Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Satellite Control

  • Finn AnkersenEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_21-1

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 signal-based two degree-of-freedom control architectures. The equations are given for the relative motion dynamics between spacecraft on elliptical orbits with the generic Yamanaka-Ankersen 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 spacecraft 

Introduction

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 state-of-the-art control techniques and architectures are addressed.

Spacecraft are classically well-known 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 model-based 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 milli-arc 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 Degree-Of-Freedom (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

Fundamentally the three attitude angles \(\boldsymbol{\theta }\) and angular rates \(\boldsymbol{\omega }\) need to be controlled to a certain reference. See Fig. 1 for definition.
Fig. 1

Spacecraft body (black) and reference (red) frames. The frames coincide for \(\boldsymbol{\theta }= \mathbf{0}\)

The general rigid body dynamics expressed in a rotating frame( ∗ ), which is mostly the case when orbiting a central body, can be expressed as
$$\mathbf{N} = \frac{d^{{\ast}}(\mathbf{I}\boldsymbol{\omega }^{{\ast}})} {dt} +\boldsymbol{\omega } \times \mathbf{I}\boldsymbol{\omega }^{{\ast}}$$
(1)
where I is the constant inertia matrix, \(\boldsymbol{\omega }\) is the inertial angular velocity, and N is the torque acting on the spacecraft (Wie 1998).

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.

Most modern spacecraft have large flexible appendices in the form of solar panels and large antennae reflectors. Fuel sloshing is a similar lightly damped oscillatory phenomena, which often needs to be taken into consideration. The incorporation of dynamic elements such as flexible panels, antennae, and sloshing fuel can be modeled by Eqs. (2) and (3) provided the overall rotation rate \(\boldsymbol{\omega }\) and linear accelerations \(\ddot{\mathbf{x}}\) are not too large.
$$\mathbf{M}_{T}\left [\begin{array}{c} \ddot{\mathbf{x}}\\ \dot{\boldsymbol{\omega }} \end{array} \right ] = \left [\begin{array}{c} \mathbf{F}\\ \mathbf{N}\end{array} \right ]-\mathbf{L}\ddot{\boldsymbol{\eta }}$$
(2)
$$\ddot{\eta }_{k}+2\zeta _{k}\Omega _{k}\dot{\eta }_{k}+\Omega _{k}^{2}\eta _{ k} = - \frac{1} {m_{k}}\mathbf{L}^{\text{T}}\left [\begin{array}{c} \ddot{\mathbf{x}}\\ \dot{\boldsymbol{\omega }} \end{array} \right ]$$
(3)
where
For attitude only the second row of Eq. (2) is needed, but translation is included here for the sake of completeness and later use.

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 high-precision 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 off-loading. Thrusters are also used to control the attitude directly on many spacecraft. They are mostly of on-off type, though continuous ones exist, and will need to be Pulse Width Modulated (PWM) to obtain a quasi-linear behavior. The nonlinear on-off 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 optimization-based selection algorithm is applied to the controller output.

Before using the plant model in Eq. (2) for a flexible spacecraft, a simpler multivariable model of a rigid spacecraft is used as in Eq. (4):
$$\dot{\mathbf{x}} = \left [\begin{array}{cc} \mathbf{0}&\mathbf{B}_{k} \\ \mathbf{0}&\mathbf{A}_{d} \end{array} \right ]\mathbf{x}+\left [\begin{array}{c} \mathbf{0}\\ \mathbf{B} _{d} \end{array} \right ]\mathbf{N}$$
(4)
where \(\mathbf{x} = [\theta _{x},\theta _{y},\theta _{z},\omega _{x},\omega _{y},\omega _{z}]^{\text{T}}\), B k is identity, \(\mathbf{B}_{d} = \mathbf{I}^{-1}\), and A d is the general Jacobian for the dynamics having a real Right Half-Plane (RHP) pole. See Ankersen (2011). The model describes the angular deviation from some reference frame, whose orientation can be arbitrary. It uses the Euler(3, 2, 1) rotation in the kinematics.
The state of the art of attitude control is today mostly based on \(\mathcal{H}_{\infty }\) type of robust controllers with synthesis performed in the frequency domain. Requirements are often specified in the time domain, but formal methods exist to transform them into frequency domain weighting functions (ESA Handbook 2011) enhancing both synthesis and analysis. System uncertainties can be formulated as structured Linear Fractional Transformations (LFT) with a general control configuration as illustrated in Fig. 2.
Fig. 2

Robust control formulation, where \(\boldsymbol{\Delta }\) is the structured uncertainty, K is the controller, P the partitioned formulation of the plant with weights, and w and z are exogenous inputs and outputs, respectively

