Advances in Spacecraft Technologies Part 15 pdf

For accelerating space vehicles, several thrust vector control design approaches havebeen developed to suppress the fuel slosh dynamics.. Modeling and Control of Space Vehicles with Fuel

It has been demonstrated that pendulum and mass-spring models can approximatecomplicated fluid and structural dynamics; such models have formed the basis for manystudies on dynamics and control of space vehicles with fuel slosh (Peterson et al., 1989) There

is an extensive body of literature on the interaction of vehicle dynamics and slosh dynamicsand their control, but this literature treats only the case of small perturbations to the vehicledynamics However, in this chapter the control laws are designed by incorporating thecomplete nonlinear translational and rotational vehicle dynamics

For accelerating space vehicles, several thrust vector control design approaches havebeen developed to suppress the fuel slosh dynamics These approaches have commonlyemployed methods of linear control design (Sidi, 1997; Bryson, 1994; Wie, 2008) andadaptive control (Adler et al., 1991) A number of related papers following a similarapproach are motivated by robotic systems moving liquid filled containers (Feddema et al.,1997; Grundelius, 2000; Grundelius & Bernhardsson, 1999; Yano, Toda & Terashima, 2001;Yano, Higashikawa & Terashima, 2001; Yano & Terashima, 2001; Terashima & Schmidt, 1994)

In most of these approaches, suppression of the slosh dynamics inevitably leads to excitation

of the transverse vehicle motion through coupling effects; this is a major drawback which hasnot been adequately addressed in the published literature

In this chapter, a spacecraft with a partially filled spherical fuel tank is considered, andthe lowest frequency slosh mode is included in the dynamic model using pendulum andmass-spring analogies A complete set of spacecraft control forces and moments is assumed to

be available to accomplish planar maneuvers Aerodynamic effects are ignored here, althoughthey can be easily included in the spacecraft dynamics assuming that they are canceled by thespacecraft controls It is also assumed that the spacecraft is in a zero gravity environment,but this assumption is for convenience only These simplifying assumptions render theproblem tractable, while still reflecting the important coupling between the unactuated sloshdynamics and the actuated rigid body motion of the spacecraft The control objective, as istypical for spacecraft orbital maneuvering problems, is to control the translational velocityvector and the attitude of the spacecraft, while attenuating the slosh mode Subsequently,mathematical models that reflect all of these assumptions are constructed These problems areinteresting examples of underactuated control problems for multibody systems In particular,the objective is to simultaneously control the rigid body degrees of freedom and the fuel sloshdegree of freedom using only controls that act on the rigid body Control of the unactuatedfuel slosh degree of freedom must be achieved through the system coupling Finally, linearand nonlinear feedback control laws are designed to achieve this control objective

It is shown that a linear controller, while successful in stabilizing the pitch and slosh dynamics,fails to control the transverse dynamics of a spacecraft A Lypunov-based nonlinear feedbackcontrol law is designed to achieve stabilization of the pitch and transverse dynamics as well

as suppression of the slosh mode while the spacecraft accelerates in the axial direction Theresults of this chapter are illustrated through simulation examples

In this section, we formulate the dynamics of a spacecraft with a spherical fuel tank andinclude the lowest frequency slosh mode We represent the spacecraft as a rigid body(base body) and the sloshing fuel mass as an internal body, and follow the development

in our previous work (Cho et al., 2000a) to express the equations of motion in terms ofthe spacecraft translational velocity vector, the angular velocity, and the internal (shape)coordinate representing the slosh mode

Modeling and Control of Space Vehicles with Fuel Slosh Dynamics 3

To summarize the formulation in (Cho et al., 2000a), let v∈ 3,ω∈ 3, andη∈ denote thebase body translational velocity vector, the base body angular velocity vector, and the internal

coordinate, respectively In these coordinates, the Lagrangian has the form L=L(v,ω,η, ˙η),

which is SE(3)-invariant in the sense that it does not depend on the base body position and

attitude The generalized forces and moments on the spacecraft are assumed to consist ofcontrol inputs which can be partitioned into two parts: τt∈ 3 (typically from thrusters) isthe vector of generalized control forces that act on the base body andτr∈ 3(typically fromsymmetric rotors, reaction wheels, and thrusters) is the vector of generalized control torquesthat act on the base body We also assume that the internal dissipative forces are derivablefrom a Rayleigh dissipation function R Then, the equations of motion of the spacecraft withinternal dynamics are shown to be given by:

d dt

∂L

d dt

∂L

∂ω+ωˆ ∂ω ∂L +ˆv∂L

d dt

In our previous research (Reyhanoglu et al., 1996, 1999), we have developed theoreticalcontrollability and stabilizability results for a large class of underactuated mechanical systemsusing tools from nonlinear control theory We have also developed effective nonlinearcontrol design methodologies (Reyhanoglu et al., 2000) that we applied to several examples

of underactuated mechanical systems, including underactuated space vehicles (Reyhanoglu,2003; Cho et al., 2000b)

