International Journal of Aerospace Engineering

Volume 2018 (2018), Article ID 5947521, 14 pages

https://doi.org/10.1155/2018/5947521

## Time-Optimal Attitude Scheduling of a Spacecraft Equipped with Reaction Wheels

Universidad Rey Juan Carlos, Camino del Molino s/n, 28943 Fuenlabrada, Madrid, Spain

Correspondence should be addressed to Ernesto Staffetti; se.cjru@itteffats.otsenre

Received 7 August 2017; Accepted 7 December 2017; Published 8 April 2018

Academic Editor: Kenneth M. Sobel

Copyright © 2018 Alberto Olivares and Ernesto Staffetti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The time-optimal control problem of a spacecraft equipped with reaction wheels has been studied, in which the spacecraft is constrained to sequentially assume a set of attitudes, whose order is not specified. This attitude scheduling problem has been solved as a multiphase mixed-integer optimal control problem in which binary functions have been introduced to model the choice of the optimal sequence of target attitudes and to enforce the constraint of adopting once and only once each attitude. Given the dynamic model of the spacecraft, the initial and final attitudes, and a set of target attitudes, solving this problem consists in finding the control inputs, the sequence of attitudes with the corresponding passage times, and the resulting trajectory of the spacecraft that minimize the time of the maneuver. The multiphase mixed-integer optimal control problem has been converted into a mixed-integer nonlinear programming problem first making the unknown passage times through the target attitudes part of the state, then introducing binary variables to discretize the binary functions, and finally applying a fifth-degree Gauss-Lobatto direct collocation method to tackle the dynamic constraints. The resulting problem has been solved using a nonlinear programming-based branch-and-bound algorithm.

#### 1. Introduction

This paper focuses on time-optimal attitude scheduling of spacecraft. This problem entails planning a sequence of slew maneuvers that join a set of attitudes of the spacecraft. It is important to point out that the order of these target attitudes to be assumed by the spacecraft is not specified and must be determined. Additionally, some constraints on the angular velocity of the spacecraft at the target attitudes must be satisfied. It is assumed that the spacecraft is rigid, asymmetric, and equipped with reaction wheels, which are subject to bound constraints on their torque and angular velocity. Since the reaction wheels generate torque on the spacecraft by accelerating, the latter assumption implies that they cannot operate when they reach their maximum angular velocity. Moreover, the spacecraft is assumed to be subject to bound constraints on its angular velocities. This kind of problem arises, for instance, in observation scheduling of Earth observation satellites and space telescopes with momentum actuators [1].

The problem of minimum time reorientation of a spacecraft between two attitudes has been studied extensively. See for example [2], in which the spacecraft is assumed to be equipped with reaction wheels, and references therein. In the rest of this section, only previous works on optimal control of spacecraft among several attitudes will be reviewed.

In [3], the Pontryagin’s maximum principle has been extended to the multipoint case and applied to the problem of attitude scheduling of a satellite equipped with reaction wheels. In this work, the satellite is constrained to assume a set of attitudes whose order is specified. On the contrary, the passage times through the attitudes of the set are not specified and must be determined. A weighted combination of energy expended and elapsed time has been used as optimality criterion. In the problem formulation, bounds on the control torque have been taken into account. However, both the angular velocity of the satellite and the angular velocity of the reaction wheels are unconstrained.

In [4], the Legendre pseudospectral method has been employed to plan minimum-time slew maneuvers among several attitudes for image acquisition and minimum-time transition maneuvers for scanning operations of a space telescope equipped with reaction wheels. In both cases, the order of the target attitudes is specified. The formulation of the optimal control problem accounts for all of the relevant spacecraft actuator, sensor, and operational constraints, which include the requirement that the proper reaction wheel bias momentum be maintained at the beginning and end of each maneuver, and bounds on the reaction wheel torques, on the reaction wheel momentum, and on spacecraft body rates. The ability of this method to reliably generate flight-implementable shortest-time maneuvers has been demonstrated by means of flight implementation of the technique over multiple operational scenarios.

