Design of Sliding Mode Attitude Control for Communication Spacecraft

Control problem of a spacecraft is an important topic in automatic control engineering. A body orbiting the Earth in geosynchronous orbit has instabilities in attitude dynamics and disturbances caused by the Earth, the Moon, the Sun and other bodies in space. These effects force the body to lose initial orbit and attitude. Here the control system takes important part of spacecraft missions where it keeps the body in designed orbit and desired attitude. The control system consists of control elements and control algorithms which are developed for the mission by a control engineer [1]. The commonly used control elements for a spacecraft in geosynchronous orbit are thrusters, reaction or momentum wheels, etc.


Introduction
Control problem of a spacecraft is an important topic in automatic control engineering.A body orbiting the Earth in geosynchronous orbit has instabilities in attitude dynamics and disturbances caused by the Earth, the Moon, the Sun and other bodies in space.These effects force the body to lose initial orbit and attitude.Here the control system takes important part of spacecraft missions where it keeps the body in designed orbit and desired attitude.The control system consists of control elements and control algorithms which are developed for the mission by a control engineer [1].The commonly used control elements for a spacecraft in geosynchronous orbit are thrusters, reaction or momentum wheels, etc.
The sliding mode theory has an attention in the aerospace field.The technique permits the use of a lower order system model for generating control commands.On the other hand, the system is robust to the external disturbances and includes unmodelled dynamics, as well.The theory and methods of sliding mode control design principles are investigated [1]- [3], etc. Variable structure systems with nonlinear control techniques and dead-band on switching function for sliding mode controllers are introduced [4].
A variable structure control design for rigid body spacecraft attitude dynamics with quaternion representation for optimal sliding mode control which consists of three parts: equivalent control, sliding variable, and relay control where simulation results illustrate that the motion along the sliding mode is insensitive to parameter variations and unmodeled effects is given [5].An automatic controller for active nutation damping in momentum biased stabilized spacecraft is introduced [6], where robust feedback stabilization of roll and yaw angular dynamics are achieved with prescribed qualitative characteristics for a spinning satellite.A smooth sliding mode control which requires well-estimated initial condition for quaternion based spacecraft attitude tracking maneuver is studied [7] where the chattering is eliminated by replacing saturation instead of signum function.A class of uncertain nonlinear systems decoupled by state variable feedback with sliding mode approach for attitude control of an orbiting spacecraft is considered [8] where simulation results show that precise attitude control is accomplished in spite of the uncertainty in the system.As seen from simulations spacecraft is stabilized approximately in 10 seconds.However there is a chattering in control action and thrusters are operating after stabilization of the spacecraft attitude dynamics.A reference book for various spacecraft attitude and orbit dynamics, orbit transfer methods, and different control strategies such as PID and robust control, pulse modulation of thruster control etc. is covered [9].An attitude control with reaction wheels is evaluated in [10].
There are many papers concerning control of flexible spacecraft.A maneuvering of a flexible spinning spacecraft is treated with variable structure control [11] where system is stabilized between 60-100 seconds for small and large angle maneuvers.An application with one sided dead-band for robust closed-loop control design for a flexible spacecraft slew maneuver using on-off thrusters is studied [12] where analytical simulations and experimental results demonstrate that the proposed switching function provides significant improvement in slew maneuver performance.The size of single-sided dead-band in switching function provides the capability of a tradeoff between maneuver time and fuel expenditure.Rotational maneuver and vibration suppression of an elastic spacecraft is considered [13] where pitch angle trajectories are asymptotically tracked by an adaptive controller.Variable structure control and active suspension of flexible spacecraft during attitude maneuver is studied [14] where positive position feedback technique is used to suspend vibration and variable sliding mode with pulse-width pulse-frequency modulation to eliminate chattering.An adaptive variable structure control of spacecraft dynamics with command input shaping which eliminate residual vibration is studied [15] where PD, conventional and adaptive variable structure output feedback controllers with and without input shaping are compared and simulated.Vibration suspension of flexible spacecraft during attitude maneuvers is considered [16] where PD controller with pulse-with pulse-frequency modulation with positive position feedback is considered for vibration reduction during on-off operation of thrusters.However, as seen from simulation results chattering occurs in control action.A comparison between linear and sliding mode controllers with reaction wheels is studied [17] where only small angle orientations are considered.Designed sliding mode controller stabilizes spacecraft attitude dynamics 30 times faster than output feedback controller with reaction wheels.Station keeping chattering free sliding mode controller design is designed in [18].
A body orbiting the Earth in geosynchronous orbit has instabilities in attitude dynamics and disturbances caused by the Earth, the Moon, the Sun and other bodies in space.These effects force the body to lose initial orbit and attitude.Here the control system takes important part in spacecraft missions where it keeps the body in designed orbit and desired attitude.The commonly used control elements for a spacecraft in geosynchronous orbit are thrusters, reaction, and momentum wheels.Dynamic model of a spacecraft is nonlinear, includes the rigid and flexible mode interaction, and the parameters of the spacecraft are not precisely known.The performance criteria for a spacecraft are fuel expenditure and vibration of flexible structures.The sliding mode technique permits usage of lower order system model for generating control commands, which includes unmodeled dynamics or uncertainties, and stabilizes the plant faster and robustly under bounded disturbance.The chattering at high frequencies is not desired because it may cause vibration.Chattering may be eliminated by replacing saturation instead of signum function.However, in that case nonzero tracking errors exist, which can be made small by taking a tiny region for saturation and also, saturation is limited with hardware capability and reduction of accuracy and robustness as introduced [7] and [8].On the other hand, chattering may be eliminated by pulse modulation as done [14].
The chapter is organized as follows.Section 1 gives an introduction to sliding mode control of a satellite.Section 2 gives the system description of rigid body nonlinear attitude dynamics.Section 3 evaluate rigid body in circular orbit with internal torquers and introduces equation of motion for a flexible spacecraft.Section 4 includes design of variable structure control systems for nonlinear attitude dynamics of a spacecraft and suspension of vibration of flexible solar arrays.Also design examples and performances comparison are studied.Section 5 concludes the chapter.