The formulation using a pendulum analogy can be summarized as follows Consider a rigidspacecraft moving in a fixed plane as indicated in Fig 1 The important variables are the axial

and transverse components of the velocity of the center of the fuel tank, v x , v z, the attitudeangleθ of the spacecraft with respect to a fixed reference, and the angle ψ of the pendulum with respect to the spacecraft longitudinal axis, representing the fuel slosh A thrust F, which

is assumed to act through the spacecraft center of mass along the spacecraft’s longitudinal

axis, a transverse force f , and a pitching moment M are available for control purposes The constants in the problem are the spacecraft mass m and moment of inertia I (without fuel), the

551Modeling and Control of Space Vehicles with Fuel Slosh Dynamics

Fig 1 A single slosh pendulum model for a spacecraft with throttable side thrusters.

fuel mass m f and moment of inertia I f (assumed constant), the length a>0 of the pendulum,

and the distance b between the pendulum point of attachment and the spacecraft center of

mass location along the longitudinal axis; if the pendulum point of attachment is in front of

the spacecraft center of mass then b>0 The parameters m f , I f and a depend on the shape of

the fuel tank, the characteristics of the fuel and the fill ratio of the fuel tank

Let ˆi and ˆk denote unit vectors along the spacecraft-fixed longitudinal and transverse axes, respectively, and denote by (x, z) the inertial position of the center of the fuel tank The

position vector of the center of mass of the vehicle can then be expressed in the spacecraft-fixedcoordinate frame as

where we have used the fact that v x= ˙x+z ˙ θ and vz=˙z−x ˙ θ.

Similarly, the position vector of the center of mass of the fuel lump in the spacecraft-fixedcoordinate frame is given by

r f= (x−a cos ψ)ˆi+ (z+a sinψ)ˆk,and the inertial velocity of the fuel lump can be computed as

˙

r f= [˙x+a ˙ ψsinψ+˙θ(z+a sin ψ)]ˆi+ [˙z+a ˙ ψcosψ−˙θ( x−a cosψ)]ˆk

= [vx+a(˙θ+ψ˙)sinψ]ˆi+ [vz+a(˙θ+ψ˙)cosψ]ˆk (5)The total kinetic energy can now be expressed as

2I ˙ θ2+1

Modeling and Control of Space Vehicles with Fuel Slosh Dynamics 5

M a

θ

v z Z

X

v x

δ F

Now consider the single slosh pendulum model for a spacecraft with a gimballed thrustengine as shown in Fig 2, whereδ denotes the gimbal deflection angle, which is considered

as one of the control inputs It is clear that, as in the previous case, the total kinetic energy

is given by equation (6) and the Lagrangian equals the kinetic energy Applying equations(1)-(3) with

and rearranging, the equations of motion can be obtained as:

(m+m f)(˙v x+˙θv z) +m f a(¨θ+ψ¨)sinψ+mb ˙ θ2+m f a(˙θ+ψ˙)2cosψ=F cos δ, (11)(m+m f)(˙v z−˙θv x) +m f a(¨θ+ψ¨)cosψ+mb ¨ θ−m f a(˙θ+ψ˙)2sinψ=F sin δ, (12)(I+mb2)¨θ+mb(˙v z−˙θv x) − ˙ψ=M+F(b+p)sinδ, (13)(I f+m f a2)(¨θ+ψ¨) +m f a[(˙v x+˙θv z)sinψ+ (˙v z−˙θv x)cosψ] + ˙ψ=0 (14)

553Modeling and Control of Space Vehicles with Fuel Slosh Dynamics

Fig 3 A single slosh mass-spring model for a spacecraft with throttable side thrusters.The control objective is to design feedback controllers so that the controlled spacecraftaccomplishes a given planar maneuver, that is a change in the translational velocity vectorand the attitude of the spacecraft, while attenuating the slosh mode The work in (Cho et al.,

