Switching time-optimal control of spacecraft equipped with reaction wheels and gas jet thrusters

Nonlinear Analysis: Hybrid Systems 29 (2018) 261–282

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Switching time-optimal control of spacecraft equipped with reaction wheels and gas jet thrusters Alberto Olivares, Ernesto Staffetti * Rey Juan Carlos University, Camino del Molino s/n, 28943 Fuenlabrada, Madrid, Spain

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Article history: Received 31 December 2016 Accepted 2 March 2018

Keywords: Spacecraft attitude control Hybrid actuation Switched systems Switching time-optimal control Numerical optimal control

a b s t r a c t This paper studies the time-optimal control problem of a rigid spacecraft equipped with both reaction wheels and gas jet thrusters, in which the reaction wheels are the main actuators and the gas jet thrusters act only after saturation or to prevent future saturation of the reaction wheels. It is assumed that the control torques are generated about the principal axes of the spacecraft. The presence of both reaction wheels and thrusters gives rise to two operating modes for each axis. Since this system can change dynamics, it can be regarded as a switched dynamical system. The time-optimal control problem for this system is solved using the embedding approach. With this technique the switched system is embedded into a larger set of systems and the optimal control problem is formulated in the latter. The main advantages of this technique are that assumptions about the number of switches are not necessary, integer or binary variables do not have to be introduced to model switching decisions between modes, and the optimal values of the switching times between modes are obtained without introducing them as unknowns of the optimal control problem. As a consequence, the resulting problem is a classical optimal control problem. Feasibility of the obtained solution is validated through propagation of the initial state. Optimality of the obtained solutions is verified by checking the compliance with Pontryagin’s Maximum Principle. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, the time-optimal control problem of a rigid spacecraft equipped with both reaction wheels and gas jet thrusters is studied. Bound constraints on both torque of the actuators and angular momentum of the reaction wheels are taken into account. To cope with the saturation of the reaction wheels, the spacecraft is equipped with thrusters which are able to generate torques about the axes of the reaction wheels. Thus, there are two operating modes for each axis, and, as a consequence, 23 operating modes for the spacecraft. The problem can be stated as follows: given an initial state and a final state, find the sequences of modes, the corresponding trajectory and control inputs that satisfy the dynamic equation of the spacecraft, and steer the system between the initial and the final states minimizing the duration of the maneuver. Since the optimal sequence of modes has to be determined, this problem is actually a time-optimal control problem of a switched dynamical system. In spite of the apparent similarities between reaction wheels and gas jet thrusters actuations, the corresponding control properties of the system are very different. In the first case, the spacecraft is controlled by momentum exchange devices subject to momentum conservation law, whereas in the second case it is controlled by means of external control torques.

*

Corresponding author. E-mail addresses: [email protected] (A. Olivares), [email protected] (E. Staffetti).

https://doi.org/10.1016/j.nahs.2018.03.003 1751-570X/© 2018 Elsevier Ltd. All rights reserved.

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The most common actuators for spacecraft are gas jet thrusters. The main disadvantage of using gas jet thrusters is fuel consumption. In contrast, although the torque that reaction wheels can offer is typically one order of magnitude lower than that of thrusters, reaction wheels are capable of finer control, and most importantly, they only consume electrical energy. The main disadvantage of reaction wheels is their saturation, that is, since they generate torque on the spacecraft by accelerating, they cannot operate when they reach their maximum angular speed. In this paper, a novel control logic for mixed actuation is investigated, more specifically, the possibility of using reaction wheels as the main actuators, which are supposed to operate with the support of the thrusters that act only after saturation or to prevent future saturation of the reaction wheels. Since the time-optimal control is of bang–bang type and reaction wheels are subject to saturation, the torque on the reaction wheels, their angular momenta, and the torque of the gas jet thrusters have been constrained in this work in order to get practical solutions. In [1], the necessary and sufficient conditions for the controllability of a rigid body in the case of one, two and three independent control torques are provided. If the spacecraft is controlled by three independent torques it is completely controllable, although in the case of momentum wheel actuators a certain minimum control effort is required. In general, works on minimum-time spacecraft reorientation [2–7], consider bounds on the control inputs, but do not include the dynamics of spacecraft, reaction wheels or control moment gyros. In these works, the Pontryagin’s Maximum Principle is used to derive the optimality conditions to solve the optimal control problem. The Pontryagin’s Maximum Principle gives the first-order necessary conditions for the solution of optimal control problems and includes the classical necessary condition given by the Euler–Lagrange equation, a second-order differential equation originally obtained by means of the theory of calculus of variations. The functions that satisfy the first-order necessary conditions are called extremals. However, the first-order necessary conditions of the Pontryagin’s Maximum Principle depend on the constraints of the problem and change when some constraints become active, i.e., when the state or the control reach the boundary of their admissible sets. This corresponds to discontinuities of the first derivative of the extremals. Similarly, the Euler–Lagrange equation holds only at those points at which the extremals are smooth and therefore at points where their first derivatives are not continuous, which are called corners, the so-called Weierstrass–Erdmann corner conditions must be satisfied. The main drawback of this setting is that it is not possible to know in advance the number of corners and their location in time, and therefore it is not easy to derive a general numerical method from these conditions. A numerical method to overcome these difficulties has been presented in [8]. It is based on the Euler–Lagrange necessary condition in integral form and has been applied to several optimal control problems with holonomic, nonholonomic and bound constraints in [9] and [10]. In [11], a version of the Maximum Principle for hybrid optimal control problems under weak regularity conditions has been presented. In particular, autonomous systems have been considered in which the dynamical behavior and the cost are invariant under time translations. The Maximum Principle has been stated for both, problems where the dynamics, the Lagrangian, the cost functions for the switchings, and the endpoint constraints are differentiable along the reference arc, and problems involving nonsmooth maps. In [12] the Hybrid Maximum Principle has been stated and necessary conditions for hybrid optimization have been studied. In particular, optimization problems on fixed compact intervals of time have been considered, in which the set of equivalence classes is finite in such a way that it is possible to drive strong conclusions from the Maximum Principle. In [13], necessary conditions of optimality, in the form of a Maximum Principle, have been presented for a broad class of hybrid optimal control problems in which the dynamics takes the form of differential equations with control terms, and restrictions on the switches between operating modes are described by collections of functional equality and inequality constraints. A wide range of possible autonomous and controlled switching strategies have been provided by different choices of the constraint functionals. Whether the eigenaxis maneuver is optimal or not depends on the definition of the set of admissible control torques. Consider the time-optimal reorientation problem of a rigid inertially symmetric spacecraft. Assume that the control axes are aligned with the principal axes and that the control torques for each axis are bounded. Early works on time-optimal spacecraft reorientation are described in [14]. It has been shown in [2] that eigenaxis rotations, which provide the minimum angular path between two orientations, are not time optimal even in inertially symmetric spacecraft. On the contrary, it has been shown in [3] that for rigid inertially symmetric spacecraft in which the total magnitude of the control torque is constrained, the eigenaxis maneuver is indeed the time-optimal solution. In [15], the minimum-time reorientation problem of a rigid axisymmetric spacecraft is solved numerically using a direct collocation method [16]. In [17], the pseudospectral technique was employed to solve the time-optimal reorientation problem of nonsymmetric rigid bodies. This technique was applied to more realistic cases of time-optimal reorientation of spacecraft in [18,19,6,20]. It is worth noting that the first orbital time-optimal maneuver performed on a real spacecraft was implemented on the NASA Transition Region and Coronal Explorer (TRACE) telescope in 2010, after twenty years of research on this topic. The design and flight implementation of time-optimal attitude maneuvers of this telescope are described in [21], which considers bounds on control and state variables, and the dynamics of the reaction wheels. In [22], a time-optimal controller for reorientation maneuver of a spacecraft with saturation constraints on both torque and angular momentum of the reaction wheels is presented. The proposed control scheme consists in generating an openloop minimum-time reorientation trajectory using the Legendre pseudospectral method which is tracked using a closed-loop control strategy. In [23], the minimum-time reorientation of a spacecraft with both reaction wheels and gas jet thrusters is considered. First, the problem is studied assuming that only reaction wheels are present. Then, gas jet thrusters are considered together

