Passive Decomposition Approach to Formation ... - Semantic Scholar

Dongjun Lee Department of Mechanical, Aerospace and Biomedical Engineering, University of Tennessee at Knoxville, 502 Dougherty Hall, 1512 Middle Drive, Knoxville, TN 37996 e-mail: [email protected]

Perry Y. Li Department of Mechanical Engineering, University of Minnesota, 111 Church Street SE, Minneapolis, MN 55455 e-mail: [email protected]

Passive Decomposition Approach to Formation and Maneuver Control of Multiple Rigid Bodies A passive decomposition framework for the formation and maneuver controls for multiple rigid bodies is proposed. In this approach, the group dynamics of the multiple agents is decomposed into two decoupled systems: The shape system representing internal group formation shape (formation, in short), and the locked system abstracting the overall group maneuver as a whole (maneuver, in short). The decomposition is natural in that the shape and locked systems have dynamics similar to the mechanical systems, and the total energy is preserved. The shape and locked system can be decoupled without the use of net energy. The decoupled shape and locked systems can be controlled individually to achieve the desired formation and maneuver tasks. Since all agents are given equal status, the proposed scheme enforces a group coherence among the agents. By abstracting a group maneuver by its locked system whose dynamics is similar to that of a single agent, a hierarchical control structure for the multiple agents can be easily imposed in the proposed framework. A decentralized version of the controller is also proposed, which requires only undirected line communication (or sensing) graph topology. 关DOI: 10.1115/1.2764507兴 Keywords: multi-agent formation control, rigid-body dynamics, passive decomposition, hierarchical control, decentralized control

1

Introduction

In many applications, a team of multiple agents holds eminent promises to achieve a level of performance, capability, and robustness beyond what a single agent can provide. Some examples include space interferometry 关1–3兴, reconnaissance/surveillance using multiple UAVs 共uninhibited aerial vehicles兲关4兴, underwater assessment 关5兴, mobile sensor network 关6兴, and multirobot cooperation 关7,8兴, to name a few. A basic requirement for such multi-agent systems is to achieve certain desired internal group formation and overall group maneuver at the same time. As a simple example, consider two particles moving on the x-axis. A desired group behavior can be achieved by controlling their formation 共e.g., x1 − x2兲 and their maneuver 1 共e.g., 2 共x1 + x2兲兲, simultaneously, where xi 苸 Re is the x-position of the ith particle 共i = 1 , 2兲. Many control schemes have been proposed for this multi-agent formation control problem, and, according to 关1兴, most of them can be classified into three categories: Behavior-based approach, leader-follower approach, and virtualstructure approach. In the behavior-based approach 关7–11兴, each agent’s control is designed to be the sum of 共or switchings among兲 several prescribed behaviors invoked by local external stimuli 共e.g., distances to neighboring agents兲. The dynamics are designed such that 共s.t.兲, collectively, certain desired formation and maneuver behavior will emerge. Unfortunately, the convergence/stability proof of such collective behavior is often very complicated and difficult. Moreover, typically, only equilibria that correspond to simple behaviors 共e.g., constant heading 关9兴兲 are provable. These approaches are, therefore, not easily adaptable for complicated formations and maneuvers 共e.g., time-varying formation兲. In the leader-follower approach 关12–15兴, an agent of a group is designated as the group leader and its motion alone represents the Contributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 2, 2006; final manuscript received February 15, 2007. Review conducted by Tal Shima.

662 / Vol. 129, SEPTEMBER 2007

overall group maneuver. The leader’s motion is controlled to achieve a desired maneuver, while a desired formation is achieved by controlling the remaining agents 共followers兲 to maintain certain relative distances with respect to 共w.r.t.兲 the leader. The main drawback of the usual leader-follower approach is that a leader has no feedback from the followers 共i.e., formation feedback 关1兴 is absent兲. Thus, the group can show such incoherent behaviors as run-away of a fast leader with slower followers left behind. Another consequence of this unidirectional feedback is poor disturbance rejection 共e.g., a disturbance on the leader can be amplified while propagating throughout the followers 关16兴兲. In the virtual-structure approach 关5,17–19兴, a virtual dynamical system 共simulated in software兲 is used to abstract the group maneuver. To avoid the incoherent group behavior of the leaderfollower approach, they incorporate formation feedback into the virtual system’s dynamics s.t. its evolution is also affected by each agent’s behavior. The desired maneuver and formation are then achieved by controlling the virtual system and agents as a leader and followers. However, simulation of such a virtual system 共that often requires additional differential equation solvers兲 might restrict applicability of this approach when available computing power is scarce 共e.g., distributed control implementation兲. More fundamentally, by relying on artificially simulated dynamics rather than real agents’ states, this virtual system may not capture real group maneuver. For example, consider a virtual point mass which, through a kinematic feedback 共i.e., formation feedback兲, is coupled with a group of point masses that are revolving on a circle periodically. Then, in general, the orbit and phase of this virtual mass would be different than those of the overall group 共e.g., group center of mass兲. As this difference becomes larger 共e.g., with smaller radius and higher speed兲, it would become less suitable to use this virtual system to abstract the overall group maneuver. In this paper, a passive decomposition approach is proposed for the formation and maneuver control of multiple rigid bodies 共e.g., satellites, submarines, and robots兲. Passive decomposition was originally proposed for tele-operators 关20,21兴 and was later generalized for general mechanical systems 关22,23兴. In this paper,