2000b) considers a constant thrust F>0 and develops a feedback law (using a backsteppingapproach) to stabilize the system (7)-(10) to a relative equilibrium defined by a constantacceleration in the axial direction A slightly modified and relatively simpler feedbackcontroller that uses a Lyapunov approach (without resorting to backstepping) can be found

in (Reyhanoglu, 2003) In the subsequent development, we will develop feedback controllers

to achieve the same control objective using the system (11)-(14)

The mass-spring analogy is related to the pendulum analogy, in which the oscillationfrequency of the mass-spring element represents the lowest frequency sloshing mode (Sidi,1997)

Consider a rigid spacecraft moving on a plane as indicated in Fig 3, where v x , v zare the axialand transverse components, respectively, of the velocity of the center of the fuel tank, andθ

denotes the attitude angle of the spacecraft with respect to a fixed reference The slosh mode

is modeled by a point mass m f whose relative position along the body z-axis is denoted by s; a

restoring force−ks acts on the mass whenever the mass is displaced from its neutral position

s=0 As in the previous model, a thrust F, which is assumed to act through the spacecraft center of mass along the spacecraft’s longitudinal axis, a transverse force f , and a pitching moment M are available for control purposes The constants in the problem are the spacecraft mass m and moment of inertia I, the fuel mass m f , and the distance b between the body z-axis and the spacecraft center of mass location along the longitudinal axis The parameters m f , k and b depend on the shape of the fuel tank, the characteristics of the fuel and the fill ratio of

the fuel tank

The position vector of the fuel mass m f in the spacecraft-fixed coordinate frame is given by

r f=xˆi+ (z+s)ˆk,

Modeling and Control of Space Vehicles with Fuel Slosh Dynamics 7

and the inertial velocity of the fuel can be computed as

˙

r f= [˙x+ (z+s)˙θ]ˆi+ [˙z−x ˙ θ+˙s]ˆk

= (v x+s ˙ θ)ˆi+ (v z+˙s)ˆk. (15)Thus, under the indicated assumptions, the Lagrangian can be found as

m f(¨s+ ˙v z− ˙θv x−s ˙ θ2) +ks+c ˙s=0 (20)The control objective is again to design feedback controllers so that the controlled spacecraftaccomplishes a given planar maneuver, that is a change in the translational velocity vectorand the attitude of the spacecraft, while suppressing the slosh mode

In this section, we restrict the development to the single slosh pendulum model of a spacecraftwith a gimballed thrust engine, i.e we design feedback controllers for the system (11)-(14)only In particular, we study the problem of controlling the system to a relative equilibriumdefined by a constant acceleration in the axial direction

To obtain the linearized equations of motion, assume small gimbal deflection so that cosδ≈1and sinδ≈δ, and we rewrite (11)-(14) as:

(m+m f)ax+m f a(¨θ+ψ¨)sinψ+mb ˙ θ2+m f a(˙θ+ψ˙)2cosψ=F , (21)(m+m f)az+m f a(¨θ+ψ¨)cosψ+mb ¨ θ−m f a(˙θ+ψ˙)2sinψ=Fδ, (22)

(I f+m f a2)(¨θ+ψ¨) +m f a(a xsinψ+a zcosψ) + ˙ψ=0 , (24)

where l=b+p and(ax , a z) = (˙v x+ ˙θv z , ˙v z− ˙θv x)are the axial and transverse components

of the acceleration of the center of the fuel tank The number of equations of motion can

555Modeling and Control of Space Vehicles with Fuel Slosh Dynamics

be reduced to two by solving equations (21) and (22) for a x and a z, and eliminating theseaccelerations from equations (23) and (24).