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 optimization-based 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 μ-synthesis 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 general relative position dynamics between Centers Of Mass (COMs) is in Eq. (5), where it is observed that the in-plane motion (x, z) is decoupled from the out-of-plane motion (y).
$$\displaystyle\begin{array}{rcl} \ddot{x} -\omega ^{2}x - 2\omega \dot{z} -\dot{\omega } z + k\omega ^{\frac{3} {2} }x& =& \frac{1} {m_{c}}F_{x} \\ \ddot{y} + k\omega ^{\frac{3} {2} }y& =& \frac{1} {m_{c}}F_{y} \\ \ddot{z} -\omega ^{2}z + 2\omega \dot{x} +\dot{\omega } x - 2k\omega ^{\frac{3} {2} }z& =& \frac{1} {m_{c}}F_{z}\end{array}$$
(5)
where ω =  ω(t) is the orbital angular rate, m c is the chaser mass, F xyz is the force on the chaser, and k is a constant determined by the orbit and is valid for any Keplerian orbit with eccentricity \(\epsilon < 1\).
Fig. 3

Definition of COM-to-COM and port-to-port positions, s and s pp , respectively, between two spacecraft

The Yamanaka-Ankersen equations (Yamanaka and Ankersen 2002) provide the generalized homogeneous solution in the form of the transition matrix \(\boldsymbol{\Phi }\), where the solution can be written as
$$\mathbf{x}(t) =\boldsymbol{ \Lambda }^{-1}(\nu )\boldsymbol{\Phi }(\nu )\boldsymbol{\Phi }_{ 0}^{-1}(\nu _{ 0})\boldsymbol{\Lambda }(\nu _{0})\mathbf{x}(t_{0})$$
(6)
where ν is the orbital true anomaly and \(\Lambda \) are transformation matrices to and from the time domain. The elements of \(\boldsymbol{\Phi }\) in Eq. (6) are detailed in (Ankersen 2011), where relevant particular solutions are also to be found. Equation (6) reduces to the well-known Clohessy-Wiltshire equations for circular orbits (\(\epsilon = 0\)) (Clohessy and Wiltshire 1960). Equation (6) is used for feedforward control and trajectory propagation in the guidance function. During the final approach (see Fig. 3), a model accounting for the docking port-to-port relative position and the couplings from the relative attitude to the position is utilized and formulated in Eqs. (7) and (8) (Ankersen 2011):
$$\mathbf{\dot{x}} = \left [\begin{array}{cc} \mathbf{A}_{p}& \mathbf{0} \\ \mathbf{0} &\mathbf{A}_{c} \end{array} \right ]\mathbf{x}+\left [\begin{array}{cc} \mathbf{B}_{p}& \mathbf{0} \\ \mathbf{0} &\mathbf{B}_{c} \end{array} \right ]\mathbf{u}$$
(7)
$$\mathbf{y} = \left [\begin{array}{cccc} \mathbf{I} &\mathbf{0}&\mathbf{B}_{dc_{1}} & \mathbf{0} \\ \mathbf{0}& \mathbf{I} & \mathbf{0} &\mathbf{B}_{dc_{2}} \\ \mathbf{0}&\mathbf{0}& \mathbf{I} & \mathbf{0}\\ \mathbf{0} &\mathbf{0} & \mathbf{0} & \mathbf{I} \end{array} \right ]\mathbf{x}$$
(8)
where \(\mathbf{x} = [\mathbf{x}_{p},\dot{\mathbf{x}}_{p},\boldsymbol{\theta }_{c},\) \(\boldsymbol{\omega }_{c}]^{\text{T}}\), \(\mathbf{y} = [\mathbf{x}_{pp},\dot{\mathbf{x}}_{pp},\boldsymbol{\theta }_{c},\) \(\boldsymbol{\omega }_{c}]^{\text{T}}\), index p refers to COM positions, index c to chaser attitude, index pp to port-to-port position, and \(\mathbf{B}_{dc_{1}},\mathbf{B}_{dc_{2}}\) are the coupling matrices of the docking port.
A relative motion scenario for a typical RVD mission looks like in Fig. 4. During the final approach ( < 300 m range), the chaser relative attitude and relative position are controlled. During the other phases, the chaser attitude is Earth pointing and the relative position is controlled at the Station-Keeping (SK) points, s 0, ⋯ , s 4 in Fig. 4. The trajectories are typically open loop feedforward controlled (often with midcourse corrections).
Fig. 4

This figure shows the phases of typical relative motion approach. The shaded area is a keep-out zone (KOZ) defined for safety reasons. V-bar is the x-axis and R-bar is the z-axis

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 Line-Of-Sight (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.