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with two different thrusting procedures for time-optimal attitude maneuver. Finally, the minimum-time reorientation of a spacecraft using both reaction wheels and gas jet thrusters is addressed. In [24], a quaternion-based technique for high precision, large angle rapid reorientation of rigid spacecraft is presented. A hybrid thrusters and reaction wheels control strategy is used, where thrusters are used to provide primary torque in open loop, while reaction wheels provide fine control torque to achieve high precision in closed-loop control. The optimal control problem of switched dynamical systems can be formulated as a multi-phase optimal control problem which includes integer or binary variables to model the choice of the mode in each phase and to generate the optimal mode sequence. Optimal control problems involving binary or integer variables are called Mixed-Integer Optimal Control (MIOC) problems [25–27]. The main drawbacks of this approach are that the number of switches between modes has to be estimated before building the model, the switching times between modes must be included as unknowns of the optimal control problem, and the presence of integer or binary variables which adds a combinatorial complexity to the optimal control problem making it particularly difficult to solve. This is the case of the problem studied in this paper in which the switched dynamical system has 8 operating modes. In [28], the optimal control problem of switched dynamical systems has been solved via embedding into a continuous optimal control problem. It has been shown that for quite a general class of optimal control problems of switched dynamical systems, the computational complexity of the Embedded Optimal Control (EOC) method is no greater than that of continuous optimal control problems. With this technique the switched dynamical system is embedded into a larger set of systems and the optimal control problem is formulated in the latter. The main advantages of this method are that assumptions about the number of switches are not necessary, integer or binary variables do not have to be introduced to model switching decisions between modes, and the optimal values of the switching times between modes are obtained without introducing them as unknowns of the optimal control problem. As a consequence, the resulting problem is a classical continuous optimal control problem [29–31]. This greatly reduces the computation time to find a solution when compared with the MIOC technique. A comprehensive treatment of all the theoretical results obtained with the EOC approach to the optimal control problem of switched mechanical systems can be found in [32]. In this paper, the time-optimal control of a spacecraft has been studied using the EOC approach. The paper is organized as follows. In Section 2, the dynamic model of a spacecraft equipped with both gas jet thrusters and reaction wheels is described. The optimal control problem for this switched dynamical system is stated in Section 3, the embedding approach is introduced in Section 4, and in Section 5, the optimal control problem is reformulated using this technique. In Section 6, the results of the application of the embedding method to several instances of the time-optimal control problem of a rigid asymmetric spacecraft with switching dynamics are reported. Finally, Section 7 contains the conclusions. 2. Model of the system Suppose that the spacecraft is actuated by three reaction wheels and three opposing pairs of gas jet thrusters. For i = 1, 2, 3, assume that the ith reaction wheel is spinning about an axis bi , fixed with respect to the spacecraft, such that the center of mass of the ith wheel lies on it. Suppose that a torque Tui is supplied to the ith wheel about the axis bi by a motor fixed with respect to the spacecraft. Consequently, an equal and opposite torque is exerted by the wheel on the spacecraft. Assume that bi is a principal axes of the spacecraft and that it coincides with the principal axis of the ith wheel about which it is symmetric. Suppose that the axis of the control torque generated by the gas jet thrusters, Tti , coincides with the axis bi . Axes X , Y , and Z of the spacecraft reference frame are chosen to be coincident with axes bi , i = 1, 2, 3, respectively. In this paper, the attitude of the spacecraft with respect to the world frame is represented by quaternions [33]. In space applications, quaternions are in general organized as a vector q = [q1 , q2 , q3 , q4 ]T in which the real part is the last element. This convention will be used to derive the dynamic model of the spacecraft. For convenience, in Section 6, together with the quaternion representation, the roll, pitch and yaw representation of attitudes, with angles (φ, θ, ψ ), will be given. In the rest of this section, Ref. [22] will be followed. The attitude kinematic equation in terms of quaternion is q˙ =

1

Q (ω)q

2

(1)

where q = [q1 , q2 , q3 , q4 ]T is the quaternion vector that represents the attitude of the spacecraft, ω = [ω1 , ω2 , ω3 ]T is the angular velocity vector of the spacecraft, and

[ × −ω Q (ω ) = −ωT

ω

]

0

[ , ω× =

0

ω3 −ω2

−ω3 0

ω1

] ω2 −ω1 .

(2)

0

The dynamic equation of a rigid spacecraft considering the dynamics of the reaction wheels and the gas jet thrusters is

ω˙ = Is−1 (−ω× Is ω − ω× Iw Ω − Tu − Tt + Tex )

(3)

where Is is the inertia momentum matrix of the spacecraft, which, since axes bi , i = 1, 2, 3 coincide with axes X , Y , and Z of the spacecraft reference frame, can be expressed as Ixx 0 0

[ Is =

0 Iyy 0

0 0 Izz

] (4)

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where Ixx , Iyy and Izz are the principal moments of inertia of the spacecraft. Iw is the inertia momentum matrix of the reaction wheels Iw 1 0 0

[ Iw =

0 Iw2 0

0 0

] (5)

Iw3

Ω = [Ω1 , Ω2 , Ω3 ]T is the vector of angular speeds of the reaction wheels, Tu = [Tu1 , Tu2 , Tu3 ]T is the vector of torques at the reaction wheels, Tt = [Tt1 , Tt2 , Tt3 ]T is the vector of torques produced by the gas jet thrusters, and Tex = [Tex1 , Tex2 , Tex3 ]T is the vector of external disturbance torques. The disturbance torques of Tex are assumed to be of small amount and will be neglected. Thus, the dynamic equation of the spacecraft becomes

ω˙ = Is−1 (−ω× Is ω − ω× Iw Ω − Tu − Tt ).