[I+m∗(b2−ab cos ψ)]¨θ−m∗ab ¨ ψcosψ+m∗ab(˙θ+ψ˙)2sinψ− ˙ψ=M+b∗F δ, (25)[I f+m∗(a2−ab cos ψ)]¨θ+ (I f+m∗a2)ψ¨+ (a∗F−m∗ab ˙ θ2)sinψ+ ˙ψ= −a∗Fδcosψ, (26)

I1¨θ−I2ψ¨− ˙ψ=M+b∗Fδ, (27)

I3¨θ+I4ψ¨+a∗Fψ+ ˙ψ= −a∗Fδ. (28)where

I1=I+m∗(b2−ab), I2=m∗ab ,

I3=I f+m∗(a2−ab), I4=I f+m∗a2.For the linearized system (27)-(28), the state variables are the attitude angleθ, the slosh angle

ψ, and their time derivatives The collection of these state variables is defined as the partial

state vector given by

Modeling and Control of Space Vehicles with Fuel Slosh Dynamics 9

where Q is a symmetric positive-semidefinite weighting matrix and R is a positive-definite

weighting matrix

The optimal control gain matrix K is found by solving the corresponding matrix Riccati

equation (or using MATLAB’s lqr function) This controller is then applied to the actual

nonlinear system (11)-(14) The simulation results show that the linear controller (31) results

in undesirable steady-state errors in transverse velocity (see Figure 4)

Consider the single slosh pendulum model of a spacecraft with a gimballed thrust engine

shown in Fig 2 If the thrust F is a positive constant, and if the gimbal deflection angle and

pitching moment are zero,δ=M=0, then the spacecraft and fuel slosh dynamics have arelative equilibrium defined by

m+m fsinψ+m f a(˙v z−˙θv x(t))cosψ+ ˙ψ=0 , (36)

where v x(t)is considered as an exogenous input

Define the error variable

˜v x=vx(t) −v∗x(t).Then, the equations of motion can be written in the following form

and(u1, u2)are new control inputs defined as

the origin of the closed loop defined by the equations (37)-(40) and the feedback control laws

(45)-(46) Note that the positive gains r i , i=1, 2, 3, 4 and l j , j=1, 2, can be chosen arbitrarily

to achieve good closed loop responses

The feedback control laws developed in the previous sections are implemented here for a

spacecraft The physical parameters used in the simulations are m=600 kg, I=720 kg/m2,

m f =100 kg, I f =90 kg/m2, a=0.2 m, b=0.3 m, p=0.2 m, F=2300 N and =0.19 kg·m2/s.

We consider stabilization of the spacecraft in orbital transfer, suppressing the transverse andpitching motion of the spacecraft and sloshing of fuel while the spacecraft is accelerating Inother words, the control objective is to stabilize the relative equilibrium corresponding to a

constant axial spacecraft acceleration of 3.286 m/s2and v z=θ=˙θ=ψ=ψ˙=0

Modeling and Control of Space Vehicles with Fuel Slosh Dynamics 11

In this section, an LQR controller of the form (31) is applied to the complete nonlinear system

(11)-(14) Using the physical parameters given above, the A and B matrices defined by

equation (30) were computed as

K= 0.31530.7627 1.22142.8027 −0.34960.6462 −0.26960.0764



This gain matrix yields the following eigenvalues for the closed-loop system matrix A−BK:(−0.3312±0.8233 i, −0.3297±0.3294 i).

Time responses shown in Fig 4 and Fig 5 correspond to the initial conditions v x0=10000 m/s, vz0=0,θ0=2o, ˙θ0=0.57o /s, ψ0=15o, and ˙ψ0=0 As can be seen in the figures, the LQRcontroller stabilizes the pitch and slosh dynamics, but fails to stabilize the transverse velocity

to zero The controller results in a steady-state error of v z= −349.1 m/s

In this section, we demonstrate the effectiveness of the Lyapunov-based controller (45)-(46) byapplying to the complete nonlinear system (11)-(14)

Time responses shown in Fig 6 and Fig 7 correspond to the initial conditions v x0=10000 m/s,

v z0=350 m/s, θ0=2o, ˙θ0=0.57o /s, ψ0=30o, and ˙ψ0=0 As can be seen in the figures, thetransverse velocity, attitude angle and the slosh angle converge to the relative equilibrium at

zero while the axial velocity v x increases and ˙v x tends asymptotically to 3.286 m/s2 Note thatthere is a trade-off between good responses for the directly actuated degrees of freedom (thetransverse and pitch dynamics) and good responses for the unactuated degree of freedom

(the slosh dynamics); the controller given by (45)-(46) with parameters r1=10−7, r2=10, r3=

102, r4=10−2, l1=103, l2=1 represents one example of this balance

559Modeling and Control of Space Vehicles with Fuel Slosh Dynamics

0 5 10 15 20 25 30 0.99

1 1.01x 10

v z

−5 0 5

−20 0 20

−0.05 0 0.05 0.1 0.15

vz

0 2 4

−50 0 50

Modeling and Control of Space Vehicles with Fuel Slosh Dynamics 13

−15

−10

−5 0 5

−50 0 50 100 150

Time (s)

Fig 7 Gimbal deflection angleδ and pitching moment M (Lyapunov-based controller).