In [5], a new approach for designing shortest-time maneuvers of a control moment gyroscope Earth observation satellite has been proposed. In this work, the boresight axis of the satellite is required to move among several presorted collection regions and traverse through the center of each of them. A feedback scheme is employed in which, to eliminate the negative effects of unpredictable interactions between the open-loop and the feedback control laws, a Riemann Stieltjes optimal control approach with a tychastic constraint has been used. This constraint ensures that the variations in the control moment gyroscope singularity index caused by the feedback control can be appropriately managed. Experimental results obtained from tests performed on a momentum control system testbed have been reported to show the effectiveness of the method on a real attitude control system.

In this paper, the time-optimal attitude scheduling of a spacecraft is formulated as a multiphase mixed-integer optimal control problem (MIOCP) which can be stated as follows: given the dynamic model of a rigid spacecraft and a set of target attitudes, find the control inputs that steer in minimum time the spacecraft from an initial attitude to a final one adopting all the target attitudes of the set and satisfying some constraints on the angular velocity of the spacecraft at the target attitudes.

In this formulation of the time-optimal spacecraft attitude scheduling problem, the order in which the target attitudes must be assumed by the spacecraft is not specified and must be determined together with the times at which they are assumed. The fact that the order of the target attitudes is not specified characterizes this formulation with respect to previous ones and makes the resulting problem very difficult to solve.

Phases have been considered to model the rotation of the spacecraft between two target attitudes and to impose constraints at these attitudes. These constraints depend on the specific application. In general, they are constraints on the angular velocity of the spacecraft. Binary functions have been introduced to model the choice of the optimal sequence of target attitudes and to enforce the constraint of adopting once and only once each attitude.

This problem can be regarded as an extension of the classical traveling salesman problem to dynamic systems, in which the problem consists in finding the inputs of a controlled dynamic system such that the resulting state trajectory visits all states of a given finite set in minimum time.

In [6], a numerical method for MIOCPs has been developed. This algorithm is based on direct multiple shooting method, convexification and relaxation of the original problem, adaptive refinement of the underlying discretization grid, and on both deterministic and heuristic integer methods. In [7], an algorithm for mixed-integer nonlinear model-predictive control has been presented. It is a combination of the direct multiple shooting method, a reformulation based on partial outer convexification and relaxation of the integer controls, a rounding scheme, and a real-time iteration scheme.

The main contribution of this paper is the solution of a spacecraft attitude scheduling problem as a multiphase MIOCP. To the best knowledge of the authors, the present paper is the first to employ and test this technique on this problem. In this approach, the multiphase MIOCP has been converted into a mixed-integer nonlinear programming (MINLP) problem first making the unknown passage times through the target attitudes part of the state, then introducing binary variables to discretize the binary functions, and finally applying a fifth-degree Gauss-Lobatto direct collocation method [8] to tackle the dynamic constraints. High-degree collocation permits the number of variables of the problem to be reduced for a given numerical precision, thus reducing the computational cost needed to solve the resulting MINLP. An important feature of the algorithm employed is a practical MINLP solution technique that combines the power of a state-of-the-art nonlinear programming (NLP) solver with a branch-and-bound (BB) strategy. Note that, even if the order in which the target attitudes are visited is fixed, the problem remains an optimal control problem which is very difficult to solve to global optimality. Our approach is based on the assumption that state-of-the-art direct numerical methods for optimal control problems [9] are able to compute locally optimal solutions that are good approximations of the optimal trajectory. The emphasis of this paper is on the algorithm used to find the discrete part of the solution, that is, the optimal sequence of target attitudes. In the numerical results presented in this paper, a fifth-degree Gauss-Lobatto direct collocation method [8] has been used, which permits the number of the variables of the problem to be reduced for a given numerical precision and, as a consequence, the computation times to be shortened. However, any numerical method for optimal control could be employed. Based on the above assumption, a BB algorithm has been used, which aims at finding the sequence of target attitudes for which the local optimal solution found by the NLP solver is best. In particular, the MINLP solver BONMIN [10] has been used, which includes a BB algorithm based on the NLP solver IPOPT [11]. Several numerical experiments have been carried out over different instances of the problem with an increasing number of target attitudes.