Rigid body dynamics
Consider a rigid-body with as body-fixed reference frame B with its origin at the center of mass of the rigid body as shown in Figure 1.C R  is the position vector of the center of mass from an inertial origin of N, and R  is the position vector of dm from an inertial origin N.    be the angular velocity vector of the rigid body in an inertial reference frame N. The angular momentum vector H  of rigid body about its center of mass can be defined as [9]: The position vector   of very small mass element dm from the center of mass is defined as and finally angular velocity vector / BN     of a rigid body in an inertial reference frame N can be writen as: The angular momentum vector (1) can be rewritten as: and the matrix form of ( 5) is: The rotational equation of motion of a rigid body in an inertial reference frame N about center of mass is [9]: Form ( 1) and ( 7) we can write a relation between external moment acting on the body about its mass center and angular momentum as [9]- [11]: The relation above with an angular momentum H  and external moment M  is the rotational equation of motion of a rigid body in a circular orbit [9]- [10]: Substituting (1) into rotational equation of motion (9) we obtain: which is known as Euler's rotational equation of motion.

Rigid body in circular orbit
Consider a rigid body orbiting the Earth with a constant radius.A local horizontal and local vertical reference frame A at the center of the mass of an orbiting spacecraft with unit vectors 123 ,, aaa  as given in Figure 2. 1 a  is along the orbital direction, 2 a  is perpendicular to the orbital plane and 3 a  is always pointing the Earth.The angular velocity of A with where n is the orbital rate defined as www.intechopen.com The angular velocity of the body-fixed reference frame B with basis vectors 123 ,, bbb   is given as [9]: where sin The gravity gradient torque can be written as [9]: The basis vector on axis 3 of local horizontal and local vertical reference frame A is . The rotational equation of motion of a rigid body with an angular momentum in a circular orbit can be obtained by substituting   and 3 a  can be expressed in terms of basis vectors of the body-fixed reference frame B as: 31 3 12 3 23 3 3 The equation of motion with control torque and disturbance can be written as [5], [7], [9], [17]: where B u is the body (3×1)-control vector and () dt is an external disturbance (solar radiation, interaction with other bodies in space, etc.).Let us define gravity gradient torque (16) where C is the third column of direction cosine (3×3)-dimensional matrix given in ( 15), ()   and () C  are (3×3)-dimensional skew symmetric matrices defined as: Relation between angular velocity  and attitude angles and their rates for circular orbit can be written as [9]: or vice versa of (25): Then the nonlinear equation of motion of three axes stabilized rigid body spacecraft (20) reduces to: where

cc ss nc s s c s s c s s c s c s ccs ccs sss ss cc
Note that, (27) will be used in simulation of attitude dynamics.

