c,t u b c,t }. (3) G t = {x R n : C s,t x b s,t }, (4) given an initial state x and a control sequence {u t } U seq, be defined by τ(x, {u t }) = inf{t Z + : x t / G t x G }. () Then the DCOC problem of finding an admissible control sequence {u t } that maximizes the first exit-time is as follows max τ(x, {u t }) {u t} U seq subject to x t+ = A t x t + B t u t + d t, x G. In the following, we denote a solution to (6), if one exists, by {u t } and τ(x, {u t }) is the optimal exit-time. Note that {u t } may not be unique. (6) 4

5 Optimal Solution An optimal solution to problem (6), if one exists, can be obtained by solving the following MILP with time horizon N 9 min N δ t {x t},{u t},{δ τlb,...,δ N } t=τ lb subject to x t+ = A t x t + B t u t + d t, x G δ t δ t C s,t x t b s,t, t =,..., τ lb C s,t x t b s,t + Mδ t, t = τ lb,..., N (7) u t U t δ t {, } Z, t = τ lb,..., N, where M is a large positive constant, denotes the n-dimensional vector of ones, and the binary variable δ t is an indicator variable for the condition x t / G t. Thus, due to the constraint δ t δ t, minimizing the sum of δ t is equivalent to maximizing the first exit-time. The parameter τ lb Z + in (7) is a lower bound on the optimal exit-time, i.e., τ(x, {u t }) τ lb. Note that N τ(x, {u t }) is required when using the MILP (7) to solve problem (6). MILP is in the class of NP-complete problems and the worst-case computation time increases exponentially with the number of integer variables., However, we can relax the MILP and obtain good-quality suboptimal solutions by replacing the binary variables δ τlb,..., δ N with non-negative variables ε t R. 9 The resulting LP formulation in (8) may provide a better balance between computation time and performance of the solution than the MILP (7) and is therefore used in the MPC scheme outlined in Algorithm. The relaxed problem has the following form min N ε t {x t},{u t},{ε τlb,...,ε N } t=τ lb subject to x t+ = A t x t + B t u t + d t, x G ε t ε t C s,t x t b s,t, t =,..., τ lb C s,t x t b s,t + ε t, t = τ lb,..., N u t U t, (8) where τ lb Z + is a lower bound on the optimal exit-time as in (7). In order to obtain a solution that violates the constraints, the time horizon N needs to be greater than the optimal first exittime τ(x, {u t }). In addition, finding a good solution to the DCOC problem (6) requires N to be sufficiently close to τ(x, {u t }). Since τ(x, {u t }) is a priori unknown, a procedure was developed that iteratively adapts N until a proper solution is found. 9 This procedure is outlined in Algorithm. At Step of Algorithm, the lower bound τ lb is initialized using the zero-control solution τ(x, {,,..., }) (assuming U t ; otherwise, any admissible control sequence can be used here). Moreover, the time horizon N is set in Step by adding a constant N add Z + to τ lb. With

6 these parameters, LP (8) is solved. If no constraint violation occurs, N + is used as a new lower bound (Step 6) and the LP is solved for an increased N. This procedure is repeated until constraint violation occurs. A threshold N ub can be specified by the user to terminate the procedure (Step ) when a strict upper bound on the computation time needs to be satisfied (for example, for an MPC implementation). However, the solution may be suboptimal in case of N ub + N add < τ(x, {u t }). Algorithm Iterative procedure based on LP to solve (6) : τ lb min{τ(x, {,,..., }), N ub } : N τ lb + N add, N add Z + 3: {x t }, {u t }, {ε τlb,..., ε N } solution of LP (8) 4: τ max{t N : ε t = } + : if τ = N AND N < N ub then 6: τ lb τ 7: Go to Step 8: end if Strategy to Recover Constraint Satisfaction When constraint violation occurs, i.e., x t / G t for some t, in some cases it may be possible to recover and return to G t. In this regard, we modify the LP in (8) by removing the constraints ε t ε t. Hence, the LP in (9) is obtained, which tries to drive the state vector back into the set G t, see (4), due to the constraints C s,t x t b s,t + ε t and ε t. If x t G t is not possible, the LP solution brings x t as close as possible to G t, assuming that the time horizon N recover is sufficiently small. N recover min {x t},{u t},{ε t} subject to t= ε t x t+ = A t x t + B t u t + d t, x / G (9) ε t C s,t x t b s,t + ε t u t U t. MPC Implementation In order to compensate for unmodeled effects, a DCOC state feedback controller is obtained by employing MPC techniques and recomputing the control, using either LP (8) or (9), over a receding time horizon based on the current state. However, initial numerical results showed that, compared to the linear solution, this MPC strategy may use an excessive amount of control (RW accelerations) when applied to the nonlinear spacecraft model. Hence, in order to minimize the control effort, the absolute values of the control inputs are added to the respective objective functions 6

