Kinematic planning of slew manoeuvres after actuator failure for low-cost satellites

Abstract

A kinematic scheme is presented which allows for planning attitude manoeuvres of satellites after actuator failure. The scheme relies on the identification of admissible rotation axes around which the control system can deliver torque components in spite of the failure. Two techniques discussed in the literature, based on a single rotation that either minimizes the angular displacement from a given target attitude or aims a sensor exactly along a prescribed direction, are compared with a new technique based on a two-step approach, which allows for achieving any prescribed attitude by means of a sequence of two feasible rotations. Dynamic simulation is used for analysing potential capabilities and limits of the considered kinematic approaches to manoeuvre planning of under-actuated satellites.

Introduction

This paper discusses and compares three different kinematic planning techniques for attitude manoeuvres of under-actuated spacecraft, that is, spacecraft where the control system can deliver torque control components around two axes only. In general, three-axis stabilisation and control requires the capability of delivering arbitrary control torque components about three axes within the available saturation limits, so that a satellite equipped with a minimal set of three attitude effectors cannot perform arbitrary slews in case of failure of one actuator. The considered manoeuvre planning schemes can be used for providing failure-modes of operations for minimal attitude control systems where, in spite of actuator failure, attitude slews can still be performed with a sufficient level of accuracy for performing (at least part of) mission operations. Considering that all the three considered approaches hinge on relatively simple analytical solutions and, consequently, the resulting computation burden is truly modest, the kinematic manoeuvre planning technique becomes particularly interesting for small, low-cost spacecraft, where operational requirements may not be extreme, while cost and weight penalties associated with redundancy may result unacceptable.

Many satellite systems, from sensors, to communication antennas and solar panels, rely on the attitude control system in order to be oriented towards a prescribed direction within a given accuracy, so that the capability of pointing them arbitrarily in space and stabilise spacecraft attitude in the presence of external disturbances is usually mandatory for the mission (Wertz & Larson, 1999). As a consequence, attitude actuator failure still represents one of the most serious threats to spacecraft operations. At the same time requirements on pointing accuracy can be extremely different, depending on the mission itself or the particular system considered. Solar panels must lie in a plane normal to the Sun vector to maximize the energy generated by solar cells, the accuracy required being approximately 1°. Earth observation sensors have similar requirements, if sizable portions of the Earth surface are to be monitored (as for meteorological satellites), but the requirement can go down to small fractions of a degree for high resolution scanning of specific targets or large communication satellites on Geostationary orbits (GEO). Astronomy payloads can be even more demanding, requiring pointing accuracies of only fractions of an arcsecond, that is, 2 and 4 orders of magnitude more accurate than the cases considered above, respectively. Also mission phase plays a role on attitude accuracy, where stringent requirements during nominal operations may be forcedly reduced during spin-wheel desaturation or during orbit trim manoeuvres, when the thruster nozzle for orbit control must be aimed in the direction of the required $\Delta \overrightarrow{\mathit{v}}$ within approximately 0.5°.

Different classes of actuators are available in order to stabilise the spacecraft on the desired attitude and manoeuvre it when necessary (Wie, 1998, chap. 7; Wertz & Larson, 1999). Control torque can be directly applied to the spacecraft by pairs of thrusters or magnetic torque rods. Thrusters are positioned in symmetric pairs in order to deliver a control torque, with no resulting net force that would perturb the orbit trajectory of the spacecraft. A total of at least eight thrusters is necessary for full three-axis control. On the converse, magnetic torquers can deliver two torque components only, inasmuch as the resultant torque lies on a plane perpendicular to the local direction of the Earth magnetic field. Full three-axis control thus requires some other form of attitude effector. Momentum exchange devices produce torque components by exchanging angular momentum between the satellite platform and a set of spinning wheels. Momentum exchange can be obtained by either accelerating or decelerating a spinning wheel around its body-fixed spin axis (momentum and reaction wheels), or by tilting the spin axis of single or double-gimbal Control Moment Gyroscopes (CMGs). In the first case the control torque is the reaction torque produced by the wheel electric motor on the spacecraft bus around the wheel spin axis, whereas an apparent gyroscopic torque, approximately perpendicular to both the spin and the gimbal axes is produced by tilting the spin axis of a CMG.

When momentum-exchange devices are employed, three-axis control is obtained by means of a momentum wheel and two reaction wheels, with mutually perpendicular spin axes, the angular momentum bias of the momentum wheel providing also gyroscopic stabilisation. Three magnetic torque rods can be mounted together with a momentum wheel, where magnetic torques are used for roll and yaw control, while the momentum wheel provides gyroscopic stabilisation, pitch control torque and compensation for perturbations induced by magnetic torques around the pitch axis. Single gimbal CMGs are employed in clusters of 4 or more elements, in different configurations (parallel, orthogonal, skewed, pyramidal, etc.). Only one double-gimbaled momentum wheel is sufficient for three-axis control.

