A Numeric Study in Coordinated Attitude Control of aFormation of Spacecraft

Matthew Clark VanDyke

October 22, 2003

Advanced Spacecraft Dynamics and ControlDepartment of Aerospace and Ocean Engineering

Virginia Polytechnic Institute and State University

1 Introduction

The recent successful formation flying maneuver of Earth Orbiter 1 and Landsat 7 has demon-strated the feasibility of formation flying.[3] NASA currently has 35 formation flying concepts underconsideration.[10] A spacecraft formation consists of two or more spacecraft in specific relative po-sitions and orientations. A formation of spacecraft provides two primary benefits over a singlespacecraft. Dispersing the functions of a single spacecraft over a formation of spacecraft produces amore robust and fault-tolerant space system architecture. A formation facilitates greater resolutionthrough the use of spatially distributed simultaneous measurements.[10] Algorithms for the preciserelative and absolute alignment of the spacecraft in the formation are required to gain the most fromthe distributed measurements.

Attitude control of a spacecraft formation presents an interesting challenge. The problem is com-plicated by the fact that the desired attitude cannot be treated as a priori information.[6] Environ-mental disturbances and model uncertainties cause each spacecraft to have its own tracking error.[5]A coordinated control algorithm is required for successful spacecraft formation attitude control.

Coordinated control algorithms can be split into centralized and decentralized schemes. Centralizedcontrol requires that all control decisions are made by a single global control agent. The global controlagent orders the actions of the local control agents based on all available information about the stateof the system. The most prominent centralized control scheme is the leader-follower coordinationscheme. In a leader-follower scheme, the spacecraft in the formation are divided into leaders andfollowers. The leaders in the formation are tasked with tracking a given reference condition, whileeach of the followers track one of the leaders in the formation. The leader-follower schemes includemany variants such as the chain, tree, and virtual structure topologies.[9] Decentralized control is thecontrol of a system using multiple local control agents. The local control agents use local observationsand any other information available about the system to determine local control actions.

A useful tool for the spacecraft formation attitude control problem is behavior-based control. Behavior-based control is implemented when a control system has multiple, and sometimes competing, ob-jectives or behaviors. The behaviors could include goal attainment, collision avoidance, obstacleavoidance, and formation keeping. The overall control action is determined by taking a weighted

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average of the control actions for each of the behaviors.[9] For the spacecraft formation attitude prob-lem, behavior-based control is used to arrive at a compromise between the control actions requiredfor the formation-keeping and station-keeping behaviors.

A brief discussion of the possible applications of this research and a quick review of rigid-bodydynamics are given as background for the reader in the next two chapters. The development of atracking attitude controller to provide attitude stabilization and control for the individual spacecraftis presented in Chapter 4. The main contribution of the paper is the evaluation of three centralizedand two decentralized controllers through numeric simulation.

2 Applications

Two NASA formation flying missions under consideration that would benefit from improved forma-tion attitude control are the Laser Interferometer Space Antenna (LISA) and the Terrestrial PlanetFinder (TPF). The LISA consists of three spacecraft in an equilateral triangle formation, each sep-arated by 5 million kilometers. A laser ranging system is used to accurately monitor the distancebetween the spacecraft. It is hoped that the system will be able to detect small variations of thedistance caused by a gravitational wave. The great distance required by the mission precludes theuse of a single spacecraft. A concept being considered for the TPF uses a formation of spacecraft,separated by hundreds of meters, equipped with inferred telescopes acting as an interferometer. Theformation would be able to produce imagery with a resolution equal to a single telescope with amirror the size of the formation. The success of these and other formation flying missions rely onprecise pointing of the spacecraft formation.

Coordinated attitude control could also allow spacecraft in Low Earth Orbit (LEO) to maintaintighter formations. Spacecraft formations in LEO experience an atmospheric drag force. The dragforce has a disturbing influence on spacecraft formations. Even if the spacecraft are identical, thedrag force experienced by each is slightly different. The difference is caused by the relative attitudetracking errors of the spacecraft. The relative attitude error causes the spacecraft have differentattitudes with respect to their velocity vectors, and therefore the free-stream. The result is differingballistic coefficients, which directly relates to the drag force. Therefore, a spacecraft formation witha smaller relative attitude error is less effected by the drag force disturbance, thereby lessening thepropulsive force required to maintain the formation.

3 Literature Review

Centralized control is a type of coordinated control where a single control agent, called the globalcontrol agent, determines the control actions for a distributed system. The greatest benefit ofcentralized coordinated controllers is that global convergence can be analytically proven. It is farmore difficult to develop an analytic proof of convergence for decentralized controllers.

Leader-follower based coordinated control dominate the centralized control literature. Some leader-follower variants include the chain, tree, and virtual structure coordination topologies. Lawton,Beard, and Hadaegh [9] develop a relatively simple leader-follower coordinated controller, and prove

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that it guarantees global formation convergence. The leader-follower coordinated controller is usedprimarily as baseline by the authors to determine the performance of a behavior-based decentralizedcontroller also developed in the paper.[9] Kang, Yeh, and Sparks [5] develop several interestingleader-follower type coordinated controllers. They develop two leader-follower schemes that usebehavior-based control to balance the formation-keeping and station-keeping control behaviors aredeveloped. The balanced control action is determined by augmenting the absolute desired attitudewith the current attitude of other spacecraft in the formation.[5] Two of the authors, Kang andSparks [6], apply this same type of controller to a coordinated tracking problem in a subsequentpaper.

The most cited paper on the topic of behavior-based control is by Balch and Arkin.[2] In the paperbehavior-based control is applied to a formation of autonomous robots. One of the coordinationschemes demonstrated is unit-center-referenced. The scheme requires all robots in the formationknow the positions of the other robots, and use the knowledge to compute the unit-center of theformation. The control action is then determined by comparing the current and desired positions ofthe robot relative to the unit-center.

Decentralized control is the stabilization and control of a large-scale system using multiple localcontrol agents. The local control agents use local observations and any information communicated bythe other control agents to determine control actions. Proving the stability of a decentralized controlsystem is a prominent topic in much of the literature. The two primary benefits of decentralizedcontrol over centralized control are robustness and relatively simple control laws. The failure of asingle local control agent in a decentralized controlled system does not lead to the destabilizationof the entire system.[4] The failure is therefore confined to the region of the failed local controlagent. Decentralized control results in relatively simple control laws, because the design of theglobal controller can be broken up into smaller control agents. The local control agents are designedso that they perform their local control tasks, as well as coordinate with one another to control theglobal system.[4] The coordination is implemented by means of communication between the localcontrol agents. Centralized controllers require greater information and information processing thenwhat is required by the local control agents.[1]

The primary topics discussed in the literature on decentralized control are global stability andthe communication and coordination of local control agents. The majority of the papers focus onderiving necessary and sufficient conditions for global stability. The communication between andcoordination of the local control agents is mentioned by many authors, but is not discussed in detailby most, with the exception of Speyer [12].

Although the majority of the papers limit themselves to linear time-invariant systems, Davison[4] investigates nonlinear time-varying systems. Sufficient conditions for global stability using adecentralized control scheme are presented. High gain adaptive controllers are used at the locallevel. Davison points out in the conclusion of the paper that a decentralized controller provides a”fail-safe” feature, meaning that the failure of a local control agent does not lead to global instabilityof the system.[4]

A decentralized coordinated controller developed specifically for the spacecraft formation attitudeproblem is presented in [9]. The controller is developed by Lawton, Beard, and Hadaegh [9] andmakes use of behavior-based control. The authors call the controller the coupled dynamics controller.The coupled dynamics controller uses a ring coordination topology, where each spacecraft knows thestate of two other spacecraft in the formation. The desired state and the state of the two other

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spacecraft are used to determine the appropriate control torque. A convergence proof is provided,however the proof does not ensure global convergence of the formation attitude. Appendix B of thepaper gives an interesting proof demonstrating that the geodesic metric between two unit quaternionscan be approximated by the Euclidian distance between them, which is a useful result for developinga formation attitude error measures. A subsequent paper by Lawton and Beard [8] use a passivity-based controller for the spacecraft formation attitude control problem. The authors also determinethe domain of attraction for the passivity-based controller and the coupled dynamics controller.

The majority of the literature on coordinated control, as with most other control topics, investigatesthe stability of distributed systems using a variety of control schemes. Although many papersdiscuss centralized coordination schemes, most of the contributions of the later papers fall underdecentralized control. Decentralized control has garnered much of the attention because of its easyscalability and robustness. Literature on spacecraft formation attitude control has emanated from,primarily, two groups of authors. Both groups use behavior-based control to achieve better formationattitude characteristics, however, they differ in the type of coordination schemes they use.

4 Rigid Body Kinematics and Dynamics

A brief review of rigid body kinematics and dynamics is presented to define the notation used bythe author, and provide a refresher for the reader. Modified Rodriguez Parameters (MRP), σ, arethe attitude representation used in this research. MRP are defined by:

σ = e tan(

φ

4

)(1)

where, e is the eigenaxis and φ is the rotation angle. The time derivative of σ in terms of ω, theangular velocity vector, is:

σ = G(σ)ω (2)

where, the matrix function G(σ) is defined as:

G(σ) =12

(1 + σ× + σσT − 1 + σT σ

21)

(3)

The rigid body dynamics of the individual spacecraft are formulated in terms of angular momentum.The total angular momentum of the spacecraft, hb, is:

hb = Iωb + AIsωs (4)

where, I is the moment of inertia matrix, and ωb is the angular velocity of the spacecraft. Is isa diagonal matrix whose diagonal elements are the moments of inertia of the momentum wheels(Is = diag [Is1, · · · , Isn]), and ωs is a column matrix whose elements are the angular velocities ofthe momentum wheels. The columns of A are the axial unit vectors of the momentum wheels. Thetotal angular momentum of the momentum wheels is:

ha = IsAT ωb + Isωs (5)

4

The rigid body dynamic equations of motion for the attitude of the individual spacecraft are writtenin the terms of hb and ha.

hb = ge − ω×b hb (6)

ha = ga (7)

where, ge is the external torque and ga is the torque applied to the momentum wheels. If aninertia-like matrix, J is defined as:

J = I−AIsAT (8)

the angular velocity of the body, ωb, becomes:

ωb = J−1(hb − ha) (9)

and, the equations of motion can be rewritten in the form:

hb = h×b J−1(hb −Aha) + ge (10)

ha = ga (11)

This is the form of the equations of motion that is used in the derivation of the attitude controllerfor the individual spacecraft.

4.1 Error Kinematics and Dynamics

The derivation of the individual attitude controller for the spacecraft in the formation requiresdefining some attitude state error quantities. The attitude error is given by the MRP that representsthe rotation from the reference to body frame. The error MRP and its time derivative are definedby:

δσ = σ(Rbr) (12)δσ = G(δσ)δω (13)

where, δω is the error angular velocity vector and is defined as:

δω = ωb −Rbrωr (14)

The error dynamics can now be found by taking the time derivative of Eq. 14.

δω = ωb −Rbrωr − δω×Rbrωr (15)

These equations define the error in the attitude state and how it develops over time.

4.2 Summary

A brief review of rigid body dynamics was presented. The error kinematic and dynamics conceptsrequired to develop a tracking controller were introduced. The notation and equations introducedin this chapter are used extensively in the following chapter to develop a robust attitude trackingcontroller for the individual spacecraft.

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5 Individual Spacecraft Attitude Controller

Coordinated attitude controllers require that the individual spacecraft in the formation have theability to accurately control and stabilize their attitudes. To this end, a robust attitude trackingcontroller is developed to ensure the controllability of the individual spacecraft. The controller isdeveloped using Liaponuv control theory.

5.1 Control Law Derivation

The first step in the development is to pick a Liaponuv candidate function. The candidate functionfor the individual spacecraft attitude controller is:

V =12δωT Jδω + 2k1 ln(1 + δσT δσ) +

12zT K2z (16)

where,

z =∫ t

0

Jδω + k1δσ dτ (17)

The Liaponuv candidate function is taken from Schaub and Junkins [11], who use it to develop anattitude controller that uses external control torques. In this work, the same candidate functionis used, however the derivation differs somewhat because momentum-exchange devices are usedas the torque actuators. The candidate function is positive definite and radially unbounded. Acontrol torque is now derived to make the first time derivative of the candidate function, V , negativesemi-definite. The first time derivative of the function is:

V =12δωT Jδω + 4k1

δσT δσ

1 + δσT δσ+ zT K2z (18)

Using Eq. 12, the function is rewritten as:

V = δωT Jδω + 4k1δσT G(δσ)δω1 + δσT δσ

+ zT K2 (Jδω + k1σ)

Equation 3 is used to derive the identity:

4k1δσT G(δσ)δω1 + δσT δσ

= k1δωT δσ

The identity simplifies the V equation to:

V = δωT Jδω + k1δωT δσ + zT K2 (Jδω + k1σ)

= δωT (Jδω + k1δσ) + zT K2 (Jδω + k1σ)

=(δω + KT

2 z)T

(Jδω + k1σ) (19)

V is set equal to a known negative semi-definite function:

V = −(δω + KT

2 z)T

K3

(δω + KT

2 z)

(20)

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Plugging in the equations of motion and solving for Aha we arrive at:

Jδω + k1σ = δω + KT2 z

δh−Aha + k1σ = −K3

(δω + KT

2 z)

Aha = δh + k1δσ + K3KT2 z + K3δω

Using this result and Eq. 7, the control torque, ga, is:

ga = A−1(h×b J−1(hb −Aha)− JRbrωr − Jδω×Rbrωr + ge −K3KT

2 Jδω0

)+A−1

(k1δσ + k1K3KT

2

∫ t

−∞δσdτ + K3(1 + K2J)δω

)(21)

5.2 Proof of Global Asymptotic Stability

Two methods are used to prove the global asymptotic stability of the attitude tracking controller ofthe individual spacecraft. The first utilizes LaSalle’s invariance principal. The second proof uses thehigher order derivatives of the candidate Liaponuv function, and follows the proof given in Schauband Junkins [11].

5.2.1 Proof Using LaSalle’s Invariance Principal

A positive definite and radially unbounded Liaponuv function with a negative definite first timederivative ensures global asymptotic stability.[7] Due to the V function being negative semi-definite,only global stability is proven for the control law derived above. LaSalle’s Invariance Principal isused to prove global asymptotic stability of the controller. Equation 20 implies:

limt→∞

(δω + KT2 z) = 0 (22)

Local stability is proven about the equilibrium points; δω = z = 0 and δω + KT2 z = 0. However,

the goal is to prove global asymptotic stability of the origin, δσ = δω = 0.

The first equilibrium condition is proven to be only stable at the origin using the equations of motionof the system. Equation 9 is plugged into Eq. 15 to give:

δω = J−1(hb −Aha)−Rbrωr − δω×Rbrωr

Inserting the previously derived control law, and the first equilibrium condition (δω = z = 0) theequation becomes:

J−1(K3K2z + K3δω + k1δσ) = 0k1J−1δσ = 0 (23)

The result implies that:

limt→∞

δσ = 0 (24)

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The first equilibrium condition is proved to be only stable about the the origin.

The proof for the second equilibrium condition begins by taking the first time derivative of theequilibrium condition:

δω = −KT2 z

= −k1(1 + K2J)−1KT2 δσ (25)

Equations 25 and 23 are equated to arrive at:

k1J−1δσ = −k1(1 + K2J)−1KT2 δσ

The only possible solution for the above equation is δσ = 0, because a positive definite matrixcannot be equal to negative definite matrix. The result implies:

limt→∞

δσ = 0

and,

limt→∞

δω = 0

due to the kinematic relationship between δσ and δω. Thus, the origin, δσ = δω = 0, is proven tobe globally asymptotically stable.

5.2.2 Proof Using the Higher Order Derivatives of the Liaponuv Function

Earlier in the section it was shown that the first time derivative of the Liaponuv function is:

V = −(δω + KT

2 z)T

K3

(δω + KT

2 z)

which is a negative semi-definite function that implies that δω+KT2 z = 0 is an equilibrium condition

of the system. The result only proves that the states are Liaponuv stable, and does not guarantee theconvergence of the system to the reference condition, δσ = δω = 0. The convergence of the systemhas already been proven using LaSalle’s Invariance Principal. The higher order time derivatives arenow inspected to further reinforce those stability results.

To begin the V function is rewritten using the identity:

α = δω + KT2 z (26)

The second time derivative of the Liaponuv function is:

V = −2αT K3α

It is easy to see that V is equal to zero on the set of interest, α = 0. Therefore, the function is onlynegative (or positive) semi-definite and does not add to our knowledge of the stability of the system.The third time derivative of the Liaponuv function is:

...V = −2αT K3α− 2αT K3α

8

Applying the equilibrium condition to the function, it simplifies to:...V = −2αT K3α

The function is negative-definite as desired. Using the definition of z, the first time derivative of αbecomes:

α = δω + KT2 z

By plugging in the equations of motion for the controlled system, the α equation becomes:

α =(1 + KT

2 J)J−1

(−k1δσ −K3KT

2 z−K3δω)

+ k1KT2 δσ

= −k1

(1 + KT

2 J)J−1δσ + k1KT

2 δσ

= −k1J−1δσ

The equilibrium condition is enforced to arrive at:

α = −k1

(1 + KT

2 J)J−1δσ + k1KT

2 δσ

= −k1J−1δσ

The...V function can be rewritten as:

...V = −2k2

1(J−1)

TδσT J−1δσ (27)

The...V function is a negative definite function that is explicitly dependant on δσ, which implies:

limt→∞

δσ = 0

The kinematic relationship between δσ and δω (Eq. 12) also implies that:

limt→∞

δω = 0

Global asymptotic stability of the attitude of the spacecraft is proven.

5.3 Simulation Results

The attitude controller is tested numerically to validate the stability results obtained above, and todemonstrate the usefulness of the integral term used in the controller derivation. The spacecraft inthe simulation has the following physical properties.

I = diag([2 3 4]T

)kg m2

Is = diag([0.1 0.1 0.1]T

)kg m2

A = 1

The initial state is set to be:

q = [0.5 0.5 0.5 0.5]T

ωb = [10 − 30 5]T deg/sec

9

Figure 1: Attitude simulation results of a spacecraft performing slew maneuver subjected to adisturbance torque.

The spacecraft is commanded to perform a 60o slew maneuver. An un-modelled environmentaldisturbance torque of 1 Nm about the body 3 axis is included in the simulation. Two simulationsare performed. The first simulation uses the following control gains:

k1 = 5K2 = 03×3

K3 = 5 13×3

The integral control gain, K2, is zeroed to demonstrate its effectiveness in counteracting the constantdisturbance torque in the simulation. The results of the first simulation are shown in Figure 1.

The steady state attitude of the spacecraft is a 22.6o 3-axis rotation away from the commandedattitude. The error is due to the constant disturbance torque. The controller contains the integralcontrol terms to combat these types of disturbances. The integral gain is now set to:

K2 = 0.01 13×3

The simulation is re-run with the same initial conditions and disturbance torque. The results arepresented in Figure 2. The plot clearly shows that the spacecraft’s attitude converges with thecommanded attitude. Complete convergence of the attitude occurs about 50 seconds after thesimulation begins.

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Figure 2: Attitude simulation results of a spacecraft performing slew maneuver subjected to adisturbance torque using the attitude controller with integral control.

5.4 Summary

A tracking attitude controller for the individual spacecraft of the formation was developed. Thecontroller was proven to be globally asymptotically stable using LaSalle’s invariance principal andby inspecting the third time derivative of the Liaponuv function. The ability of the controller toreject disturbance torques was demonstrated through the use of numeric simulation. The coordinatedcontrollers described in the next section will utilize the individual controllers of each of the spacecraftin the formation to obtain desired formation attitudes.

6 Coordinated Control

The coordination schemes investigated in this work include three centralized and two decentralizedschemes. The three centralized schemes are topological variants of the leader-follower coordinationscheme: the one-leader, chain, and tree topologies. The two decentralized schemes investigated usebehavior-based control. The two decentralized coordination schemes differ through the coordinationtopologies used. The first uses a ring topology while the second uses a web topology.

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6.1 Leader-Follower Coordination

Leader-follower controllers split the local control agents of a distributed system into leaders andfollowers. The leaders are tasked with following a desired reference trajectory. The followers aretasked with following the leaders.

The primary advantage of leader-follower controllers is that analytical convergence proofs are gener-ally available. The proof can easily be seen through the use of a spacecraft formation as an example.The dynamics of the spacecraft in the formation are only interconnected due to the decisions of theirindividual controllers and not due to any physical connection. The individual spacecraft each usetracking controllers that are globally asymptotically stable. Therefore, the leaders of a formationwill always converge to the desired reference trajectory. The followers of a formation will then alwaysconverge to the trajectory of the leader, which is the desired reference trajectory. In this way itcan be seen that the spacecraft formation using a leader-follower coordinated control scheme willalways converge. The greatest disadvantage of leader-follower controllers is that the system has asingle-point of failure, the leader or leaders. If the leader suffers a failure, the entire system becomesunstable and fails

Leader-follower controllers differ primarily by the coordination topologies used. The coordinationtopology defines how the local control agents communicate and cooperate to meet global systemgoals. Below three leader-follower coordination strategies are presented.

6.1.1 One Leader Coordination Topology

The most basic leader-follower coordination topology, the one-leader topology, designates one localcontrol agent as the leader. The other control agents, the followers, are then tasked with following thetrajectory of the leader, possibly by a given offset that could be time-varying. Because this topologyis the most basic, it is often used as a baseline to determine the performance of more complex andexotic coordinated control schemes. Figure 3 depicts the one leader coordination topology as appliedto a formation of spacecraft. The red cylinder represents the leader and the blue cylinders denotefollowers. This color code is used throughout the rest of the chapter.

The arrows represent lines of communication that are required by the one leader coordination topol-ogy. As can be seen in the figure, all of the lines of communication are unidirectional. The leaderneeds to only have the ability to broadcast to the followers.

Mathematically the one leader coordination topology is expressed as:

x1 = xR (28)xi = x1 + ∆i(t) for i = 2, 3, ..., n

where, xi is the desired state of spacecraft i, and ∆(t) is the time-varying offset of spacecraft i. Thedesired reference trajectory of the formation is represented by xR.

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Figure 3: A diagram of the one-leader leader-follower coordination topology.

6.1.2 Chain Coordination Topology

The second leader-follower coordination topology is the chain topology. Just as the name implies,in this topology the local control agents are strung together to form a communication chain. Theresulting chain is formed mathematically as:

x1 = xR (29)xi = xi−1 + ∆i(t) for i = 2, 3, ..., n

Figure 4 depicts the structure of the chain topology. The chain topology introduces the interestingconcept of having local control agents that are both leaders and followers simultaneously. The purplecylinders in Figure 4 represent these control agents. An undesired effect of the chain topology isthat the later parts of the chain lag behind earlier parts of the chain.

6.1.3 Tree Coordination Topology

An interesting generalization of the chain topology is the tree topology. In the tree topology leaderstypically command multiple followers. The result is the hierarchical command structure shown inFigure 30. As in the chain topology some of the local control agents are simultaneously leaders andfollowers. Because the tree topology is a generalization of the chain topology, it also suffers fromthe same lag-time response issues. The tree topology lends itself well to distributed systems where

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Figure 4: A diagram of the chain leader-follower coordination topology.

Figure 5: A diagram of the tree leader-follower coordination topology.

a natural hierarchy exists. Mathematically the tree topology for a distributed system consisting of7 local control agents can be expressed as:

x1 = xR (30)xi = x1 for i = 2, 3xi = x2 for i = 4, 5xi = x3 for i = 6, 7

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6.2 Decentralized Coordination

Decentralized coordinated control, as in centralized coordinated control, uses local control agentsto stabilize and control a distributed system. The two types of coordinated control differ in wherecontrol decisions are made. In centralized control, all control decisions are made by a single globalcontrol agent, while in decentralized control the control decisions are made by the local controlagents.

The key problem in decentralized control is that the decisions by the local control agents are madewithout complete knowledge of the current state of the system. This leads to difficulty in providinganalytic convergence proofs. However, decentralized control does help to make a robust distributedsystem. Systems using decentralized control do not have single-points of failure, and thereforeany failures result in a graceful degradation of system performance instead of the global instabilitysuffered by centralized controllers.

A decentralized control using a ring coordination topology, and one using a web coordination topol-ogy are investigated. The two control schemes utilize behavior-based control ideas to determine thecontrol actions of the local control agents. Behavior-based control is useful in systems where the con-trol actions required to reach different and simultaneous desired goals or behaviors might conflict. Inthe case of spacecraft formations, the desired behaviors are station-keeping and formation-keeping.Station-keeping is the desired behavior that drives an individual spacecraft to its desired absoluteattitude alignment. The formation-keeping behavior attempts to align the individual spacecraft withthe other spacecraft in the formation. The decentralized coordinated controllers described in thissection use a linear weighting of the desired states of the behaviors to calculate a pseudo-desiredstate. The pseudo-desired state is feed into the controllers to determine the appropriate control ac-tion. The coordinated controllers are therefore essentially decoupled from the individual controllersof the spacecraft.

6.2.1 Ring Coordination Topology

The first decentralized coordination topology discussed is the ring topology. Each spacecraft in theformation is connected to two other spacecraft in the formation. The resulting communication ringis shown in Figure 6.

In contrast to the leader-follower coordination schemes, the lines of communication are bi-directionalin the decentralized controllers. The lines connecting the spacecraft in the figure represent the bi-directional communication by having arrows at both ends of the lines.

The ring coordination topology is described mathematically as:

x1 = (1− ρ)xR + ρ12(x2 + xn) for i = 1 (31)

xi = (1− ρ)xR + ρ12(xi+1 + xi−1) for i = 2, 3, ..., n− 1

xn = (1− ρ)xR + ρ12(x1 + xn−1) for i = n

15

Figure 6: A diagram of the ring decentralized coordination topology.

The variable, ρ, is the behavior weighting factor, and varies from 0 to 1. A large value of ρ indicates anincreased importance in the formation-keeping behavior, while a small value places more importanceon the station-keeping behavior.

A modification of Eq. 31 is required to account for the fact that average value of a set of quater-nions may violate the quaternion length constraint. The equation for the desired quaternion of theindividual spacecraft becomes:

q1 =(1− ρ)qR + ρ 1

2 (q2 + qn)‖(1− ρ)qR + ρ 1

2 (q2 + qn)‖for i = 1 (32)

qi =(1− ρ)qR + ρ 1

2 (qi+1 + qi−1)‖(1− ρ)qR + ρ 1

2 (qi+1 + qi−1)‖for i = 2, 3, ..., n− 1

qn =(1− ρ)qR + ρ 1

2 (q1 + qn−1)‖(1− ρ)qR + ρ 1

2 (q1 + qn−1)‖for i = n

The ring topology is used by Lawton, Beard, and Hadaegh [9] to develop a decentralized spacecraftformation attitude controller. In the paper, the authors are able to prove stability and convergence oftheir controller analytically with a few constraints. The proof relies on the initial formation attitudeerror being below a certain value, and on the spacecraft in the formation beginning with no angularvelocity.

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6.2.2 Web Coordination Topology

A generalization of the ring topology is the web coordination topology. It requires that the spacecraftin the formation have complete knowledge of the state of the formation. Figure 7 depicts the webtopology.

Figure 7: A diagram of the web decentralized coordination topology.

The web topology requires the most computation of any of the coordination schemes discussed inthis chapter, because it uses the largest amount of information to determine control actions.

The web coordinated control can be expressed mathematically as:

xi = (1− ρ)xR + ρ1n

n∑i=1

xi for i = 1, 2, ..., n (33)

where, ρ is the behavior weighting factor. This coordination topology runs into the same problemas the ring topology, because it also requires some measure of the average attitude of the formation.Therefore, a similar modification of the equation is required.

qi =(1− ρ)qR + ρ 1

n

∑ni=1 qi

‖(1− ρ)qR + ρ 1n

∑ni=1 qi‖

for i = 1, 2, ..., n (34)

6.3 Summary

Three centralized controllers and two decentralized controllers were described. The coordinationtopologies were described using diagrams and equations. The controllers are tested through numericsimulation in the next chapter.

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7 Numeric Simulation Results

The five coordinated controllers presented in the previous chapter are tested through numeric simu-lation. A formation of seven spacecraft is used in each case. A slew and a spin maneuver is simulatedto determine if the type of maneuver affects the performance of the coordinated controller. The sim-ulations are performed in Matlab c©. The Matlab c© code for the coordinated controllers is presentedin Appendix A to provide a reference for the reader.

Each spacecraft in the formation is given a 5o initial attitude error from the nominal formationattitude. The formation initially has no angular velocity. The spacecraft are initialized with a 5o/sangular velocity error. To simulate differing tracking errors of the spacecraft, a different constantdisturbance torque is applied throughout the maneuver to each of the spacecraft. The controllersusing behavior-based control used a behavior weighting, ρ, of 0.5

There are many different methods to quantify attitude error. For this work the rotation anglebetween the desired and actual reference frame is used. The absolute formation error and relativeformation error are calculated by first determining an average formation attitude. The relativeformation error represents the average angular difference between the average formation attitudeand the attitude of each of the individual spacecraft. The absolute formation error is the angulebetween the average formation attitude and the desired formation attitude. The average of theabsolute and relative formation error is taken to arrive at a total error measure for the spacecraftformation

7.1 Slew Maneuver

The first maneuver simulated is a slew maneuver. The formation is initialized so that the nominalattitude of the formation is aligned with the inertial reference frame. The formation is then com-manded to perform a 90o slew maneuver. The results of the simulation are presented in the Figures8 and 9.

Plot A of figure 8 shows the absolute angular formation error, and plot B shows the relative angularformation. Each curve in the two plots represent the error results for a different coordinated con-troller. Figure 9 shows a plot of the total formation error for each of the controllers throughout thesimulation.

Despite which coordinated controller is used, the formation converges to the new desired attitudeafter approximately 100 seconds. The errors of chain and tree controllers are higher than the errorsof the other controllers. The one-leader controller has a consistently smaller error value than theother leader-follower controllers throughout the simulation. The results for the two decentralizedcontrollers follow each other closely throughout the maneuver. During the first ∼ 10 seconds ofthe simulation the two match very closely, but then start to deviate slightly with the ring controllererror being slightly less than the error for the web controller. The decentralized controllers havesmaller error values than the centralized controllers throughout the simulation.

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Figure 8: The absolute(A) and relative(B) angular formation error of the spacecraft formation whileexecuting a slew maneuver using the various coordinated controllers.

7.2 Steady Spin Maneuver

The second maneuver simulated is a steady spin maneuver. The formation is initialized so thatthe nominal attitude of the formation is aligned with the inertial reference frame. The formation iscommanded to follow the attitude of a rigid body that starts at an attitude that is a 90o rotationfrom the initial formation attitude and is in a steady 3o/s spin about its body 2 axis. Figure 10contains the plots of the absolute and relative formation errors.

The absolute formation error results are shown in plot A, and the relative formation error resultsare shown in plot B. Figure 11 shows the total formation error throughout the simulation for eachof the controllers.

The results for the spin maneuver simulation are similar to the results for the slew maneuver. Thesame general trends can be seen. The major difference between the results is the error curve forthe web controller. Although the curve begins as in the slew maneuver to closely match the ringcontroller, the curve splits off and remains about 2 to 3 degrees higher than the ring controller errorcurve for a significant portion of the simulation.

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Figure 9: The total angular error of the spacecraft formation while executing a slew maneuver usingthe various coordinated controllers.

7.3 Summary and Supposition

Two numeric simulations were performed using the five coordinated controllers described in theprevious chapter. The first simulation was an attitude slew maneuver, while the second had theformation track a rigid body in a steady spin. Plots depicting the absolute, relative, and totalformation error were presented.

The plots of the resulting errors contained several interesting trends. The controllers all convergedin about the same amount of time. In both simulations the chain and tree controllers performedthe worst, and the ring controller had the best performance. The relative formation error plots ofboth maneuvers show the lag problems associated with the chain and tree controllers. Surprisingly,the web controller was outperformed by the ring controller in both simulations. The web controllerused complete knowledge of formation’s state and was not able to attain the performance of the ringcontroller, which used only partial knowledge of the formation’s state. Another interesting resultis that the behavior-based controllers did not have lower relative formation error than the one-leader controller, even though one of the behaviors in the design of the behavior-based controllers isformation-keeping.

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Figure 10: The absolute(A) and relative(B) angular formation error of the spacecraft formationwhile executing a steady spin maneuver using the various coordinated controllers.

8 Conclusions

The use of formations of spacecraft will provide future space missions with reliable systems thatare able to provide higher resolution data than missions using a single large spacecraft. Spacecraftformations require coordinated controllers to maintain desired formation configurations and align-ments. Three centralized and two decentralized coordinated controllers were investigated in thiswork. Behavior-based control was implemented in the decentralized controllers.

Numeric simulations were used to compare the performance of the coordinated controllers. It wasfound that the decentralized controller using the ring coordination topology produced the smallesttotal formation error for both the slew and spin maneuvers. A surprising result in the simulationswas that the web controller did not perform as well as the ring controller even with completeknowledge of the formation’s state. The centralized one-leader controller performed the best out ofthe centralized controllers, and was able to match the accuracy of the ring controller ∼ 30 secondsinto both simulations.

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Figure 11: The total angular formation error of the spacecraft formation while executing a steadyspin maneuver using the various coordinated controllers.

References

[1] Masano Aoki and Mu T. Li, Partial reconstruction of state vector in decentralized dynamicsystems, IEEE Transactions on Automatic Control (1973), 289–292.

[2] Tucker Balch and Ronald C. Arkin, Behavior-based formation control for multirobot teams,IEEE Transactions on Robotics and Automation 14 (1998), no. 6, 926–939.

[3] Goddard Space Flight Center, Earth observer 1, http://eo1.gsfc.nasa.gov/miscPages/home.html,February 2002.

[4] E.J. Davison, The decentralized stabilization and control of a class of unknown non-linear time-varying systems, Automatica 10 (1974), 309–316.

[5] His-Han Yeh Kang, Wei and Andrew Sparks, Coordinated control of relative attitude for satelliteformation, Guidance, Naviagtion, and Control Conference and Exhibit (Montreal, Canada),AIAA, August 2001, pp. AIAA–2001–4093.

[6] Wei Kang and Andrew Sparks, Coordinated attitude and formation control of multi-satellitesystems, Guidance, Naviagtion, and Control Conference and Exhibit (Monterey, California),AIAA, August 2002, pp. AIAA–2002–4655.

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[7] Hassan K. Khalil, Nonlinear systems, MacMillan, New York, New York, 1992.

[8] Jonathan R. Lawton and Randal W. Beard, Synchronized multiple spacecraft rotations, Auto-matica 38 (2002), 1359–1364.

[9] Randal W. Beard Lawton, Jonathan and Fred Y. Hadaegh, Elementary attitude formationmaneuvers via leader-following and behavior-based control, Guidance, Naviagtion, and ControlConference and Exhibit (Denver, Colorado), AIAA, August 2000, pp. AIAA–2000–4442.

[10] F. Bauer D. Folta M. Moreau R. Carpenter Leitner, J. and J. How, Formation flight in space,GPS World (2002).

[11] Hanspeter Schaub and John L. Junkins, Analytical mechanics of space systems, AIAA EducationSeries, American Institute of Aeronautics and Astronautics, Reston, Virginia, 2003.

[12] Jason L. Speyer, Computation and transmission requirements for a decentralized linear-quadratic-gaussian control problem, IEEE Transactions on Automatic Control AC-24 (1979),no. 2, 266–269.

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Appendix A: Coordinated Controller Matlab c© Code

function xrn = OneLeader(t,x,xr,n)if n == 1

xrn = xr;else

xrn = [x(1:7); 0; 0; 0];end

function xrn = Chain(t,x,xr,n)xrn = xr;if n == 1

xrn = xr;else

xrn = [x(10*(n-1)-9:10*(n-1)-3); 0; 0; 0];end

function xrn = Tree(t,x,xr,n)xrn = xr;if n == 1

xrn = xr;end if (n == 2) | (n == 3)

xrn = [x(1:7); 0; 0; 0];end if (n == 4) | (n == 5)

xrn = [x(10*2-9:10*2-3); 0; 0; 0];end if (n == 6) | (n == 7)

xrn = [x(10*3-9:10*3-3); 0; 0; 0];end

function xrn = Ring(t,x,xr,i)% Determine the number of SC in the formation.n = length(x) / 10;

% Set relative attitude weightingrho = 0.5;

% Set angular range for formation lockang = 10 * pi/180;

% Check difference between the absolute desired and formation% desired attitude.qn = x(10*i-9:10*i-6);qr = xr(1:4); Rrn = q2R(qr) * q2R(qn)’;[a,phi] = R2aphi(Rrn); xrn = xr;

% Determine desired reference state. If the formation and absolute% attitude are close augment the reference attitude, otherwise% just use the reference attitude.

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if phi < angif i == 1

q_avg = UnitV(x(10*(i+1)-9:10*(i+1)-6) + x(10*n-9:10*n-6));qrn = UnitV((1-rho) * qr + rho * q_avg);wrn = (1-rho) * xr(5:7) + rho * 0.5 * (x(10*(i+1)-5:10*(i+1)-3) + x(10*n-5:10*n-3));xrn = [qrn; wrn; 0; 0; 0];

else if i == nq_avg = UnitV(x(1:4) + x(10*(i-1)-9:10*(i-1)-6));qrn = UnitV((1-rho) * qr + rho * q_avg);wrn = (1-rho) * xr(5:7) + rho * 0.5 * (x(5:7) + x(10*(i-1)-5:10*(i-1)-3));xrn = [qrn; wrn; 0; 0; 0];

elseq_avg = UnitV(x(10*(i+1)-9:10*(i+1)-6) + x(10*(i-1)-9:10*(i-1)-6));qrn = UnitV((1-rho) * qr + rho * q_avg);wrn = (1-rho) * xr(5:7) + rho * 0.5 * (x(10*(i+1)-5:10*(i+1)-3) + x(10*(i-1)-5:10*(i-1)-3));xrn = [qrn; wrn; 0; 0; 0];

endend

function xrn = Web(t,x,xr,i)% Determine the number of SC in the formation.n = length(x) / 10;

% Set relative attitude weightingrho = 0.5;

% Set angular range for formation lockang = 10 * pi/180;

% Check difference between the absolute desired and formation% desired attitude.qn = x(10*i-9:10*i-6); qr = xr(1:4); Rrn = q2R(qr) * q2R(qn)’;[a,phi] = R2aphi(Rrn);

xrn = xr;

% Determine desired reference state. If the formation and absolute% attitude are close augment the reference attitude, otherwise% just use the reference attitude.q_avg = zeros(4,1); w_avg = zeros(3,1); if phi < angfor k = 1:n

q_avg = UnitV(q_avg + x(10*k-9:10*k-6));w_avg = w_avg + 1/n * x(10*k-5:10*k-3);

endqrn = UnitV((1-rho) * qr + rho * q_avg);wrn = (1-rho) * xr(5:7) + rho * 0.5 * w_avg;xrn = [qrn; wrn; 0; 0; 0];

end

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