7 in LP (8) and (9), i.e., LP (8) min N ε t + N u t, t=τ lb LP (9) min Nrecover t= t= ε t + Nrecover t= u t, yielding nonlinear programs (as a consequence of the absolute values). The nonlinear programs are approximated with standard LP by introducing the variables γ t R m as substitutions for u t and adding the constraints γ t u t and γ t u t. Thus, LP (8) and (9), respectively, are extended for the MPC implementation as follows min {x t},{u t},{ε τlb,...,ε N },{γ t} subject to N t=τ lb ε t + w x t+ = A t x t + B t u t + d t, x G N t= T γ t ε t ε t C s,t x t b s,t, t =,..., τ lb C s,t x t b s,t + ε t, t = τ lb,..., N () u t U t γ t u t γ t u t, N recover N recover min ε t + w recover T γ t {x t},{u t},{ε t},{γ t} subject to t= t= x t+ = A t x t + B t u t + d t, x / G ε t () C s,t x t b s,t + ε t u t U t γ t u t γ t u t, where w > and w recover > are weights that we set to w = w recover =. in this paper. The resulting MPC scheme, emphasizing reduced control effort in addition to the DCOC objective, is outlined in Algorithm. At each time instant t sys, the control sequence is recomputed based on the current state. If the current state does not satisfy the constraints (x / G ), LP () is used for control computation to try to recover constraint satisfaction, see Step 8 of Algorithm. Otherwise, the control sequence is computed by the iterative scheme in Algorithm, where LP (8) is replaced by LP (), see Step. The first element, u, of the computed control sequence is applied to the system at Step and the procedure is repeated for the next sampling interval. Note that the moving time horizon approach requires shifting the time-varying constraints and dynamics for control computation. To address this, the original constraints and dynamics, initially stored in Step 3, are used in Steps and 6 to shift the constraints and dynamics based on the current time instant t sys. 7

8 Algorithm MPC scheme based on Algorithm : t sys : t G max{n recover, N ub + N add } 3: {G st,t }, {U st,t }, {A st,t }, {B st,t }, {d st,t } {G t }, {U t }, {A t }, {B t }, {d t } 4: x current state x(t sys ) : for t {,..., t G }: {G t }, {U t } {G st,t+tsys }, {U st,t+tsys } 6: for t {,..., t G }: {A t }, {B t }, {d t } {A st,t+tsys }, {B st,t+tsys }, {d st,t+tsys } 7: if x / G then 8: {u t } solution of () 9: else : {u t } output of Algorithm with LP () instead of LP (8) : end if : Apply u as input u(t sys ) to the system 3: t sys t sys + 4: Go to Step 4 SIMULATION RESULTS Parameter Units Value Parameter Units Value J kg m diag(43,, 3) J w kg m.43 L x m L y m. L z m l x m l y m. l z m ˆq sun I - [, /, / ] T Φ sun W/m 367 c m/sec 99,79,48 C diff -. Table : Model parameters. 6 Numerical results are presented for an example spacecraft with parameters listed in Table. Note that the vector representing the direction of the Sun is resolved in the inertial frame I in Table and needs to be transformed to the B frame continuously using the current attitude of the spacecraft. Throughout this section we assume minimum/maximum angular accelerations of ± rad/sec for each wheel, i.e., U t {u R m : α i [, ] rad/sec, i {,,..., m}}. The receding horizon drift counteraction control scheme in Algorithm is used to maximize the time until the spacecraft violates prescribed constraints on its attitude and RW spin rates. For the Euler angles, we consider constraints of the form φ min φ φ max, θ min θ θ max, ψ min ψ ψ max, () with φ min <, θ min <, ψ min <, φ max >, θ max >, and ψ max >. Similarly, for the spin rate of the ith wheel, ν i,min ν i ν i,max, where ν i,min, ν i,max R and i {,,..., m}. 8