We have shown that a linear controller, while successful in stabilizing the pitch and sloshdynamics, fails to control the transverse dynamics of a spacecraft We have designed aLyapunov-based nonlinear feedback control law that achieves stabilization of the pitch andtransverse dynamics as well as suppression of the slosh mode, while the spacecraft accelerates

in the axial direction The effectiveness of this control feedback law has been illustratedthrough a simulation example

The many avenues considered for future research include problems involvinghigher-frequency slosh modes, multiple propellant tanks, and three dimensional maneuvers.Future research also includes designing nonlinear control laws that achieve robustness,insensitivity to system and control parameters, and improved disturbance rejection Inparticular, we plan to explore the use of sliding mode controllers to accomplish this

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Synchronization of Target Tracking Cascaded

Leader-Follower Spacecraft Formation

Rune Schlanbusch and Per Johan Nicklasson

Department of Technology, Narvik University College, PB 385, AN-8505 Narvik

Norway

1 Introduction

In recent years, formation flying has become an increasingly popular subject of study This

is a new method of performing space operations, by replacing large and complex spacecraftwith an array of simpler micro-spacecraft bringing out new possibilities and opportunities

of cost reduction, redundancy and improved resolution aspects of onboard payload One

of the main challenges is the requirement of synchronization between spacecraft; robustand reliable control of relative position and attitude are necessary to make the spacecraftcooperate to gain the possible advantages made feasible by spacecraft formations Forfully autonomous spacecraft formations both path- and attitude-planning must be performedon-line which introduces challenges like collision avoidance and restrictions on pointinginstruments towards required targets, with the lowest possible fuel expenditure The systemmodel is a key element to achieve a reliable and robust controller

1.1 Previous work

The simplest Cartesian model of relative motion between two spacecraft is linear and known

as the Hill (Hill, 1878) or Clohessy-Wiltshire (Clohessy & Wiltshire, 1960) equations; alinear model based on assumptions of circular orbits, no orbital perturbations and smallrelative distance between spacecraft compared with the distance from the formation tothe center of the Earth As the visions for tighter spacecraft formations in highly ellipticorbits appeared, the need for more detailed models arose, especially regarding orbital

perturbations This resulted in nonlinear models as presented in e.g (McInnes, 1995; Wang

& Hadaegh, 1996), and later in (Yan et al., 2000a) and (Kristiansen, 2008), derived forarbitrary orbital eccentricity and with added terms for orbital perturbations Most previouswork on reference generation are concerned with translational trajectory generation for fueloptimal reconfiguration and formation keeping such as in (Wong & Kapila, 2005) where

a formation located at the Sun-Earth L2 Langrange Point is considered, while (Yan et al.,2009) proposed two approaches to design perturbed satellite formation relative motion orbitsusing least-square techniques Trajectory optimization for satellite reconfiguration maneuverscoupled with attitude constraints have been investigated in (Garcia & How, 2005) where

a path planner based on rapidly-exploring random tree is used in addition to a smootherfunction Coupling between the position and attitude is introduced by the pointing constrains,

and thus the trajectory design must be solved as a single 6N Degrees of Freedom (DOF) problem instead of N separate 6 DOF problems.

25

Ground target tracking for spacecraft has been addressed by several other researchers, such

as (Goerre & Shucker, 1999; Chen et al., 2000; Tsiotras et al., 2001) and (Steyn, 2006) whereonly one spacecraft is considered The usual way to generate target tracking reference is

to find a vector pointing from the spacecraft towards a point on the planet surface wherethe instrument is supposed to be pointing, and then the desired quaternions and angularvelocities are generated to ensure high accuracy tracking of the specified target point

Due to the parameterization of the attitude for both Euler angles and the unit quaternion weobtain a set of two equilibria of the closed-loop system of a rigid body, and possibilities of theunwinding phenomenon One approach to solve the problem of multiple equilibria is the use

of hybrid control (cf (Liberzon, 2003), (Goebel et al., 2009)), and different solutions have been

presented, as in (Casagrande, 2008) for an underactuated non-symmetric rigid body, and by(Mayhew et al., 2009) using quaternion-based hybrid feedback where the choice of rotationaldirection is performed by a switching control law

The nonlinear nature of the tracking control problem has been a challenging task in robotics

and control research The so called passivity-based approach to robot control have gained much

attention, which, contrary to computed torque control, coupe with the robot control problem

by exploiting the robots’ physical structure (Berghuis & Nijmeijer, 1993) A simple solution tothe closed-loop passivity approach was proposed by (Takegaki & Arimoto, 1981) on the robotposition control problem The natural extension the motion control task was solved in (Paden

& Panja, 1988), where the controller was called PD+, and in (Slotine & Li, 1987) where the controller was called passivity- based sliding surface The control structure was later applied for

spacecraft formation control in (Kristiansen, 2008)

For large systems, e.g complex dynamical systems such as spacecraft formations, the expression divide and conquer may seem appealing, and for good reasons; by dividing a

system into smaller parts, the difficulties of stability analysis and control design can be greatly

reduced A particular case of such systems is cascaded structure which consists of a driving system which is an input to the driven system through an interconnection (see (Lor´ıa & Panteley,

2005) and references therein)

The topic of cascaded systems have received a great deal of attention and has successfullybeen applied to a wide number of applications In (Fossen & Fjellstad, 1993) a cascadedadaptive control scheme for marine vehicles including actuator dynamics was introduced,while (Lor´ıa et al., 1998) solved the problem of synchronization of two pendula through use

of cascades The authors of (Jankovi´c et al., 1996) studied the problem of global stabilisability

of feedforward systems by a systematic recursive design procedure for autonomous systems,while time-varying systems were considered in (Jiang & Mareels, 1997) for stabilization ofrobust control, while (Panteley & Lor´ıa, 1998) established sufficient conditions for UniformGlobal Asymptotical Stability (UGAS) for cascaded nonlinear time-varying systems Theaspects of practical and semi-global stability for nonlinear time-varying systems in cascadewere pursued in (Chaillet, 2006) and (Chaillet & Lor´ıa, 2008) A stability analysis of spacecraftformations including both leader and follower using relative coordinates was presented in(Grøtli, 2010), where the controller-observer scheme was proven input-to-state-stable

1.2 Contribution

In this paper we present a solution for real-time generation of attitude referencesfor a leader-follower spacecraft formation with target tracking leader and followerscomplementing the measurement by pointing their instruments at a common target on theEarth surface The solution is based on a 6DOF model where each follower generates the

Synchronization of Target Tracking Cascaded Leader-Follower Spacecraft Formation 3

attitude references in real-time based on relative translational motion between the leaderand its followers, which also ensures that the spacecraft are pointing at the target duringformation reconfiguration We are utilizing a passivity-based sliding surface controller forrelative position and Uniform Global Practical Asymptotic Stability (UGPAS) is proven forthe equilibrium point of the closed-loop system The control law is also adapted for hybridswitching control with hysteresis for attitude tracking spacecraft in formation to ensurerobust stability when measurement noise is considered, and avoid unwinding, thus achievingUniform Practical Asymptotical Stability (UPAS) in the large on the set S3×R3 for theequilibrium point of the closed-loop system Simulation results are presented to show howthe attitude references are generated during a formation reconfiguration using the derivedcontrol laws

The rest of the paper is organized as follows In Section 2 we describe the modeling of relativetranslation and rotation for spacecraft formations; in Section 3 we present a scheme werethe attitude reference for the leader and follower spacecraft is generated based on relativecoordinates; in Section 4 we present continuous control of relative translation and hybridcontrol of relative rotation where stability of the overall system is proved through use ofcascades; in Section 5 we present simulation results and we conclude with some remarks inSection 6

2 Modeling

In the following, we denote by ˙x the time derivative of a vector x, i.e ˙x=dx/dt, and moreover,

¨x=d2x/dt2 We denote by·the Euclidian norm of a vector and the inducedL2norm of a

matrix The cross-product operator is denoted S(·), such that S(x)y=x×y Reference frames

are denoted byF (·), and in particular, the standard Earth-Centered Inertial (ECI) frame is

denotedF iand The Earth-Centered Earth-Fixed (ECEF) frame is denotedF e We denote by