(6)

The dynamic equation of the reaction wheels is

˙ = Iw−1 Tu . Ω

(7)

The state and control variables of the problem are x = [q1 , q2 , q3 , q4 , ω1 , ω2 , ω3 , Ω1 , Ω2 , Ω3 ]T

(8)

u = [Tu1 , Tu2 , Tu3 , Tt1 , Tt2 , Tt3 ]

(9)

T

respectively. The maximum torque and angular momentum of the reaction wheels are actually control and state constraints, respectively. The constraint on angular momentum can be transformed into a constraint on the angular speed of the reaction wheels

|Ωi | ≤ Ω , i = 1, 2, 3

(10)

where Ω is the maximum angular speed of the reaction wheels. The constraints on the maximum torque at the reaction wheels can be expressed as

|Tui | ≤ T u , i = 1, 2, 3

(11)

where T u is the maximum torque at the reaction wheels. Likewise, the constraints on the maximum torque of the gas jet thrusters can be expressed as

|Tti | ≤ T t , i = 1, 2, 3

(12)

where Tt is the maximum torque of the thrusters. The four components of the quaternion vector must satisfy the following condition: q21 + q22 + q23 + q24 = 1.

(13)

Moreover, all the components of quaternion vector are bounded in the interval [−1, 1]. Eqs. (1), (6) and (7) can be rewritten in the general form x˙ = f (x, u)

(14)

which can be regarded as a differential constraint. The performance index for minimum-time problems is defined as tF − tI , where tI is the initial time, which is usually known, and tF is the unknown final time of the maneuver to be determined. The performance index for minimum energy problems is usually defined as the squared norm of the control vector [Tu , Tt ]T . For numerical reasons, it is useful to scale state and control variables of the optimal control problem as follows

˜ ω=

ω Ω Tu Tt ˜ = ,˜ ,Ω Tu = ,˜ Tt = . ω Ω Tu Tt

(15)

The state equations with scaled state and control variables take the following form q˙ =

1 2

ωQ (˜ ω)q

T Tt ˜ − u˜ ˜ ω˙ = Is−1 (−ω˜ ω × Is ˜ ω − Ω˜ ω × Iw Ω Tu − ˜ Tt ) ω ω

˙ = T u I −1˜ ˜ Ω w Tu Ω

(16)

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265

which can be expressed as

˜ x˙ = f (˜ x, ˜ u)

(17)

˜1 , Ω ˜2 , Ω ˜3 ]T ˜ x = [q1 , q2 , q3 , q4 , ˜ ω1 , ˜ ω2 , ˜ ω3 , Ω

(18)

˜ u = [˜ Tu1 , ˜ Tu2 , ˜ Tu3 , ˜ Tt1 , ˜ Tt2 , ˜ Tt3 ]T .

(19)

with

In the dynamic model (16) both operating modes of each control axis of the spacecraft are considered. Thus, the dynamic equations corresponding to a specific operating mode for control axis i can be easily obtained from this general model by setting ˜ Tui or ˜ Tti to zero. In the performance index of the minimum-time optimal control problem, J = tF − tI

(20)

without loss of generality, it can be set tI = 0. The scaled state and control variables are subject to the following bound constraints

˜ xmin ≤ ˜ x ≤˜ xmax

(21)

˜ umin ≤ ˜ u ≤˜ umax

(22)

where

˜ xmax = −˜ xmin = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]T ˜ umax = −˜ umin = [1, 1, 1, 1, 1, 1]T .

(24)

3. The switched optimal control problem Following [32], in this section the optimal control problem of a two-switched dynamical system is formalized. The dynamical model of a two-switched dynamical system can be represented by x˙ S (t) = fvS (t) (t , xS (t), uS (t)), xS (tI ) = xI ∈ Rn , vS (t) ∈ {0, 1}, t ≥ tI .

(25)

In this model the continuously differentiable vector fields, f0 , f1 : R × R × R → R , specify the dynamics of the two possible modes of the system. The control input uS (t) ∈ Ω ⊂ Rm is constrained to belong at each time instant to the bounded and convex set Ω . The binary variable vS (t) is the mode selection variable that identifies which of the two possible system modes, f0 or f1 , is active. Thus, both uS (t) and vS (t) can be regarded as control variables. The initial time tI , the initial state xS (tI ), the final time tF , and final state xS (tF ) are assumed to be restricted to a boundary set B as follows: (tI , xS (tI ), tF , xS (tF )) ∈ B = TI × BI × TF × BF ⊂ R2n+2 . The performance index of the optimal control problem has the following form n

JS (xI , uS , vS ) = g(tI , xI , tF , xF ) +

tF

∫

FvS (t) (t , xS (t), uS (t))dt

m

n

(26)

tI

where the function g is defined on a neighborhood of B, and F0 and F1 are real-valued continuously differentiable functions that represent the cost of operation of the system in each mode. The switched optimal control problem (SOCP) is thus stated as follows min

uS ∈Ω ,vS ∈{0,1}

JS (xI , uS , vS )

(27)

subject to Eq. (25) and endpoint constraints (tI , xS (tI ), tF , xS (tF )) ∈ B. 4. The embedding approach Eq. (25) can be rewritten as follows x˙ E (t) = [1 − vE (t)]f0 (t , xE (t), uE0 (t)) + vE (t)f1 (t , xE (t), uE1 (t)).

(28)

Eqs. (25) and (28) formally coincide under the conditions vE (t) = vS (t) and uE0 = uE1 = uS (t). However, in the latter vE (t) ∈ [0, 1] whereas in the former vS (t) ∈ {0, 1}, which is the key aspect of the embedding technique [28,32]. This method

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is based on solving the SOCP defined above using only continuous control variables vE (t) ∈ [0, 1], uE0 , uE1 ∈ Ω , in which the performance index (26) is rewritten as JE (xI , uE0 , uE1 , vE ) = g(tI , xE (tI ), tF , xE (tF )) tF

∫ +

{ } [1 − vE (t)]F0 (t , xE (t), uE0 (t)) + vE (t)F1 (t , xE (t), uE1 (t)) dt .

(29)

tI

The embedded optimal control problem (EOCP) is thus formulated as min

uE ,uE ∈Ω ,vE ∈[0,1] 0

JE (xI , uE0 , uE1 , vE )

(30)

1

subject to Eq. (28) and endpoint constraints (tI , xE (tI ), tF , xE (tF )) ∈ B. This is an optimal control problem without binary variables. Therefore, classical techniques from optimal control theory can be applied to solve it. However, to guarantee the existence of a solution of the EOCP some additional conditions are needed [28,32], which are the following (i) (ii) (iii) (iv) (v)

The set of admissible pairs state-control inputs, (xE , uE0 , uE1 , vE ), is not empty. There is a compact set that includes all the points (t , xE (t)) for all t ∈ [tI , tF ]. The terminal constraint set B is compact. The input constraint set Ω × Ω × [0, 1] is compact. The vector fields for the two modes of operation are linear in their control inputs, that is, f0 (t , xE , uE0 ) = A0 (t , xE ) + B0 (t , xE )uE0 f1 (t , xE , uE1 ) = A1 (t , xE ) + B1 (t , xE )uE1