9 The control strategy (Algorithm ) is based on the linear discrete-time dynamics in (). However, all closed-loop simulations are performed on the continuous-time nonlinear model given by () and (4), where the control input is constant in each sampling interval (zero-order hold). Due to the discrete-time formulation of the DCOC algorithm and unmodeled nonlinear effects, slight constraint violation may occur between the discrete time instants when applying the control inputs to the nonlinear continuous-time model. To address this, we tighten the attitude constraints in () and compute the control inputs for the modified constraints. Thus, if possible, we avoid constraint violation between discrete time instants as illustrated in Figure. Linear solution (discrete-time) Nonlinear solution with linear-based control (continuous-time) Constraint violation Linear solution (discrete-time) Tightened constraints Nonlinear solution with linear-based control (continuous-time) Figure : Illustration of effect of constraint tightening when linear-based control (discrete-time) is applied to the nonlinear continuous-time model. Top part: State trajectories without constraint tightening. Bottom part: State trajectories with constraint tightening. With (), the set defining the state constraints is given by G t (η) {x R 6+m : φ [( η)φ min, ( η)φ max ], θ [( η)θ min, ( η)θ max ], ψ [( η)ψ min, ( η)ψ max ], ν i [ν i,min, ν i,max ], i {,,..., m}}, (3) where η is a parameter to tighten the attitude constraints. Note that, for the nonlinear closedloop simulations, η =, and we terminate the simulation and report the corresponding exit-time once the current state vector is not inside G t () anymore. On the other hand, for the control computations according to Algorithm, we set η =. (see Figure ). In addition, the numerical conditioning of each LP is improved by normalizing the state vector x as follows x = Ox + o, (4) where O R 6+m 6+m and o R 6+m transform the state constraints given by G t (.) in (3) to G t = {x : φ, θ, ψ, ν i [, ], i {,,..., m}}. Moreover, ω j = 3 rad/s corresponds to ω j = and ω j = 3 rad/s corresponds to ω j =, where j {,, 3}. 9

10 8, (rad/sec )!! 3 A (rad)!? (rad) 3 (rad) The sampling time for the linear discrete-time model is chosen as t =. sec. The other parameters of the controller are set to N add =, N ub = 3, and N recover =. We use MATLAB a on a laptop with an i-63 processor and 8 GB RAM for computations. The Hybrid Toolbox 3 (lpsol function with default settings) is employed for LP solving. One RW # -3 LP: open-loop linear model (discrete-time) MPC: closed-loop nonlinear model (continuous-time) # # # # Figure : One operable RW: state and control trajectories vs. time. The case of one operable RW (m = ) with spin axis ḡ = [/ 3, / 3, / 3] T is considered. The attitude and spin rate constraints for this case are given by φ min = θ min =.7 rad, ψ min =.7 rad, φ max = θ max =.7 rad, ψ max =.7 rad, ν,min = rad/sec, and ν,max = rad/sec. Note that the lower bound on ν is chosen to avoid zero speed crossings and increase in wear and power consumption at low speeds. Example state and control trajectories are plotted with dotted black lines for an initial condition