2.1 Cartesian coordinate frames

Basically there are two different approaches for modeling spacecraft formations: Cartesiancoordinates and orbital elements, which both have their pros and cons The orbital elementmethod is often used to design formations concerning low fuel expenditure because of therelationship towards natural orbits, while Cartesian models often are used where an orbitwith fixed dimensions are studied, which is the case in this paper

The coordinate reference frames used throughout the paper are shown in Figure 1, and defined

as follows:

the center of mass of the leader spacecraft The eraxis in the frame coincide with the vector

rl ∈R3from the center of the Earth to the spacecraft, and the ehaxis is parallel to the orbital

angular momentum vector, pointing in the orbit normal direction The eθaxis completes theright-handed orthonormal frame The basis vectors of the frame can be defined as

Follower orbit reference frame: The follower orbit frame has its origin in the center of mass

of the follower spacecraft, and is denotedF f The vector pointing from the center of the Earth

to the frame origin is denoted rf ∈R3, and the frame is specified by a relative orbit position

vector p= [x, y, z] expressed inF l components, and its unit vectors align with the basisvectors ofF l Accordingly,

2.2 Quaternions and kinematics

The attitude of a rigid body is often represented by a rotation matrix R∈ SO(3)fulfilling

SO(3) = {R∈R3×3 : RR=I, detR=1}, (3)

which is the special orthogonal group of order three, where I denotes the identity matrix.

A rotation matrix for a rotation θ about an arbitrary unit vector kkk ∈R3 can be angle-axis

parameterized as –cf (Egeland & Gravdahl, 2002),

Quaternions are often used to parameterize members of SO(3)where the unit quaternion is

defined as q= [η,  ] ∈ S3= {x∈R4 : xx=1}, whereη=cos(θ/2 ) ∈R is the scalar

p

Fig 1 Reference coordinate frames

Synchronization of Target Tracking Cascaded Leader-Follower Spacecraft Formation 5

part and=k sin(θ/2 ) ∈R3is the vector part The set S3 forms a group with quaternionmultiplication, which is distributive and associative, but not commutative, and the quaternionproduct is defined as

Ra b=Ra bS



ω b a,b

where fsd ∈R3is the perturbation term due to external effects, fsa ∈R3is the actuator force,

m s is the mass of the spacecraft, and super-/sub-script s denotes the spacecraft in question, so

s=l, f for the leader and follower spacecraft respectively The spacecraft masses are assumed

to be small relative to the mass of the Earth M e, soμ ≈ GM e , where G is the gravitational

constant According to (2) the relative position between the leader and follower spacecraftmay be expressed as

m f¨p+Ct( ω l

i,l)˙p+Dt(ω˙l

i,l,ω l i,l , r f)p+nt(rl , r f) =Fa+Fd, (15)

567Synchronization of Target Tracking Cascaded Leader-Follower Spacecraft Formation

is a nonlinear term The composite perturbation force Fdand the composite relative control

force Faare respectively written as

Note that all forces f are presented in the inertial frame If the forces are computed in another

frame, the rotation matrix should be replaced accordingly The orbital angular velocity andangular acceleration can be expressed asω i

i,l=S(rl)vl/r l rl, and

˙

ω i i,l=r l rlS(rl)al −2v l rlS(r

i,sb)Js ω sb i,sb+τττ sb

ω sb s,sb=ω sb i,sb −Rsb i ω i

where Js=diag{ J sx , J sy , J sz } ∈R3×3 is the spacecraft moment of inertia matrix,τττ sb

sd ∈R3isthe total disturbance torque,τττ sb ∈R3 is the total actuator torque andω i

i,s=S(rs)vs/r s rsisthe orbital angular velocity Rotation from the leader body frame to the inertial frame are

denoted qi

lb, while rotation from the follower body frame to the inertial frame are denoted

qi f b Relative rotation between the follower and leader body frame is found by applying the

quaternion product (cf (7)) expressed as

and with a slightly abuse of notation we denote ql=qi lband qf =qlb f b The relative attitude

dynamics may be expressed as (cf (Yan et al., 2000b; Kristiansen, 2008))

Jf ω˙ +JfS(Rlb f b ω lb

i,lb)ω −JfRlb f bJ−1

l S(ω lb i,lb)Jl ω lb

+S(ω+Rlb f b ω lb

i,lb)Jf(ω+Rlb f b ω lb

i,lb) =Υd+Υa,