As stated before, failure of any single actuator prevents the possibility of performing arbitrary slews along a prescribed attitude path, if the satellite is equipped with a minimal set of actuators. Redundancy is the most widely approach for increasing mission safety with respect to control hardware failures. Thrusters require duplication of the whole system, which doubles overall actuator system weight and cost, but no control law reconfiguration is necessary, as the safe-mode in case of actuator failure simply relies upon transferring control authority to the redundant system, so that it is relatively easy to implement. Reaction wheel clusters are made of 4 or 5 wheels, where the spare wheel(s) are at an angle with respect to all the others. This means that there is no need for complete duplication of the system, but control reconfiguration is required after failure and stronger coupling between the three control axes is present. Redundancy for a double-gimbaled wheel requires duplication of the system whereas single-gimbal CMG clusters are made of N ≥ 4 momentum wheels, which makes the system intrinsically redundant, but in this latter case control system reconfiguration is necessary if one wheel fails. Moreover, as the number of CMGs is reduced, it is harder to tackle the effects of cluster singular states (Wie, 1998, chap. 7).

In all the considered scenarios, combination of fault detection algorithms, redundancy and reconfiguration make the system fault-tolerant at the expenses of serious penalties in terms of weight, cost and complexity, penalties often unacceptable for low-cost mission scenarios. As outlined by Bille, Kane, and Cox (1998), the low-cost microsat paradigm relies on the ability to correlate requirements to current and projected state of technology, where acceptance of a reasonable failure rate plays a crucial role. This is particularly true for University nano-sats, where self-built hardware elements and/or materials not qualified for space operations are often employed, with obvious reduction of system reliability. To avoid the penalties associated with redundancy, the new approaches to subsystems health management goes into the direction of (i) putting the spacecraft into a safe-mode when a failure is detected while (ii) trying to maintain the nominal attitude by using the residual controllability (Davis, Polites, & Trevino, 2005). Thus, an attitude control system which allows for performing at least part of mission operations by means of a simple reconfigurations of the attitude manoeuvre sequence is extremely interesting. Past experience clearly shows how the capability of performing these tasks autonomously before it can be serviced greatly enhances satellite survivability, while allowing for potentially significant overall mission cost reduction (Lee & Santo, 1998). For low-cost missions the possibility of handling a failure without the need for redundant elements is even more appealing: the actuator system remains minimal and, assuming that the failure can be detected, a sufficient degree of attitude dexterity may be available at the negligible cost of a control law reconfiguration, provided that the resulting manoeuvre accuracy is compatible with mission constraints.

The problem of under-actuated satellite attitude control has thus been gaining increasing interest among researchers in the last few years. The techniques considered in the literature range from relatively simple kinematic approaches for damping out the angular velocity vector by use of external control torques, such as gas jets or magnetotorquers (Aeyels and Szafranski, 1988, Sontag and Sussmann, 1989, Andriano, 1993, Morin, 1996, Tsiotras and Schleicher, 2000) to more complex scenarios, where stabilisation of a prescribed attitude is pursued (Byrnes and Isidori, 1991, Krishnan and McClamroch, 1994, Tsiotras et al., 1995, Coron and Kerai, 1996, Morin and Samson, 1997, Spindler, 2000, Tsiotras and Luo, 2000). When spacecraft attitude control is obtained by use of internal torques generated by momentum-exchange devices, gyroscopic coupling makes the issue of under-actated attitude control even more difficult (Tsiotras and Doumtchenko, 2000, Krishnan et al., 1995, Kim and Kim, 2001, Vadali and Junkins, 1983, Hall, 1996, Bang et al., 2002).

The approaches considered in this paper for planning feasible manoeuvres in under-actuated conditions rely on the definition of a suitable (sequence of) rotation(s) about admissible rotation axes driving the spacecraft onto a final attitude characterized by interesting properties with respect to the desired target one. Admissible rotation axes are perpendicular to the direction of the failed control axis, so that the resulting attitude manoeuvre can be controlled by means of the available actuators about two axes only. This means that the manoeuvre is planned in such a way that no control torque requirement should result along the direction of the failed actuator. Three types of feasible manoeuvres will be considered: two techniques based on a single eigenaxis rotation which result in (a) minimum final alignment error with respect to the nominal target attitude (Giulietti & Tortora, 2007) or (b) exact pointing of a single body-axis along an arbitrary direction in space (Avanzini & Giulietti, 2008) are compared with a new approach that (c) takes the spacecraft exactly onto any desired final attitude by means of a sequence of two admissible rotations.

The approach proposed by Giulietti and Tortora (2007) is here revisited by using the concept of quaternion error vector, which greatly simplifies the derivation of the relevant equations and lends itself to the extension of the approach to an arbitrary number N of feasible rotations. Case (c) thus becomes the generalisation of Case (a) for N = 2. As a further contribution, the paper also discusses for the first time practical issues for the implementation of the planned feasible manoeuvre by means of a sequence of thruster pulses, demonstrating potential application and limits by means of direct numerical simulation.

In what follows, after a paragraph devoted to a brief review on attitude representation by means of quaternions, the three manoeuvre planning schemes based on purely kinematic considerations are presented in Section 3. The accuracy achieved for different nominal manoeuvres is then discussed in the following section of Results, where selected manoeuvres are simulated in an open-loop control framework. A paragraph of Conclusions ends the paper.