Rigid spacecraft with internal torquers in circular orbit
Here we have considered a three axis stabilized communication satellite with a bias momentum wheel.Some parameters of considered communication satellite are given in Table 1.Internal control torquers for the satellite are reaction wheels mounted on roll and yaw axes, and a momentum wheel is set up on pitch axis which spins along negative direction, see Figure 3.The total angular momentum of spacecraft can be written as [9]: where 1 H , 2 H , 3 H were defined in (5).We can obtain equation of motion for principle axis frame B from rotational equation of motion (9) with considered gravity gradient torque (21), external disturbances and internal torquers as: 0 () ( ) Or in term of attitude angles

Communication satellite: rigid body with flexible solar arrays
During on-orbit normal mode operations, both solar arrays always point towards the sun, whereas the main body points towards the Earth.This results in a very slow change of modal frequencies and modal shapes.For control design purposes, however, the spacecraft model will be treated as a time-invariant but nonlinear system with a known range of modal characteristics.The equation of attitude motion of the three axis stabilized with flexible solar array is given [9]- [16].Extending the rigid body equation ( 27) with the flexible solar arrays, following set of equations can be obtained for main body: or in terms of  2 23 ( ) and two solar arrays [9] with control force: Note that, two solar arrays point towards the Sun.Hence, the solar driving mechanism actuated by Sun Sensors causes the control torque to point solar arrays towards the Sun.In our case we have considered an station keeping controller that can activate the solar array driver mechanism to suspend vibration of flexible solar arrays caused by attitude angle acceleration during maneuvering.The control law will be designed in Section 4.2.

Sliding mode control
The sliding mode technique permits usage of lower order system model for generating control commands, which includes unmodeled dynamics or uncertainties, and stabilizes the plant faster and robustly under bounded disturbance.The chattering at high frequencies is not desired because it may cause vibration.Chattering may be eliminated by replacing saturation instead of signum function.However, in that case non-zero tracking errors exist, which can be made small by taking a tiny region for saturation and also, saturation is limited with hardware capability and reduction of accuracy and robustness as introduced [7] and [8].On the other hand, chattering may be eliminated by pulse modulation as done [16].In this section we suggest variable structure attitude and station keeping control system design for a communication satellite.

Sliding mode attitude control system design of a rigid spacecraft
Let s represent a sliding manifold as where k is a constant to be selected.The attitude dynamics dominated on the sliding manifold 0 s  can be written from [5], [17] Taking the norm of (40) we have Some matrix and vector norms in (41) satisfy inequalities as below: Therefore, substituting norm values (42)-( 44) into (41) we obtain: Hence, the sliding mode controller forces the system trajectories toward the sliding manifold asymptotically for 20 0 NdM  and 10 NL s   .

Example
Sliding mode control system design is performed firstly with determining proper switching function where the system trajectory is caused to follow the sliding manifold, 0 s  .Then proposed discontinuous control term is employed to model reaction wheels.Physical sliding surface consists of inputs form the Earth sensor for yaw and pitch attitude angles, from the star tracker for yaw, roll and pitch attitude angles, and from the rate gyro for attitude angle rates.Considered control function (38) and sliding manifold (36) stabilizes the dynamic equation of attitude angles presented via (31) for small attitude angles errors 1 0.3deg    .The disturbance is assumed to be as: () The control torque is produced by reaction wheels which have approximately 6000 rpm angular velocity and can produce 1.5 Nm control torque.Therefore, control inputs can be developed for each pitch, roll and yaw axes by sliding mode control approach.Thus attitude sliding mode controller can be obtained for 20 0 NdM  and 10 NL s where J is the inertia matrix of reaction wheels.
As seen from Figure 4, spacecraft attitude errors are eliminated in 120 seconds by sliding mode control approach where some chattering appears in control action.Since control function is applied to electro-motor of reaction wheel, the electro-motors are driven at nominal speeds on chattering phenomena for which sliding motion is conventional [2].In general, chattering effect can be eliminated by using a element.