The controllers are frequently of multivariable \(\mathcal{H}_{\infty }\) type. They are similar to what is described in connection with Fig. 2. Flexible modes and in particular sloshing need to be taken into account using Eq. (2). Sloshing pendulum models are used during boost maneuvers and spring mass damper models during other modes. The couplings between relative attitude and relative position in Eq. (8) can be analytically decoupled setting the matrix \(\mathbf{C}\) to identity and premultiplying with a decoupling matrix V d , such that
$$\mathbf{V}_{d}\mathbf{C} = \mathbf{I} \Leftrightarrow \mathbf{V}_{d} = \mathbf{C}^{-1}$$
(9)
and by the inversion theorem for partitioned matrices the upper right partition just changes sign. The designed controller then needs to be premultiplied by \(\mathbf{V}_{d}^{-1}\), which facilitates a simpler control design maintaining the 6DOF performance after 2 times 3DOF synthesis.
A 2 degree-of-freedom control architecture as in Fig. 5 is beneficial since much of the performance is achieved by controller K 1. The structure of the synthesis formulation is a signal-based model-reference configuration for the \(\mathcal{H}_{\infty }\) control rather than the more classical mixed sensitivity type. It has proven to have higher robustness and performance for this type of applications. As an example, consider a controller that has to follow a sawtooth motion of the docking port of the International Space Station (ISS) with an amplitude of 0. 4 m and reversal times of 8 s. The signal-based model-reference controller manages to track such a motion with errors less than 0. 01 m compared to the best operational performance of 0. 08 m.
Fig. 5

Principal structure of the 2 degree-of-freedom controller

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 high-precision ( < 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 fault-tolerant GNC designs. Model Predictive Control (MPC) will enter in particular on the guidance functions. More integrated GNC system-level designs, of multidisciplinary nature, are expected.

Cross-References

Bibliography

  1. Alfriend K, Vadali S, Gurfil P, How J, Breger L (2010) Spacecraft formation flying. Elsevier, Amsterdam/Boston/LondonGoogle Scholar
  2. Ankersen F (2011) Guidance, navigation, control and relative dynamics for spacecraft proximity maneuvers. Aalborg University, Denmark. ISBN:978-87-92328-72-4Google Scholar
  3. Bryson A (1999) Control of spacecraft and aircraft. Princeton University Press, PrincetonGoogle Scholar
  4. Clohessy W, Wiltshire R (1960) Terminal guidance system for satellite rendezvous. J Aerosp Sci 27(9):653–658CrossRefzbMATHGoogle Scholar
  5. Cornford S (2012) Evaluating a fractionated spacecraft system: a business case tool for DARPA’s F6 program. In: Aerospace conference, Big Sky. IEEE, Big Sky, MT, pp 1–20Google Scholar
  6. D’Errico M (2012) Distributed space missions for earth system monitoring. Springer, New YorkGoogle Scholar
  7. ESA Handbook (2011) ESA pointing error engineering handbook. European Space Agency. http://peet.estec.esa.int
  8. Falcoz A, Watt M, Yu M, Kron A, Menon P, Bates D, Ankersen F, Massotti L (2013) Integrated control and structure design framework for spacecraft applied to BIOMASS satellite. In: 19th IFAC conference on automatic control in aerospace, Würzburg, Germany, 2–6 Sept 2013Google Scholar
  9. Fehse W (2003) Automated rendezvous and docking of spacecraft. Cambridge University Press, Cambridge/New YorkCrossRefGoogle Scholar
  10. Hughes P (1986) Spacecraft attitude dynamics. Wiley, New YorkGoogle Scholar
  11. Kaplan M (1976) Modern spacecraft dynamics & control. Wiley, New YorkGoogle Scholar
  12. Ogata K (1970) Modern control engineering. Prentice-Hall, Englewood CliffsGoogle Scholar
  13. Sidi M (2000) Spacecraft dynamics and control: a practical engineering approach. Cambridge University Press, CambridgeGoogle Scholar
  14. Skogestad S, Postlethwaite I (1996) Multivariable feedback control. Wiley, Chichester/New YorkGoogle Scholar
  15. Wertz J (1980) Spacecraft attitude determination and control. Kluwer, DordrechtGoogle Scholar
  16. Wie B (1998) Space vehicle dynamics and control. American Institute of Aeronautics and Astronautics, RestonGoogle Scholar
  17. Yamanaka K, Ankersen F (2002) New state transfer matrix for relative motion on an arbitrary elliptical orbit. J Guid Control Dyn 25(1):60–66CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.European Space AgencyNoordwijkThe Netherlands