(31)

with A0 , B0 , A1 , B1 continuously differentiable functions. (vi) For every (t , xE ) the functions F0 (t , xE , uE0 ) and F1 (t , xE , uE1 ) are convex in the inputs uE0 and uE1 , respectively. It has been shown in [28] and [32] that, once a solution of the EOCP has been obtained, either the solution is of the switched type, that is, vE takes only the values 0 and 1, or suboptimal trajectories of the SOCP can be constructed that can approach the value of the cost for the EOCP arbitrarily closely, and satisfy the boundary conditions within ϵ , with arbitrary ϵ > 0. A thorough discussion about the relationship between SOCP’s and EOCP’s solutions can be found in [28] and [32]. As said before, once the SOCP has been reformulated as an EOCP, an optimal control problem without binary variables is obtained which can be solved using usual optimal control techniques. In this paper, a pseudospectral method [34] has been employed to transcribe the EOCP into a NonLinear Programming (NLP) problem. More specifically, a Legendre–Gauss– Lobatto pseudospectral method has been used. The set of constraints of the resulting NLP problem includes the system constraints that correspond to the differential constraint (28) and the discretized versions of the other constraints of the optimal control problem. They include the algebraic constraints (13), the state and control constraints (21) and (22), and the boundary conditions. 5. Specification of the embedded optimal control problem As said before, each control axis of the spacecraft has two operating modes, the wheel mode and the thruster mode, which will be denoted by the symbols W and T, respectively. To apply the embedding technique to the eight operating modes of the spacecraft there are two possible choices: introducing seven mode selection variables to describe which operating mode is active for the spacecraft or, since the control action for each control axis is independent of the others, introducing one mode selection variable per control axis to describe which operating mode is active for that axis. The second option requires the introduction of only three mode selection variables and therefore has been adopted for the numerical experiments. Each one of the eight operating modes for the spacecraft will be denoted as 3-tuple of symbols from the set {W, T}. For example, the WWT mode of the spacecraft corresponds to W mode active for the X and Y axes and T mode active for the Z axis. Let vEi , i = 1, 2, 3 be the mode selection variables that describe which operating mode is active for axis bi . Without loss of generality, it can be assumed that vEi = 1 corresponds to W mode active for control axis bi and vEi = 0 corresponds to T mode active for the same axis. Eq. (28) particularized for the two operating modes per control axis of the spacecraft described by Eq. (16), takes the form q˙ 1 = q˙ 2 = q˙ 3 =

1 2 1 2 1 2

ω(˜ ω1 q4 − ˜ ω2 q3 + ˜ ω3 q2 ) ω(˜ ω1 q3 + ˜ ω2 q4 − ˜ ω3 q1 ) ω(−˜ ω1 q2 + ˜ ω2 q1 + ˜ ω3 q4 )

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267

1 ω(−˜ ω1 q1 − ˜ ω2 q2 − ˜ ω3 q3 ) 2 ( ) T t˜ Tt1 T u˜ Tu1 1 ˜2˜ ˜3 − [1 − vE1 (t)] Iyy ω˜ − vE1 (t) ˜ ω˙ 1 = ω2˜ ω3 − Izz ω˜ ω2˜ ω3 + Iw2 Ω Ω ω3 − Iw3 Ω ˜ ω2 Ω Ixx ω ω q˙ 4 =

˜ ω˙ 2 = ˜ ω˙ 3 =

1

( ˜3 − Iw1 Ω Ω ˜1˜ Iw3 Ω ˜ ω1 Ω ω3 − Ixx ω˜ ω1˜ ω3 + Izz ω˜ ω1˜ ω3 − [1 − vE2 (t)]

Iyy 1 Izz

( ˜1˜ ˜2 − [1 − vE3 (t)] Ixx ω˜ ω1˜ ω2 − Iyy ω˜ ω1˜ ω2 + Iw1 Ω Ω ω2 − Iw2 Ω ˜ ω1 Ω

T t˜ Tt2

ω T t˜ Tt3

ω

− vE2 (t) − vE3 (t)

T u˜ Tu2

)

ω T u˜ Tu3

ω

) (32)

˜ ˙ = v (t) T u Tu1 ˜ Ω 1 E1 Ω Iw 1 ˜ T ˙ = v (t) u Tu2 ˜ Ω 2 E2 Ω Iw 2 ˜ T ˙ = v (t) u Tu3 ˜ Ω 3 E3 Ω Iw 3 ˜1 , Ω ˜2 , Ω ˜3 ]T , uE0 = ˜ Tu . Tt , uE1 = ˜ in which xE = [q1 , q2 , q3 , q4 , ˜ ω1 , ˜ ω2 , ˜ ω3 , Ω Being a time-optimal control problem, in this case, the functions F0 (t , xE , uE0 ) = F1 (t , xE , uE1 ) = 1 and the performance index (29) is JE (xI , uE0 , uE1 , vE ) =

tF

∫

dt = tF − tI .

(33)

tI

The set of constraints of the problem includes the algebraic constraints (13), the state constraints (21), and the control constraints (22). It is not difficult to check that the conditions for the existence of a solution of the EOCP given in [28] and [32] are fulfilled (see Appendix A). 6. Numerical results Two optimal control problems have been studied for a rigid inertially asymmetric spacecraft equipped with both reaction wheels and gas jet thrusters: the minimum-time rest-to-rest reorientation maneuver and the minimum-time detumbling maneuver. Attitude control is required for nearly all spacecrafts to point directional antennas, solar panels and observation instruments, to control heat dissipation, and to orient propulsion systems. Reorientation is the process of controlling the orientation of a spacecraft from one attitude to another. In general, the spacecraft is at rest at the beginning and at the end of the maneuver. Since imperfect launch conditions can induce tumbling motions in a spacecraft, a control tasks to be executed after deployment from the launch vehicle is to stabilize its angular rotation, which is called detumbling. Thus, detumbling is the process of controlling the rotation of a spacecraft which, in general, at the beginning of the maneuver, is not about an eigenaxis. It can be accomplished by stopping the rotation of the spacecraft or by transforming it into a rotation about an eigenaxis. Let the initial and final scaled states be denoted by ˜ xI and ˜ xF , respectively. It is possible that not all the components of ˜ xI and ˜ xF are specified. Solving the optimal control problems mentioned above, consists in finding the scaled control variables (19) the sequence of dynamic modes for each axis, and the switching times between them that steer in minimum time the spacecraft from the specified components of ˜ xI to those of ˜ xF , subject to constraints (13), (21) and (22). In particular, in the numerical experiments the following parameters have been used for the dynamic model of the spacecraft: Ixx = 6.63 kg m2 , Iyy = 8.90 kg m2 , Izz = 9.63 kg m2 , Iw1 = Iw2 = Iw3 = 0.000169 kg m2 , Ω = 710 rad/s, T u = 0.0101 N m, T t = 0.00505 N m, and ω = 0.04 rad/s. The introduction of a maximum angular speed of the spacecraft about each control axis is motivated by the fact that some attitude determination systems, such as star trackers, can provide accurate determination of spacecraft attitude only if this angular speed does not exceed a specified limit value. As mentioned in the introduction, it is assumed that the reaction wheels are the main actuators of the spacecraft and that the gas jet thrusters act only after saturation or to prevent future saturation of the reaction wheels. Therefore, mode shifting is necessary since no torque can be generated by the reaction wheels at very high angular speed. This shifting can be induced by introducing angular speed-depending efficiency functions [32]. The same behavior can be achieved by setting the maximum torque of the reaction wheels at a value higher than the maximum torque of the gas jet thrusters.

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(a) Spacecraft actuated by reaction wheels and gas jet thrusters.

(b) Spacecraft actuated by reaction wheels only.

Fig. 1. Motion of the body frame in the solution of the minimum-time rest-to-rest reorientation maneuver about the unactuated axis.

6.1. Minimum-Time rest-to-rest reorientation maneuver In this numerical experiment the following initial and final attitude parameters of the spacecraft have been assumed: q1 (tI ) = 0, q1 (tF ) = 0, q2 (tI ) = 0, q2 (tF ) = 0, q3 (tI ) = 0, q3 (tF ) = 0.5, q4 (tI ) = 1, q4 (tF ) = 0.8660254038, which correspond to φ (tI ) = θ (tI ) = ψ (tI ) = φ (tF ) = θ (tF ) = 0 rad, ψ (tF ) = π/3 rad. The initial and final angular velocities of the satellite and the initial and final angular speeds of the reaction wheels have been set to zero. Thus, this experiment is a rest-to-rest reorientation maneuver for the spacecraft in which also the reaction wheels are required to stop at the end of the maneuver. The number of intervals of the discretization has been 100. The computation time to find a solution has been 10.68 s. The minimum time to execute the maneuver has been tF = 66.24 s. The motion of the body reference frame is shown in Fig. 1a, where the initial configuration of the body frame is represented in light gray and the final configuration in black. The optimal control variables are represented in Fig. 2. The mode selection variables vE1 , vE2 and vE3 are represented in Figs. 3b, 3d, and 3f. In these figures vEi = 1, i = 1, 2, 3 correspond to W mode active for the axes X , Y , and Z , respectively. On the contrary, vEi = 0, i = 1, 2, 3 correspond to T mode active for the axes X , Y , and Z , respectively. The corresponding state variables are represented in Figs. 3a, 3c, and 3e. It can be seen from Fig. 3e that all the wheels saturate. Analyzing the solutions obtained for each control axis separately, one can observe that the control action for the X axis is quite complex. It can be seen from Fig. 3b that until time 11.63 s the W mode is active for the X axis whose rotation is controlled by wheel 1 or thruster 1. As shown in Fig. 2a, wheel 1 is accelerated with ˜ Tu1 = +1 from time 0 to time 11.63 s and, as shown in Fig. 3e, at time 11.63 s it reaches its maximum ˜1 = +1 and saturates. Thus, to continue the control action T mode is activated for the X axis until positive angular speed Ω time 21.13 s. However, as it can be seen from Fig. 2b, at time 18.40 s there is a change of sign of the control torque, from ˜ Tt1 = +1 to ˜ Tt1 = −1. At time 21.13 s, W mode for the X axis starts again with ˜ Tu1 = −1. This means that wheel 1 is ˜1 = +1 to Ω ˜1 = −1. This value of negative angular speed is reached at time 45.11 s. When this occurs, T decelerated from Ω mode for the X axis is activated until time 54.62 s. However, as it can be seen from Fig. 2b, at time 47.85 s there is a change of sign of the control torque, from ˜ Tt1 = −1 to ˜ Tt1 = +1. Afterwords, W mode for the X axis is activated with ˜ Tu1 = +1 until ˜1 = −1 to Ω ˜1 = 0. the end of the maneuver and wheel 1 is accelerated from Ω In response to this sequence of control actions, the spacecraft performs a sequence of rotations in different directions about the X axis. The final angular position of the spacecraft with respect to X axis is zero. The spacecraft reaches its maximum negative angular speed about the X axis at time 19.30 s, zero angular speed about the X axis at time 33.61 s, and its maximum positive angular speed about the X axis at time 46.94 s. The resulting control action for the Y and Z axes are simpler. For the sake of brevity, only the control action for the Z axis will be described. It can be seen from Fig. 3f that the W mode for the Z axis is active only at the beginning and at the end of the maneuver. Indeed, the maneuver starts in W mode for the Z axis with ˜ Tu3 = −1 until wheel 3 reaches its maximum negative ˜3 = −1 and saturates. This occurs at time 12.42 s. Then, the control action for the Z axis is performed in T angular speed Ω

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269

Fig. 2. Control variables in the solution of the minimum-time rest-to-rest reorientation maneuver.

mode with ˜ Tt3 = −1 until time 54.62 s. However, there is a change of sign in ˜ Tt3 from ˜ Tt3 = −1 to ˜ Tt3 = 1 at time 33.61 s. At time 54.62 s the control action continues in W mode with ˜ Tu3 = 1 until the end of the maneuver. In response to this sequences of control actions the spacecraft is accelerated and decelerated about the Z axis. It reaches its maximum angular speed at time 32.63 s. Its final angular position about the Z axis is ψ (tF ) = π/3 rad. Thus, the maneuvers about the three axes present the desired behavior. The solution is of bang–bang type about the three control axes, as expected in time-optimal control problems. Analyzing the solutions obtained jointly, 7 phases can be distinguished. They are represented in Fig. 4 together with the angular speeds of the reaction wheels. Each phase corresponds to a different operating mode of the spacecraft which is the result of the combination of the operating modes of each control axis. The operating modes of the spacecraft in the minimum-time rest-to-rest reorientation maneuver together with the time intervals in which they take place are given in

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(a) q1 (blue), q2 (purple), q3 (yellow) and q4 (green).

(b)

(c) ˜ ω1 (blue), ˜ ω2 (purple), and ˜ ω3 (yellow).

(d)

˜1 (blue), Ω ˜2 (purple), and Ω ˜3 (yellow). (e) Ω

(f)

Fig. 3. State variables and mode selection variables in the solution of the minimum-time rest-to-rest reorientation maneuver. The state variables obtained by propagation of the control variables are represented with dotted lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1. It is easy to see that 4 of the 8 possible operating modes are used in this maneuver, namely, WWW, WTW, TWT, WTT. In particular, the mode WTT is used once whereas the others are used twice. For instance, it can be seen in Fig. 4 that in the third phase, which corresponds to TWT operating mode of the spacecraft, only the angular speed of wheel 2 changes over the interval (12.42, 22.06) s. This means that W mode is active for Y control axis whereas the X and Z control axes are operated in T mode. It is worth noting that, as mentioned before, the optimal values of the switching times between modes are determined by the mode selection variables vEi , i = 1, 2, 3, and therefore are obtained without introducing them as unknowns of the EOCP. Following [6], a numerical analysis has been carried out in order to check first if the obtained solution is a feasible minimum-time rest-to-rest reorientation maneuver, and then if the solution meets the necessary optimality conditions derived from Pontryagin’s Minimum Principle [35,36]. The results of this analysis, which is described in Appendix B, show that the obtained solution is feasible and that it fulfills the necessary optimality conditions.

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271

˜1 in blue, Ω ˜2 in purple, and Ω ˜3 in yellow, with the phases and the corresponding operating modes Fig. 4. Angular speeds of the reaction wheels, with Ω in the solution of the minimum-time rest-to-rest reorientation maneuver. The optimal switching times between phases can be read-off from Table 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Operating modes of the spacecraft in the solution of the minimum-time restto-rest reorientation maneuver together with the time intervals in which they take place. Operating mode

Time interval (s)

WTW WWW TWT WTT TWT WWW WTW

(0.00, 10.09) (10.09, 12.42) (12.42, 22.06) (22.06, 45.11) (45.11, 54.62) (54.62, 56.89) (56.89, 66.24)

A similar experiment was conducted with the same boundary conditions and number of intervals of the discretization but assuming that the spacecraft is equipped only by reaction wheels, which, therefore is not a switched optimal control problem. The computation time to find a solution has been in this case 3.67 s. The minimum time to execute the maneuver has been tF = 84.36 s. As expected, in spite of the higher complexity of the switched optimal control problem solved in the previous experiment, the computation times in these two experiments are of the same order of magnitude. Moreover, in the case in which the spacecraft is equipped only by reaction wheels, since no other actuator can provide torque after their saturation, the minimum time to execute the maneuver has been higher. For the same reason, singular arcs appeared in the switching structure of the solution during the period in which the control torque oscillates around zero, which are not present when the spacecraft is equipped by both reaction wheels and gas jet thrusters. A discussion of these singular conditions [2,6] is beyond the scope of this paper. The motion of the body reference frame is shown in Fig. 1b, the optimal control variables are represented in Fig. 5, and the corresponding state variables are represented in Fig. 6. 6.2. Minimum-Time detumbling maneuver In this numerical experiment, the following initial and final attitude parameters of the spacecraft have been assumed: q1 (tI ) = 0, q1 (tF ) = 0.0182830462, q2 (tI ) = 0, q2 (tF ) = 0.2853201330, q3 (tI ) = 0, q3 (tF ) = 0.3352703443, q4 (tI ) = 1, q4 (tF ) = 0.8976925687, which correspond to φ (tI ) = θ (tI ) = ψ (tI ) = 0 rad, φ (tF ) = π/12 rad, θ (tF ) = π/6 rad, ψ (tF ) = π/4 rad. The components of the initial angular velocity vector of the spacecraft have been set as follows ˜ ω1 (tI ) = 0.1, ˜ ω2 (tI ) = 0.2, ˜ ω3 (tI ) = 0.3. The components of the final angular velocity vector of the spacecraft have been set as follows ˜ ω1 (tF ) = 0, ˜ ω2 (tF ) = 0, and ˜ ω3 (tI ) = free. The initial angular speeds of the reaction wheels have been set to zero. The final angular speeds of the reaction wheels have been set free. The number of interval of the discretization has been 100. The computation time to find a solution has been 13.46 s. The minimum time to execute the maneuver has been tF = 36.89 s. The motion of the body reference frame is shown in Fig. 7a, where the initial configuration of the body

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Fig. 5. Control variables in the solution of the minimum-time rest-to-rest reorientation maneuver for a spacecraft actuated only by reaction wheels.

frame is represented in light gray and the final configuration in black. The optimal control variables are represented in Fig. 8, the mode selection variables vE1 , vE2 and vE3 are represented in Figs. 9b, 9d, and 9f, and the corresponding state variables are represented in Figs. 9a, 9c, and 9e. It can be seen from Fig. 9e that, as in the previous experiment, all the wheels saturate, although the saturation time for wheel 1 is short and occurs at the end of the maneuver. Analyzing the solutions obtained for each control axis separately, it can be seen from Figs. 9d and 8c that the control action for the Y axis is performed in W mode with ˜ Tu2 = −1 from the beginning of the maneuver until saturation of wheel 2 which occurs at time 11.77 s. Then, it continues in T mode with ˜ Tt2 = −1 until time 19.80 s. At time 19.80 s there are two simultaneous switches for the Y axis: from T mode to W mode and from ˜ Tt2 = −1 to ˜ Tu2 = 1 in the control torque. Then, the control action for the Y axis is performed in W mode until the end of the maneuver. In response to this sequence of control actions the spacecraft is accelerated and decelerated about Y axis. It reaches its maximum angular speed at time 19.80 s. The control actions for the X and Z axes can be easily inferred from Figs. 9b, 8a, 8b, and 9f, 8e, 8f, respectively. Note that at time 36.87 s there is a phase of very short duration in which W mode is activated for the Z axis with ˜ Tu3 = 1. This control action occurs when wheel is saturated at its maximum negative angular speed, slightly reducing it. As a consequence, the angular speed of the spacecraft about the Z axis is also reduced. This control action allows the spacecraft to reach the final conditions. As required, the spacecraft reaches the final attitude with zero angular speed about both X and Y axes whereas the final angular speed about the Z axis is ˜ ω3 (tF ) = 0.81557. The initial momentum of the spacecraft is partially transferred to the reaction wheels, which, being at rest at the beginning, have nonzero angular speeds at the end of the maneuver. Analyzing the solutions obtained jointly, 6 phases can be distinguished. They are represented in Fig. 10 together with the angular speeds of the reaction wheels. The operating modes of the spacecraft in the minimum-time detumbling maneuver together with the time intervals in which they take place are given in Table 2. It is easy to see that 4 of the 8 possible operating modes are used in this maneuver, namely the modes TWW, WWT, TTW, TWT. In particular, the operating modes TWW and TTW are used once whereas the others are used twice. The validation of the feasibility of the solution and the verification of the fulfillment of the necessary optimality conditions derived from Pontryagin’s Minimum Principle have been carried out as explained in the in Appendix B, The results of this analysis show that the obtained solution is feasible and that it fulfills the necessary optimality conditions.

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(a) q1 (blue), q2 (purple), q3 (yellow) and q4 (green).

273

(b) ˜ ω1 (blue), ˜ ω2 (purple), and ˜ ω3 (yellow).

˜1 (blue), Ω ˜2 (purple), and Ω ˜3 (yellow). (c) Ω Fig. 6. State variables in the solution of the minimum-time rest-to-rest reorientation maneuver for a spacecraft actuated only by reaction wheels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a) Spacecraft actuated by reaction wheels and gas jet thrusters.

(b) Spacecraft actuated by reaction wheels only.

Fig. 7. Motion of the body frame in the solution of the minimum-time detumbling maneuver.

A similar experiment was conducted with the same boundary conditions and number of intervals of the discretization but assuming that the spacecraft is equipped only by reaction wheels. The computation time to find a solution has been in this case 5.24 s. The minimum time to execute the maneuver has been tF = 40.23 s. Again, in spite of the higher complexity of the switched optimal control problem solved in the previous experiment, the computation times in these two experiments are of the same order of magnitude, and the minimum time to execute the maneuver has been higher in the latter. Singular arcs

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Fig. 8. Control variables in the solution of the minimum-time detumbling maneuver.

appeared in the switching structure of the solution which are not present when the spacecraft is equipped by both reaction wheels and gas jet thrusters. The motion of the body reference frame is shown in Fig. 7b, the optimal control variables are represented in Fig. 11, and the corresponding state variables are represented in Fig. 12. 7. Conclusions In this paper, the minimum-time reorientation problem of a rigid spacecraft equipped with both reaction wheels and gas jet thrusters on each control axis has been studied. The presence of reaction wheels and gas jet thrusters gives rise to two operating modes for each control axis and the resulting optimal control problem for this switched dynamical systems has

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(a) q1 (blue), q2 (purple), q3 (yellow) and q4 (green).

(b)

(c) ˜ ω1 (blue), ˜ ω2 (purple), and ˜ ω3 (yellow).

(d)

˜1 (blue), Ω ˜2 (purple), and Ω ˜3 (yellow). (e) Ω

(f)

275

Fig. 9. State variables and mode selection variables in the solution of the minimum-time detumbling maneuver. The state variables obtained by propagation of the control variables are represented with dotted lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2 Operating modes of the spacecraft in the solution of the minimum-time detumbling maneuver together with the time intervals in which they take place. Operating mode

Time interval (s)

WWT TWW TTW TWT WWT TWT

(0.00, 8.77) (8.88, 12.29) (12.29, 20.34) (20.34, 30.85) (30.85, 34.81) (34.81, 36.89)

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˜1 in blue, Ω ˜2 in purple, and Ω ˜3 in yellow, with the phases and the corresponding operating modes in Fig. 10. Angular speeds of the reaction wheels, with Ω the solution of the minimum-time detumbling maneuver. The optimal switching times between phases can be read-off from Table 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Control variables in the solution of the minimum-time detumbling maneuver for a spacecraft actuated only by reaction wheels.

been solved using the embedding approach. This method converts the switched optimal control problem into a classical optimal control one avoiding the combinatorial complexity of mixed-integer optimal control formulations. It has been demonstrated that this technique can efficiently deal with planning minimum-time maneuvers of a rigid spacecraft, in which bound constraints on the torque of the actuators and on the angular momentum of the reaction wheels are taken into account. It has been shown that the computation times required to plan spacecraft maneuvers that involve complex switching sequences between reaction wheels and gas jet thrusters, are of the same order of magnitude of those required to plan maneuvers that do not imply switches between operating modes.

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277

(b) ˜ ω1 (blue), ˜ ω2 (purple), and ˜ ω3 (yellow).

(a) q1 (blue), q2 (purple), q3 (yellow) and q4 (green).

˜1 (blue), Ω ˜2 (purple), and Ω ˜3 (yellow). (c) Ω Fig. 12. State variables for the minimum-time detumbling maneuver for a spacecraft actuated only by reaction wheels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Appendix A Existence of solutions of the embedded optimal control problem In this section the conditions for the existence of solutions of the EOCP stated in Section 5 will be checked. For the sake of simplicity, the conditions will be checked for the particular case in which only control axis b1 is supposed to have two modes of operation whereas control axes b2 and b3 operate in fixed modes, for instance W mode and T mode, respectively. In this case, Eq. (32) takes the form q˙ 1 = q˙ 2 = q˙ 3 =

1 2 1 2 1 2 1

ω(˜ ω1 q4 − ˜ ω2 q3 + ˜ ω3 q2 ) ω(˜ ω1 q3 + ˜ ω2 q4 − ˜ ω3 q1 ) ω(−˜ ω1 q2 + ˜ ω2 q1 + ˜ ω3 q4 )

ω(−˜ ω1 q1 − ˜ ω2 q2 − ˜ ω3 q3 ) 2 ( ) T t˜ Tt1 T u˜ Tu1 1 ˙ ˜ ˜ ˜ ω1 = Iyy ω˜ ω2˜ ω3 − Izz ω˜ ω2˜ ω3 + Iw3 Ω Ω2˜ ω3 − Iw3 Ω ˜ ω2 Ω3 − [1 − vE1 (t)] − vE1 (t) Ixx ω ω q˙ 4 =

1 ˜ ω˙ 2 =

(

1 ˜ ω˙ 3 =

(

˜3 − Iw3 Ω Ω ˜1˜ Iw3 Ω ˜ ω1 Ω ω3 − Ixx ω˜ ω1˜ ω3 − Izz ω˜ ω1˜ ω3 −

Iyy

Izz

˜1˜ ˜2 − Ixx ω˜ ω1˜ ω2 − Iyy ω˜ ω1˜ ω2 + Iw3 Ω Ω ω2 − Iw3 Ω ˜ ω1 Ω

T u˜ Tu2

)

ω T t˜ T t3

ω

)

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Fig. 13. Control variables and the associated Karush–Kuhn–Tucker multipliers in the solution of the minimum-time rest-to-rest reorientation maneuver.

˜ ˙ = v (t) T u Tu1 ˜ Ω 1 E1 Ω Iw1 ˜ T T ˙ = u u2 ˜ Ω 2 Ω Iw2 ˙ ˜ Ω = 0.

(34)

3

It is easy to see from Eq. (28) with xE (t) = ˜ x(t) and uE1 = ˜ T u , uE 0 = ˜ Tt , and Eq. (34), that the vector fields f0 and f1 take in this case the form (31) with the following expressions for the matrices A0 , B0 , A1 , and B1

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279

Fig. 14. Hamiltonian validation of the solution of the minimum-time rest-to-rest reorientation maneuver.

⎡

1

ω(˜ ω1 q4 − ˜ ω2 q3 + ˜ ω3 q2 )

⎤

⎢ ⎥ 2 ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ω(˜ ω1 q3 + ˜ ω2 q4 − ˜ ω3 q1 ) ⎢ ⎥ 2 ⎢ ⎥ 1 ⎢ ⎥ ⎤ ⎡ ⎢ ⎥ ω(−˜ ω1 q2 + ˜ ω2 q1 + ˜ ω3 q4 ) 0 ⎢ ⎥ 2 ⎢ ⎥ ⎢ 0 ⎥ 1 ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ω(−˜ ω1 q1 − ˜ ω2 q2 − ˜ ω3 q3 ) ⎢ ⎥ ⎢ ⎥ 2 ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ) 1 ( ⎢ ⎥ ⎢ ⎥ ˜2˜ ˜3 ω2˜ ω3 − Izz ω˜ ω2˜ ω3 + Iw3 Ω Ω ω3 − Iw3 Ω ˜ ω2 Ω Iyy ω˜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Ixx ⎢− T t ⎥ ( ) ⎢ ⎥ , B = A0 = ⎢ ⎢ ⎥ 0 ⎢ ω⎥ T u˜ Tu2 ⎥ ⎢ 1 I Ω˜ ⎥ ˜ ˜ ⎢ ⎥ ω Ω − I Ω Ω ˜ ω − I ω ˜ ω ˜ ω − I ω ˜ ω ˜ ω − 0 ⎢ ⎥ w3 1 3 w3 1 3 zz 1 3 1 3 xx ⎢ ⎥ ω ⎢ Iyy ⎥ ⎥ ⎢ 0 ⎢ ( )⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ˜ ⎢ 1 T t T t3 ⎥ ⎣ ⎢ ⎥ ˜2 − ˜1˜ 0 ⎦ ω1˜ ω2 − Iyy ω˜ ω1˜ ω2 + Iw3 Ω Ω ω2 − Iw3 Ω ˜ ω1 Ω ⎢ Izz Ixx ω˜ ⎥ ω ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ T u˜ Tu2 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Ω Iw2

(35)

0

⎡

1

ω(˜ ω1 q4 − ˜ ω2 q3 + ˜ ω3 q2 )

⎤

⎢ ⎥ 2 ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ω(˜ ω1 q3 + ˜ ω2 q4 − ˜ ω3 q1 ) ⎢ ⎥ 2 ⎤ ⎡ ⎢ ⎥ 0 1 ⎢ ⎥ ⎢ ⎥ ω(−˜ ω1 q2 + ˜ ω2 q1 + ˜ ω3 q4 ) ⎢ 0 ⎥ ⎢ ⎥ 2 ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ω(−˜ ω1 q1 − ˜ ω2 q2 − ˜ ω3 q3 ) ⎢ 0 ⎥ ⎢ ⎥ 2 ⎥ ⎢ ⎢ ⎥ ⎢ Tu ⎥ ) 1 ( ⎢ ⎥ ⎥ ⎢ ˜ ˜ Iyy ω˜ ω2˜ ω3 − Izz ω˜ ω2˜ ω3 + Iw3 Ω Ω2˜ ω3 − Iw3 Ω ˜ ω2 Ω3 ⎢ ⎥ ⎢− ω ⎥ ⎢ ⎥ Ixx ⎢ ⎥ ( )⎥ , B 1 = ⎢ A1 = ⎢ ⎥. ⎢1 ⎢ 0 ⎥ T u˜ Tu2 ⎥ ⎢ ⎥ ˜ ˜ ⎥ ⎢ Iw3 Ω ˜ ω1 Ω3 − Iw3 Ω Ω1˜ ω3 − Ixx ω˜ ω1˜ ω3 − Izz ω˜ ω1˜ ω3 − ⎢ ⎥ ⎢ 0 ⎥ ω ⎢ Iyy ⎥ ⎢ ⎥ ⎢ ( )⎥ ⎢ Tu ⎥ ⎢ ⎥ ⎢ ⎥ ˜ ⎢ 1 ⎥ T T ⎢ Ω Iw1 ⎥ ⎢ ˜1˜ ˜2 − t t3 ⎥ I ω ˜ ω ˜ ω − I ω ˜ ω ˜ ω + I Ω Ω ω − I Ω ˜ ω Ω ⎢ ⎥ xx 1 2 yy 1 2 w 2 w 1 3 3 ⎢ Izz ⎥ ω ⎣ 0 ⎦ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ˜ T u Tu2 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Ω Iw2 0

(36)

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Fig. 15. Control variables and the associated Karush–Kuhn–Tucker multipliers in the solution of the minimum-time detumbling maneuver.

Thus, f0 and f1 are linear in their control input and A0 , B0 , A1 , B1 are continuously differentiable functions. The same occurs particularizing Eq. (32) for the two operating modes of control axes b2 and b3 , supposing that the other control axes are operating in arbitrary fixed modes. Thus, condition (v) given in Section 4 is met. Since for a time-optimal control problem F0 (t , xE , uE0 ) = F1 (t , xE , uE1 ) = 1, condition (vi) is fulfilled. Based on the assumptions made on the input constraint (21) and (22), and on the vector fields f0 and f1 , one can conclude that conditions (i), (ii), and (iv) are met. Finally, since a sufficiently large compact set can be found to replace the terminal set, condition (iii) is also fulfilled. Thus, the EOCP stated in Section 5 has a solution.

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281

Fig. 16. Hamiltonian validation of the solution of the minimum-time detumbling maneuver.

Appendix B Verification and validation of the solution of the minimum-time rest-to-rest reorientation maneuver First of all, the feasibility of the obtained solution of the optimal control problem has been evaluated through independent propagation of the initial state. The obtained optimal control has been introduced as input into a standard Runge–Kutta (RK) solver to check if it drives the dynamical system from the initial state to the final state. In particular, the MATLAB ODE45 routine has been used. It is a single-step medium order solver based on an explicit Runge–Kutta (4, 5) formula.1 A comparison between the state variables of the solution and those obtained by propagation of the initial state is shown in Figs. 3a, 3c, and 3e, where the state variables obtained by propagation are represented with dotted lines. It is easy to see that the state variables of the solution overlaid with the RK-propagated states. Once the feasibility of the obtained solution has been verified, the necessary conditions for optimality have been checked. Since the control is subject to the inequality constraint (22), the Karush–Kuhn–Tucker (KKT) theorems and complementarity conditions [6] can be applied. For the sake of clarity of exposition consider the control inputs ˜ Ti for axis i, with i = 1, 2, 3. These control inputs are related to the control inputs Tui and Tti through the mode selection variables vEi as follows: ˜ Tui = ˜ Ti vEi , ˜ T ti = ˜ Ti (1 − vEi ), i = 1, 2, 3. Moreover, they are bounded in the interval [−1, 1] as Tui and Tti . Let µi , i = 1, 2, 3 be the KKT multipliers associated with control inputs ˜ Ti , i = 1, 2, 3. These KKT multipliers must satisfy the complementary conditions:

⎧ ⎨≤ 0 µi ≥ 0 ⎩ =0

if ˜ Ti = −1, if ˜ Ti = 1, if − 1 ≤ ˜ Ti ≤ 1 ,

which define a relationship between the control variables and the KKT multipliers. When a multiplier is less than zero, its associated control variable should take its minimum value. Similarly, when a multiplier is greater than zero, the corresponding control variable should take its maximum value. When a multiplier crosses through zero, the associated control variable should switch from one extreme value to the other, consistently with the direction of the crossing. Similar considerations hold for mode selection variables vEi , i = 1, 2, 3, which are actually input variables bounded in the interval [0, 1], and the corresponding KKT multipliers. Fig. 13 shows that the relations between control variables ˜ Ti , i = 1, 2, 3, mode selection variables vEi , i = 1, 2, 3, and their associated multipliers meet the KKT conditions. Regarding the behavior of the Hamiltonian, H, in this case, it does not depend on time. Therefore, ∂∂Ht = 0, which implies that H must have a constant value along the optimal trajectory. Moreover, the value of the H at final time tF must be H(tF ) = −1. As a consequence, a necessary condition for optimality is that the Hamiltonian must be constant and numerically equal to −1. Fig. 14 shows that the computed Hamiltonian approximately fulfills this necessary condition. Verification and validation of the solution of the minimum-time detumbling maneuver The consideration made in the previous section for the verification and validation of the solution of the minimum-time rest-to-rest reorientation maneuver can be repeated for the verification and validation of the solution of the minimum-time detumbling maneuver. Figs. 9a, 9c, and 9e, where the state variables obtained by propagation are represented with dotted lines, show that the state variables of the solution overlaid with the RK-propagated states. Fig. 15 shows that the relations 1 http://www.mathworks.com/.

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