11 of φ =. rad, θ =.3 rad, ψ =. rad, ω, = 4 rad/sec, ω, = rad/sec, ω 3, = 4 rad/sec, and ν, = rad/sec in Figure. The state and control constraints are indicated by dashed red lines in Figure. In addition, Figure shows the open-loop solution of LP (8) for the discrete-time linear model with η =, i.e., for G t (), which violates the constraints after 4 discrete time steps corresponding to 4 t = 98.4 sec =.64 min. For comparison, this value is identical to the open-loop solution of MILP (7), which was solved with the milpsol function of the Hybrid Toolbox. 3 It can be seen that the trajectories of the closedloop nonlinear model with the MPC implementation are close to the linear open-loop solution. Constraint violation occurs after 94.4 sec (.7 min), which is as close as 4 % to the optimal exittime of the linear problem. The worst-case computation time for recomputing the control during the closed-loop simulations was.4 sec < t = sec and the average computation time was. sec. Two RWs Now two operable RWs are assumed (m = ), increasing the control authority of the spacecraft. The spin axes of the two wheels are given by ḡ = [,, ] T and ḡ = [/ 3, / 3, / 3] T. The constraints on spacecraft attitude and RW spin rates for this case are set to φ min = θ min =.7 rad, ψ min =.7 rad, φ max = θ max =.7 rad, ψ max =.7 rad, ν,min = rad/sec, ν,max = rad/sec, ν,min = rad/sec, and ν,max = rad/sec. Figure 3 shows example state and control trajectories over time for an initial condition of φ =.7 rad, θ =.9 rad, ψ =.4 rad, ω, =. rad/sec, ω, = rad/sec, ω 3, =, ν, = 4 rad/sec, ν, = 8 rad/sec. In addition to the closed-loop nonlinear continuous-time solution (dotted black lines), the open-loop solution to the linear problem (for η =, i.e., G t ()) is plotted with solid black lines in Figure 3. The optimal exit-time for the linear discrete-time problem is 7 sec = 4. min (8 steps) as obtained by both MILP (7) and LP (8). The MPC implementation generates nonlinear trajectories that are close to the linear solution. Constraint violation occurs after 63.4 sec (4.39 min), which is. % lower than the optimal exit-time of the linear problem. The longest computation times for recomputing the control occur in the beginning when the time until constraint violation is long. In this case, a worst-case computation time of.44 sec < t =. sec was recorded while the average computation time over all sampling instants was.4 sec. Three RWs Finally, the case of three RWs (m = 3) is treated, where ḡ = [,, ] T, ḡ = [,, ] T, and ḡ 3 = [,, ] T. The constraints on the attitude and RW spin rates are φ min = θ min = ψ min =.7 rad, φ max = θ max = ψ max =.7 rad, ν,min = ν,min = ν 3,min = rad/sec, and ν,max = ν,max = ν 3,max = rad/sec. The initial values for the spacecraft attitude and angular velocity are φ =.7 rad, θ = ψ =, ω, =.8 rad/sec, ω, =.9 rad/sec, and ω 3, =.3 rad/sec. For the initial RW spin rates, two cases are considered: ν = [4, 3, 8] T rad/sec and ν = [4, 3, 8] T rad/sec. For the first case, the spin rates are close to the lower boundary which reduces the control authority. Thus, constraint violation occurs in finite time. This can be seen in Figure 4 showing the trajectories of the closed-loop nonlinear continuous-time model with the

12 MPC implementation (dotted black lines) as well as the open-loop solution with linear discretetime dynamics and η = (solid black lines). The optimal exit-time for the linear problem is 34 sec =.67 min (36 time steps), whereas constraint violation occurs with the nonlinear model after 37 sec =.4 min, which is 4 % less than the optimal exit-time of the linear problem. The worst-case computation time was. sec < t =. sec with an average computation time of.6 sec. For the second case, the initial RW spin rates are further away from the boundaries. The increased control authority may enable to satisfy constraints indefinitely. Simulation results for the first sec are plotted in Figure showing that the proposed feedback controller (dotted black lines) can keep the spacecraft within the limits. A worst-case computation time of.66 sec < t =. sec was recorded while the average computation time over all sampling instants was.79 sec. Note that the open-loop solution of the linear problem in Figure (solid black line) was obtained for a time horizon of N = 4 in (8), not considering the system s evolution for t > 4. CONCLUSION This paper considered an optimal control problem of maximizing the time until prescribed constraints are violated with applications to spacecraft attitude control. The proposed state feedback controller is based on linear programming. In addition, a similar controller was developed to recover constraint satisfaction in case constraints are violated. Closed-loop simulations on the nonlinear model showed that drift from solar radiation pressure is successfully counteracted in order to satisfy constraints on the spacecraft attitude and reaction wheel spin rates for as long as possible. ACKNOWLEDGMENT This research is supported by the National Science Foundation Award Number EECS REFERENCES [] P. Romero, J. M. Gambi, and E. Patiño, Stationkeeping manoeuvres for geostationary satellites using feedback control techniques, Aerospace Science and Technology, Vol., No., 7, pp [] B.-S. Lee, Y. Hwang, H.-Y. Kim, and S. Park, East West station-keeping maneuver strategy for COMS satellite using iterative process, Advances in space research, Vol. 47, No.,, pp [3] R. A. E. Zidek and I. V. Kolmanovsky, Deterministic drift counteraction optimal control and its application to satellite life extension, 4th IEEE Conference on Decision and Control (CDC), IEEE,, pp [4] R. A. E. Zidek and I. V. Kolmanovsky, Geostationary satellite station keeping using drift counteraction optimal control, 6th AAS/AIAA Space flight mechanics meeting, 6, pp. AAS 6 7. [] P. Tsiotras and J. Luo, Control of underactuated spacecraft with bounded inputs, Automatica, Vol. 36, No. 8,, pp [6] C. D. Petersen, F. Leve, M. Flynn, and I. Kolmanovsky, Recovering linear controllability of an underactuated spacecraft by exploiting solar radiation pressure, Journal of Guidance, Control, and Dynamics, Vol. 39, No. 4,, pp [7] H. Kuninaka and J. Kawaguchi, Lessons learned from round trip of Hayabusa asteroid explorer in deep space, Aerospace Conference, IEEE, IEEE,, pp. 8. [8] R. Cowen, The wheels come off Kepler, Nature, Vol. 497, 3, pp [9] R. A. E. Zidek, A. Bemporad, and I. V. Kolmanovsky, Optimal and receding horizon drift counteraction control: linear programming approaches, American Control Conference (ACC) 7, accepted. [] R. Linares, M. K. Jah, J. L. Crassidis, F. A. Leve, and T. Kelecy, Astrometric and photometric data fusion for inactive space object mass and area estimation, Acta Astronautica, Vol. 99, 4, pp.. [] A. Bemporad and M. Morari, Control of systems integrating logic, dynamics, and constraints, Automatica, Vol. 3, No. 3, 999, pp

13 , (rad/sec ), (rad/sec ) 8 8!! 3 A (rad)!? (rad) 3 (rad) [] A. Richards and J. How, Mixed-integer programming for control, Proceedings of the, American Control Conference,., IEEE,, pp [3] A. Bemporad, Hybrid Toolbox - User s Guide, 4. bemporad/hybrid/toolbox. # -3 LP: open-loop linear model (discrete-time) MPC: closed-loop nonlinear model (continuous-time) # # # # Figure 3: Two operable RWs: state and control trajectories vs. time. 3

14 , (rad/sec ), 3 (rad/sec ) 8 3, (rad/sec ) 8 8!! 3 A (rad)!? (rad) 3 (rad).. LP: open-loop linear model (discrete-time) MPC: closed-loop nonlinear model (continuous-time) # # # Figure 4: Three operable RWs with ν = [4, 3, 8] T rad/sec: state and control trajectories vs. time. 4

15 .. 3 (rad).? (rad) !. A (rad) # LP: open-loop linear model (discrete-time) MPC: closed-loop nonlinear model (continuous-time)! #-4 - #-4 4! , ,3, Figure : Three operable RWs with ν = [4, 3, 8]T rad/sec, where the optimal exit-time may be indefinite: state and control trajectories vs. time.