Discontinuous station keeping sliding mode control design of flexible spacecraft
Station keeping sliding mode control algorithm for a three axes stabilized geosynchronous communication satellite is considered in this subsection.The spacecraft is assumed to be controlled externally with small attitude thrust jets.Classical sliding mode technique with chattering free correction and elimination of operation for small attitude angles via deadband function will be used to model thrust jets variable on-off operation for stabilization of the spacecraft.The performances, modeling and simulation are discussed on a design example by using MATLAB-Simulink programming.The attitude sliding mode controller for geosynchronous satellite with flexible solar arrays will use fuel optimally and adequately as little as possible with proposed control algorithm.Also, attitude sliding mode controller is robust to bounded external disturbances and includes unmodelled dynamics as well.
Two types of simple and easy-to-apply variable structure P+relay controllers different from existing (for example from [5] which includes the linear equivalent and sliding terms plus relay) are proposed for the stabilization of full nonlinear attitude dynamics and vibration control of flexible solar arrays during station keeping maneuvering.Variable structure P+relay control law has only two design parameters.A modified sliding function with a dead-band instead of conventional one is considered to reduce fuel expenditure for small attitude corrections that can be stabilized by internal torquers (reaction or momentum wheels).The size of dead-band provides the capability of a tradeoff between maneuver time and fuel expenditure.The limits of the dead-band of switching function can be determined from maximum available torque produced by reaction wheels.On the other hand, large angle orientation of spacecraft induces structural deformation in the flexible solar arrays.Dynamical models of satellites are nonlinear and include rigid and flexible mode interaction.Therefore, vibration suppression of flexible solar arrays is required.For this case variable structure P+relay algorithm is proposed to eliminate vibration of flexible solar panels.
Desired sliding manifold on which the system equation of motion has good transient performances need to be selected before form a control law.The switching surfaces can be selected [1]- [5], [7], [11], [13]: where 123 (, ,) Kd i a g kkk  , in general is a diagonal design matrix to be selected.Particularly, these parameters are selected 123 kkkk   .Then the sliding manifold is: After selecting sliding manifold, a variable structure P+relay control algorithm can be formed as follows: where 1 N and 2 N are design parameters to be selected.The control law consists of two terms: P term and relay term.The first term is used for the compensation of the model nonlinearities, model and parameter uncertainties.The second term is used to compensate the bounded external disturbances, flexibility effects of solar arrays, and gravity gradient torque.This controller should provide the existence of the sliding mode motion on the selected sliding manifold.So, consider a Lyapunov function candidate: Now the sliding mode existence condition for the nonlinear satellite equation of motion in large will be investigated.The time derivative of (51) along the state trajectories of dynamics system defined by nonlinear equations ( 33) or ( 34) and ( 35) can be calculated as follows: Taking the norm of (52) where () we have where and the internal disturbance is: For providing negativeness of the V  it is required that the following sliding mode existence conditions should be satisfied: In result, (56) can be evaluated as: Therefore, the sliding manifold () 0 st  is reached in finite time [2]:

Modification of switching function
Thrusters apply discontinuous external force for stabilization of the nonlinear attitude dynamics of the spacecraft in finite time with limited thrust force.Control system using attitude thrusters is operated for large attitude angle orientations and its faster stabilization.Note that, thrusters are not required to operate for small attitude correction.Hence, here sliding function s can be modified to two-side dead-band (see Figure 5) to stop thruster operation for small attitude errors: where d s  is the upper and lower limit of dead-band.Therefore, the control action (50) forces the system to the dead-band limits of sliding manifold 0 d ss   and keeps it in dead zone.As shown [4], dead-zones can have a number of possible effects on control system.Their most common effect is to decrease static output accuracy.They can actually stabilize a system or suppress self-oscillations.For example, if a dead-zone is incorporated into ideal relay, it may lead to the avoidance of the oscillation at the contact point of the relay.Note that, (64) and (66) are operating regime of the attitude sliding mode controller that produces external control torques for stabilization via thrusters.Thus, the controller (50) forces the system to reach dead-band on switching function at finite time.On the other hand, if sliding variable has a value between dd ss s   then the sliding mode controller do not take any action.So, the system behavior is determined by attitude dynamics (33) with 0 B u  .Therefore, strictly, 0 V   for two operating cases of () s  (63).This means that system is convergent.

Example
The design of attitude sliding mode controller begins with selection of appropriate sliding manifold.A design parameter k can be selected from (61): we chose 0.25 k  which usually gives better performance for sliding manifold [5].Inertia matrix can be constructed form Table 1 as: The design parameters  

Comparison
Comparison of simulation results for both designed satellite control systems are done in the table.As seen from first part of Table 3 proposed variable structure controller in Section 4.1 has a large settling time then other considered attitude control systems.Since considered internal actuators has maximum 1.5 Nm torque capability, the settling time is three time shorter than [9] where same satellite (see Table 1) has been considered.

Fig. 1 .
Fig. 1.Body fixed reference frame B at the center of mass of a rigid body.Let
gravity gradient torque and bounded external disturbances as shown in Figure4.

Fig. 4 .
Fig. 4. Time responses of attitude sliding mode control system.Required parameter k is selected as 0.3 k .The disturbance is assumed to be as: , from (53) and (54) after some evaluations it is easy to design sliding gain constant k as:
from center of mass of a small mass element R  : position vector from an inertial origin dm : a small mass element n matrix  : skew symmetric matrix of angular velocity  : skew symmetric matrix of direction matrix E d : external disturbance q : modal coordinates vector To describe orientation of the bodyfixed reference frame B with respect to the local vertical local horizontal reference frame A in terms of Euler angles where / BA   is the angular velocity of B relative to A.  to B from A. For this sequence relation is: