Satellite Control
Abstract
Spacecraft control systems are described for single and distributed space systems. The attitude dynamics is formulated including flexible and sloshing phenomena, followed by a description of attitude sensors and actuators. \(\mathcal{H}_{\infty }\) and robust controls are formulated as signalbased two degreeoffreedom control architectures. The equations are given for the relative motion dynamics between spacecraft on elliptical orbits with the generic YamanakaAnkersen state transition matrix. Formulations are provided for rendezvous and docking scenarios and formation flying control, maneuvers, avionics, and laser metrology systems together with the onboard autonomy needs.
Keywords and Phrases
Spacecraft attitude control Spacecraft position control Rendezvous and docking Formation flying Robust control \(\mathcal{H}_{\infty }\) control Flexible modes Sloshing Multivariable systems Relative dynamics Fractionated spacecraftIntroduction
This entry explains the control needs of spacecraft after they have been separated from the launch vehicle and injected onto their initial orbit.
Actuators and sensors are explained followed by the control objectives. The stateoftheart control techniques and architectures are addressed.
Spacecraft are classically wellknown physical systems that can be described by first principles. The advantage is fairly precise plant models and uncertainty characterization of physical parameters. This is well suited for a modelbased control design approach.
Mission Types
From a control point of view, space missions can be split into two main categories according to which physical states need to be controlled:
 Attitude Control:

This is needed by any spacecraft irrespective of the mission objectives. Such missions are typically Low Earth Orbit (LEO) missions for astronomy, observations, and, in higher orbits, constellations for navigation and communication. Further, there are interplanetary and planetary exploration science missions. The pointing requirements vary from a few degrees to milliarc seconds.
 Relative Position Control:

Within distributed space systems, this is relevant for RendezVous and Docking (RVD) and Formation Flying (FF) missions. It leads to a 6 DegreeOfFreedom (DOF) control problem as the relative attitude is also needed. The former is mostly for missions to space station logistics infrastructures and the latter for scientific missions. Relative position can also be required during the final stages of controlled planetary landings. Another category is missions with ultrahigh control performance requirements, where the spacecraft platform and the science instrument need to be considered as one coupled system.
Attitude Control
The kinematics can be described by one of the 12 sets of Euler angles (can have singularities) or the hypercomplex quaternion vector (no singularities) (Hughes 1986).
The dynamics and kinematics equations need to be linearized and are in the general form of a coupled 12th order system. It is the fundamental model for the rigid body spacecraft control design.
M _{ T }  :  rigid body mass/inertia matrix  
\(\ddot{\mathbf{x}},\dot{\boldsymbol{\omega }}\)  :  linear and angular acceleration  
F, N  :  forces and torques on the spacecraft  
η _{ k }  :  the kth flexible state  
ζ _{ k }  :  the kth flexible damping factor  
\(\Omega _{k}\)  :  the kth flexible eigen frequency  
m _{ k }  :  the kth modal mass (normalized to 1)  
L  :  participation matrix of the kth mode 
The sensors utilized are typically gyroscopes for measuring the inertial angular rate, sun sensors to measure orientation at low accuracy, and star trackers for highprecision angular attitude measurements. All of those sensors are linear in their normal operational range and it suffices to use bias noise models for synthesis. Gyros do need a drift estimation and compensation to function properly over longer time. All sensors utilize redundancy for providing measurements around all three axes as well as providing fault tolerance. Some scientific observatory spacecraft use their telescopes for attitude measurements in order to obtain the required precision beyond the capability of star trackers.
The actuators producing pure torques are magnetic torquers, reaction wheels, and control momentum gyros. The last can produce large torques used for rapid slew maneuvers with little power. The last two types have nonlinear issues around low to zero speed due to friction issues. They accumulate angular momentum from asymmetric disturbances. This leads to a need for thrusters for angular momentum offloading. Thrusters are also used to control the attitude directly on many spacecraft. They are mostly of onoff type, though continuous ones exist, and will need to be Pulse Width Modulated (PWM) to obtain a quasilinear behavior. The nonlinear onoff nature needs to be taken into account for the control closed loop analysis. It is done by use of the negative inverse describing function (Ogata 1970) for stability analysis and nonlinear modeling for verification simulations in the time domain. For larger numbers of thrusters, an optimizationbased selection algorithm is applied to the controller output.
Commonly the \(\mathcal{H}_{\infty }\) controller K is designed, and the lower loop in Fig. 2 is closed via a lower LFT such that N = F _{ l }(P, K) and Robust Stability (RS) and Robust Performance (RP) analysis is performed on the \(\mathbf{N},\boldsymbol{\Delta }\) system (Skogestad and Postlethwaite 1996).
On high performance pointing spacecraft, active vibration suppression of, e.g., cryocoolers is needed. The implementation of control design and recursive system identification can achieve significantly better attenuation compared to classical passive isolation techniques.
Lately optimizationbased codesign of structures and control has been performed successfully. A joint performance function is formulated (mass, stiffness, pointing, fuel, etc.) and an optimization is performed (differential evolution algorithm) iterating on control design and Finite Element Models (FEM). A μsynthesis controller is synthesized, the pointing performance is fulfilled, and 15–20 % mass saving is obtained on the flexible structures. The entire process is fully automated (Falcoz et al. 2013).
Relative Position Control
For all distributed space systems, relative dynamics is important. Rendezvous and formation flying missions need tracking or maintenance of the desired relative separation, orientation, and position between or among the spacecraft. This is common and independent of the mission type and will be described in general terms ahead of the specific RVD and FF missions.
The avionics sensors for the attitude control part are generally similar to those described earlier under attitude control in connection with Fig. 1. Active laser CCD type of sensors is used to measure the relative position (range and LineOfSight (LOS) angles) and at short range ( < 50 m) the relative attitude. They require a target pattern to provide precise measurements at short range. Accelerometers are used, particularly for pulsed maneuvers. The next generation of RVD GNC systems, test flown, will utilize Lidar, infrared cameras, and visual cameras in combination with advanced image processing providing RVD capabilities with both cooperative and passive target spacecraft.
The actuators are mostly thrusters arranged to achieve controllability for all the 6DOF maneuvers needed. Based upon the controller output, the active thrusters are selected by means of some type of fuel optimization algorithm. The selected thrusters are then Pulse Width Modulated (PWM) within the sampling time.
Formation flying usually includes more than two spacecraft with the need to be controlled relative to each other. The objective of FF is to form an instrument in space, not possible with fixed structures, like a synthetic aperture or an interferometer of large size.
The performance needs are high and require innovative highprecision ( < 1 \(\upmu\)m) metrology sensors. They are based on divergent laser beams for the coarse part to be able to transit from lower to higher accuracy. The fine metrology uses a laser beam and internal interferometers to reach the \(\upmu\)m domain. Actuators are in the range of \(\upmu\)N thrust, which can be achieved with either cold gas or electrical propulsion thrusters.
The maneuvers realized by entire formations are rotation, resizing, and slew while maintaining the formation in most cases (Alfriend et al. 2010).
Formation flying missions with the highest performance requirements have optical payloads, which need to have internal control loops at component level. To reach the performance required for applications such as optical interferometry, the formation and payload must be considered as one system. The synthesis of a multivariable controller then handles all the cross couplings in the system needed to reach performance. Beyond flexible modes, such systems might also have a need for active vibration damping for systems using cryocoolers.
The GNC architecture is often centralized for nominal science operational modes. For the formation deployment and contingency situations, a decentralized control architecture is needed. This leads to a dual architecture GNC system in general for formation flying systems. The onboard autonomy needs to be fairly high in order to cope with the contingencies in the formation without ground intervention.
Finally there is an emerging concept of fractionated spacecraft. There, a formation consists of a large number of small simple vehicles maneuvering relative to each other fully autonomously based upon the nearest neighbor knowledge and not necessarily information about the entire formation (Cornford 2012).
Summary and Future Directions
The control of spacecraft has been described for pure attitude control needs and for spacecraft performing relative proximity maneuvers like rendezvous and formation flying. The focus has been on sensors, actuators, dynamics, and the robust control methods applied today.
The further development direction of the field is expected to be increased on board autonomy with replanning capabilities and faulttolerant GNC designs. Model Predictive Control (MPC) will enter in particular on the guidance functions. More integrated GNC systemlevel designs, of multidisciplinary nature, are expected.
CrossReferences
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