The paper is organized as follows. In Section 2, the dynamic model of a rigid asymmetric spacecraft equipped with reaction wheels is described. In Section 3, the control properties of this system are reviewed and in Section 4, the spacecraft attitude scheduling problem is stated. In Section 5, the general mathematical formulation of a multiphase MIOCP is given and in Section 6, it is particularized to model the spacecraft attitude scheduling problem. In Section 7, the methodology used for its resolution is outlined and in Section 8, the results of the application of the proposed method to several instances of the spacecraft attitude scheduling problem are reported. Finally, in Section 9, the conclusions are given.

#### 2. Model of the Spacecraft

The spacecraft is assumed to be actuated by three reaction wheels. For assume that the *i*th reaction wheel is spinning about an axis , fixed with respect to the spacecraft, such that the center of mass of the *i*th wheel lies on it. Assume that a torque is supplied to the *i*th wheel about the axis 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 is a principal axes of the spacecraft and that it coincides with the principal axis of the *i*th wheel about which it is symmetric. Axes *x*, *y*, and *z* of the spacecraft reference frame are chosen to be coincident with axes , respectively.

In this paper, the attitude of the spacecraft with respect to the world frame is represented by quaternions [12]. In space applications, quaternions are in general organized as a vector 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 8, together with the quaternion representation, the roll, pitch, and yaw (RPY) representation of attitudes, with angles , will be given.

In the rest of this section, [2] will be followed. The attitude kinematic equation in terms of quaternion is where is the quaternion vector that represents the attitude of the spacecraft, is the angular velocity vector of the spacecraft, and

The dynamic model of the spacecraft can be expressed in the following form:
where is the inertia momentum matrix of the spacecraft, which, since axes , coincide with axes *x*, *y*, and *z* of the spacecraft reference frame, can be expressed as
where , , and are the principal moments of inertia of the spacecraft.

is the inertia momentum matrix of the reaction wheels

is the angular velocity vector of the reaction wheels, is the vector of torques at the reaction wheels, and is the vector of external disturbance torques about axes , respectively. The disturbance torques of are assumed to be of small amount and will be neglected. Thus, the dynamic equation of the spacecraft becomes

The dynamic equation of the reaction wheels is

The state and control variables of the problem are respectively.

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

The constraint on the angular velocity of the spacecraft can be expressed as where is the maximum angular speed of the spacecraft about each control axis.

The four components of the quaternion vector must satisfy the following condition:

Moreover, all the components of quaternion vector are bounded in the interval [−1, 1].

Equations (1), (6), and (7) can be rewritten in the general form which can be regarded as a differential constraint. The performance index for minimum-time problems is defined as , where is the initial time, which is usually known, and 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 .

For numerical reasons, it is useful to scale state and control variables of the optimal control problem as follows:

With all previous assumptions, the state equations with scaled state and control variables take the following form: which can be expressed as with

In the performance index of the minimum-time optimal control problem, without loss of generality, it can be set . The scaled state and control variables are subject to the following bound constraints: where

#### 3. Control Properties of the Spacecraft

In [13] 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.

Consider an inertially symmetric spacecraft, that is, a spacecraft whose principal moments of inertia are equal. Assume that the control torques for each axis are bounded. The so-called Euler’s principal axis or eigenaxis maneuver has often been considered as the fastest rotational maneuver, since it is the shortest angular path between two orientations. Whether the eigenaxis maneuver is optimal or not depends on the definition of the set of admissible control torques. For an inertially symmetric rigid spacecraft, it has been shown in [14] that, when the total magnitude of the control torque is constrained, the eigenaxis maneuver is indeed the time optimal maneuver. It has been shown in [15] that, when the three orthogonal components of the control torque are independently constrained, the associated optimal reorientation time is lower than in the eigenaxis maneuver case, as the nutational components can provide higher control torques about the rotation axis. An example of motion of the body frame of a spacecraft equipped with reaction wheels in a minimum-time rest-to-rest counterclockwise rotation of (rad) about the *z* axis is given in Figure 1. It can be observed from this figure, in which the traces of the *x*-, *y*-, and *z*-axes on the unit sphere are represented in yellow, blue, and purple, respectively, that the deviation from the eigenaxis rotation is large. This deviation does not appear so counterintuitive if one considers the solution to the classical brachistochrone problem in which the shortest path is not the time-optimal path.