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Fig. 1 Hierarchy by abstraction: Each group 0⌺i abstracted by its locked system 1⵱Li; group of groups 1⌺1 again abstracted by its locked system 2⵱L1; formation among agents/groups described by their shape systems „not shown…

using the passive decomposition, the actual multi-agent sixdegree-of-freedom nonlinear rigid-body dynamics are decomposed into: 共1兲 The shape system describing the internal group formation, and 共2兲 the locked system abstracting the overall group maneuver. By controlling the decoupled locked and shape systems individually, the desired maneuver and formation can then, respectively, be achieved. The shape and locked systems have dynamics similar to that of mechanical systems. Additionally, the total energy of the group’s agents is preserved in the energies of the shape and locked systems. Furthermore, the shape and locked systems can be decoupled from each other in an energetically passive manner 共i.e., does not require input of net energy兲. Recently, the authors in 关3兴 captured a similar idea of formation and maneuver decomposition. However, their decomposition is not passive, and is only applicable for linear dynamics 共e.g., translational dynamics兲. In contrast, our passive decomposition is equally applicable to linear and nonlinear dynamics 共e.g., attitude dynamics兲 and have the advantageous passivity property. Since the passive decomposition allows the locked and shape systems to be perfectly decoupled from each other, the proposed control can achieve tight formation keeping and precise maneuver management without any dynamic crosstalk between them. With this, we can avoid the situations where the internal formation wriggles as the overall group accelerates, or the overall group motion fluctuates as the formation expands/contracts. Formation and maneuver decoupling is not achieved in previous approaches with the exception of 关12–15兴, which, in essence, rely on oftenexpensive acceleration feedback. In contrast, our decoupling requires only position and velocity information, which are more readily available. The locked system abstracts the group maneuver similar to the virtual system in the virtual-structure approach. However, since the locked system 共and also the shape system兲 is directly derived from the agents’ real states and dynamics, our abstraction is free from the aforementioned problems of the virtual-structure approach due to the artificiality. In addition, our framework provides formation feedback so that the group together will automatically slow down or speed up to keep pace with slower or faster agents. Because the locked system itself has dynamics similar to that of a single agent, a group of grouped agents can, in turn, be aggregated into formation by considering the passive decomposition of the set of locked systems. In this way, a hierarchical grouping of agents can be obtained in a very natural way 共Fig. 1兲. This hierarchy Journal of Dynamic Systems, Measurement, and Control

would be useful for controlling a collection of a large number of agents and groups. One remarkable property of the passive decomposition is that the locked and shape decoupling does not change the total energy of the system. This implies that the 共energetic兲 passivity 关24兴 of the group’s open-loop dynamics is not altered by the decoupling. A consequence of this is that, as one agent is actuated or mechanically perturbed by its environment, the decoupling control serves to distribute this energy input from the environment among other agents. Such passivity property would be useful for orbital formation flying, as formation and maneuver can then be decoupled without affecting the orbits and the periodicity of the group motion. This passivity property would also enhance the coupling stability 关25兴 of the group when it interacts with other groups to form a larger group 共e.g., through the hierarchy in Fig. 1兲 or physically interacts with real external systems 共e.g., human astronaut or robonaut 关26兴兲. In this paper, we focus mainly on the formation and maneuver aspects of the passive decomposition. For applications where this passivity property is emphasized, refer to 关20,21,27兴. Direct implementation of the proposed control scheme generally requires the state information of all the agents 共e.g., via centralized communication兲. To relax this communication requirement, a decentralized controller is also proposed here, which only requires undirected line communication 共or sensing兲 graph topology among the agents. We also show that this decentralized control has performance similar to that of the centralized control when desired group behavior is not too aggressive. The rest of the paper is organized as follows. The control problem is formulated in Sec. 2. The passive decomposition is derived in Sec. 3. Design of the centralized control is given in Sec. 4, and its decentralization is presented in Sec. 5. Section 6 presents an extension of the results to the equivalence class of the formation variables and Sec. 7 presents the simulation results. Section 8 contains some concluding remarks.

2

Problem Formulation

2.1 System Modeling. We consider a group of m-agents that have the dynamics of a fully actuated 6-DOF flying rigid body 关19兴. The translational dynamics of the ith agent with respect to a common inertial frame Fo is then given by the following 3-DOF linear point-mass dynamics: i = 1 , . . . , m, 共1兲

mix¨ i = ti + fi

where mi 苸 Re is the 共constant兲 mass, and xi = 关xi , y i , zi兴苸 Re3, ti 苸 Re3, and fi 苸 Re3 are the position, control 共to be designed兲, and environmental disturbance 共e.g., gravitational force, drag, etc.兲 w.r.t. Fo, respectively. Following 关28,29兴, we also model the attitude dynamics of the ith agent w.r.t. the inertia frame Fo by the following 3-DOF nonlinear dynamics 关30兴: i = 1 , . . . , m, +

T

˙ i + Q i共 ␨ i, ␻ i兲 ␻ i = ␶ i + ␦ i H i共 ␨ i兲 ␻

共2兲

where ␨i = 关␾i , ␪i , ␺i兴苸 Re are the roll, pitch, and yaw angles with −␲ / 2 ⬍ ␪i ⬍ ␲ / 2, ␻i共t兲 = 共d / dt兲␨i共t兲苸 Re3 is the angular rate, ␶i , ␦i 苸 Re3 are the control 共to be designed兲 and disturbance w.r.t. Fo, respectively. Also, Hi共␨i兲苸 Re3⫻3 and Qi共␨i , ␨˙ i兲苸 Re3⫻3 are the symmetric and positive-definite inertia matrix and the Coriolis ˙ 共␨ 兲 − 2Q 共␨ , ␨˙ 兲 is skew symmetric. In this paper, we matrix s.t. H i i i i i mainly consider the agents whose translation and attitude dynamics are derived w.r.t. a common inertial frame Fo as in 共1兲 and 共2兲. How to extend the presented results in coordinates to the case where each agent’s dynamics is given w.r.t. its own body frame 共or other frames兲 is a topic for future research. T

3

2.2 Formation and Maneuver Control Objectives. For the group of m-agents 共1兲 and 共2兲, we define the formation and maneuver variables s.t. SEPTEMBER 2007, Vol. 129 / 663


pE共x1共t兲,x2共t兲, . . . ,xm共t兲, ␨1共t兲, ␨2共t兲, . . . , ␨m共t兲兲苸 Re6m−p 共3兲 pL共x1共t兲,x2共t兲, . . . ,xm共t兲, ␨1共t兲, ␨2共t兲, . . . , ␨m共t兲兲苸 Rep

共4兲

where pE : Re6m → Re6m−p and pL : Re6m → Rep are smooth maps with full rank Jacobian 共i.e., smooth submersion兲 and p is a positive integer less than 6m. We suppose that pE and pL are designed s.t. for a given mission purpose, they can represent formation and maneuver aspects among the group agents, respectively. The formation and maneuver control objectives can then be written as pL共t兲 → pLd共t兲

共5兲

pE共t兲 → pEd共t兲

共6兲

where pdE共t兲苸 Re6m−p, pLd共t兲苸 Rep are desired formation and maneuver trajectories, respectively. For an illustration, let us consider two agents having 3-DOF motion 共translation and yaw rotation兲 on the 共x , y兲-plane. One example of the formation and maneuver variables pair is then

冢

x1 − x2 − L cos共␺1兲

pE共t兲 ª y 1 − y 2 − L sin共␺1兲 ␺1 − ␺2

pL共t兲 ª

冣

冢冣 1 共x1 + x2兲 2 1 共y 1 + y 2兲 2 1 共 ␺ 1 + ␺ 2兲 2

苸 Re3

other structures for 共8兲 and 共9兲 would also be possible. See Sec. 6 for an extension of the results to a certain equivalence class of the formation variables 共8兲 and 共9兲. The structures of 共8兲 and 共9兲共and those in Sec. 6兲 are chosen for the following reasons: 共1兲 With these formation variables, as to be shown in Sec. 3, we can achieve a simple 共closed-form兲 expression for the passive decomposition with which concepts can be presented more efficiently and clearly; and 共2兲 we can exploit the flat properties of the translation dynamics 共1兲共i.e., Riemannian curvature vanishes everywhere 关31兴兲 and of the formation variable 共8兲共i.e., its level sets are given by flat planes兲. With these flat properties, as shown in Sec. 4.1, the locked system of the translation dynamics 共1兲 will have the usual point mass dynamics whose configuration and mass are given by the position of the center of mass and the sum of all agents’ mass, respectively. As shown in Sec. 5, the structures of the formation variables 共8兲 and 共9兲 implicitly define the required communication topology. Remark 1. Using the coordinate-independent results of 关22,23兴, instead of 共1兲, 共2兲, 共8兲, and 共9兲, we can also incorporate different formulations of the group dynamics 共e.g., derived w.r.t. an orbiting frame or unit quaternion for SO共3兲兲 and general holonomic constraints 共e.g., pL , pE in 共3兲 and 共4兲兲, respectively. However, expressions for the passive decomposition and control might be complicated.

and

3

苸 Re3

共7兲

where xi , y i , ␺i are the positions and the yaw angle of the ith agent 共i = 1 , 2兲, and L is a positive scalar defining distance between them. If we achieve the formation objective 共6兲 with pdE共t兲 = 0, then the two agents will have aligned yaw angles and form a rod-like shape of length L, whose orientation is determined by the first agent’s yaw angle. In addition, by enforcing the maneuver objective 共5兲, we can drive this rod’s 3-DOF configuration 共i.e., location on the 共x , y兲-plane and orientation兲 as specified by the target trajectory pLd共t兲. In this paper, we consider separate formation and maneuver variables for the translation and attitude dynamics 共1兲 and 共2兲 s.t. pL共t兲 = 关xLT共t兲 , ␨LT共t兲兴T 苸 Re6 and pE共t兲 = 关xTE共t兲 , ␨TE共t兲兴T 苸 Re6共m−1兲, where x쐓 and ␨쐓 共쐓 苸兵L , E其兲 are functions of 共x1 , . . . , xm兲 and 共␨1 , . . . , ␨m兲, respectively. In particular, we use the following simple but versatile formation variables:

The Passive Decomposition

In this section, we use the passive decomposition 关22,23兴 to decompose the multi-agent group dynamics into two decoupled systems: The shape system representing internal group formation as given by the formation variables 关8兴 and 关9兴, and the locked system abstracting overall group maneuver. To derive a unified decomposition for the translation and attitude dynamics 共1兲 and 共2兲 under the formation variables 共8兲 and 共9兲, we consider the following group dynamics of m n-DOF mechanical systems:

M1共q1兲q¨ 1 + C1共q1,q˙ 1兲q˙ 1 = T1 + F1 M2共q2兲q¨ 2 + C2共q2,q˙ 2兲q˙ 2 = T2 + F2 ] Mm共qm兲q¨ m + Cm共qm,q˙ m兲q˙ m = Tm + Fm

共10兲

under the following formation variable 共holonomic constraint兲

T T T xE ª 关xT1 − xT2 ,xT2 − xT3 , . . . ,xm−1 − xm 兴苸 Re3共m−1兲

共8兲

T T T − qm 兴苸 Re共m−1兲n qET ª 关qT1 − qT2 ,qT2 − qT3 , . . . ,qm−1

T T T ␨E ª 关␨T1 − ␨T2 , ␨T2 − ␨T3 , . . . , ␨m−1 − ␨m 兴苸 Re3共m−1兲

共9兲

where qi 苸 Ren and q˙ i 苸 Ren are the configuration and the velocity, Ti 苸 Ren and Fi 苸 Ren are the controls and the disturbances, and Mi共qi兲苸 Ren⫻n and Ci共qi , q˙ i兲苸 Ren⫻n are the symmetric and positive-definite inertia matrix and the Coriolis matrix s.t. 共d / dt兲Mi共qi兲 − 2Ci共qi , q˙ i兲 are skew symmetric, respectively. Following the tangent-space decomposition in 关22,23,32兴, the T T velocity 共i.e., tangent vector兲 q˙ ª 关q˙ T1 , q˙ T2 , . . . , q˙ m 兴苸 Remn of the group dynamics 共10兲 can then be decomposed into the locked system velocity vL 苸 Ren and the shape system velocity vE 苸 Re共m−1兲n s.t.

i.e., relative positions and attitudes between two consecutive agents. The maneuver variables xL共t兲苸 Re3 and ␨L共t兲苸 Re3 will be defined in Sec. 4 so that their dynamics are decoupled from the dynamics of the formation variables 共8兲 and 共9兲. pLd共t兲 , pdE共t兲 in 共5兲 and 共6兲 can also then be partitioned s.t. pLd共t兲 = 关xLdT共t兲 , ␨LdT共t兲兴T dT T 6共m−1兲苸 Re6 and pdE共t兲 = 关xdT where xLd共t兲 , ␨Ld共t兲 E 共t兲 , ␨E 共t兲兴苸 Re d d 3 3共m−1兲苸 Re and xE共t兲 , ␨E共t兲苸 Re are desired maneuver and formation for the group translation and attitude, respectively. Of course, 664 / Vol. 129, SEPTEMBER 2007

共11兲



共12兲

T T where q ª 关qT1 , qT2 , . . . , qm 兴苸 Remn = 1 , . . . , m兲 is given by

and

␾i共q兲苸 Ren⫻n

共i

␾i共q兲 ª 关M1共q1兲 + M2共q2兲 + … . + Mm共qm兲兴−1Mi共qi兲共13兲 Using the fact that ␾i共q兲 in 共13兲 is nonsingular and the property that

␾1共q1兲 + ␾2共q2兲 + ¯ + ␾m共qm兲 = I

共14兲

⌺i共q兲 = ␾i共q兲 + ␾i+1共q兲 + ¯ + ␾m共q兲苸 Ren⫻n

so that ⌺1共q兲 = I and ⌺m共q兲 = ␾m共q兲 from the definition of ␾i共q兲 in 共13兲. With the decomposition in 共12兲, the group inertia of 共10兲 is now block diagonalized s.t. S−T共q兲diag关M1共q1兲,M2共q2兲, ¯ Mm共qm兲兴S−1共q兲

we can show that S共q兲 in 共12兲 is nonsingular. From 共12兲, the shape system velocity vE 苸 Re共m−1兲n is given by d qE共q1共t兲,q2共t兲, . . . ,qm共t兲兲 dt

vE共t兲 =

共15兲

i.e., time derivative of the formation variable qE in 共11兲. Thus, having qE as its configuration, the shape system would explicitly describe the formation aspect. In addition, with the definition 共13兲, the locked system velocity vL in 共12兲 is given by

冋兺册 m

vL =

−1

Mi共qi兲

关M1共q1兲q˙ 1 + M2共q2兲q˙ 2 ¯ + Mm共qm兲q˙ m兴

i=1

共16兲 i.e., average of all agents’ velocities with each agent’s inertia being the weighting. Notice that if the inertia matrices Mi共qi兲 are all scalar constants, this vL in 共16兲 will become the velocity of center of mass, i.e., vL =

冉

d M 1q 1 + M 2q 2 + . . . M mq m dt M1 + M2 + . . . Mm

冢冣冢冣冢冣冢冣 ↑

TE

T2

ª S−T共q兲

]

↓

F1

↑

and

ª S−T共q兲

FE ↓

Tm

F2 ]

共17兲

Fm

共m−1兲n

where S 共q兲 = 共S 共q兲兲 , and FL , TL 苸 Re and FE , TE 苸 Re are the effects of environmental forcing and controls on the locked and shape systems, respectively. Here in 共17兲, the matrix S−1共q兲苸 Remn⫻mn is found to be −T

S−1共q兲 =

where

−1

冤

T

⌺2共q兲

n

⌺3共q兲

¯

⌺m共q兲

⌺3共q兲

¯

⌺m共q兲

I ⌺2共q兲 − I ⌺3共q兲 − I

⌺m共q兲

I

I ⌺2共q兲 − I ]

]

]

冋

ML共q兲

0

0

ME共q兲

册

共20兲

where ML共q兲苸 Ren⫻n and ME共q兲苸 Re共m−1兲n⫻共m−1兲n are the 共symmetric and positive definite兲 inertia matrices for the locked and shape systems, respectively. This inertia block-diagonalizing property 共20兲 implies that the total kinetic energy of the m-mechanical systems 共10兲 is also decomposed into the sum of those of the locked and the shape systems 共see Proposition 1 below兲. From 共20兲 with 共18兲, we can show that ML共q兲 ª M1共q1兲 + M2共q2兲 + ¯ . + Mm共qm兲苸 Ren⫻n 共21兲 i.e., the locked system inertia is given by the sum of those of all agents. Using the decomposition 共12兲 and 共17兲 with the inertia blockdiagonalizing property 共20兲, the group dynamics 共10兲 can then be partially decoupled s.t. 共22兲共23兲

FL

T1

¬

冊

Therefore, with vL in 共16兲 as its velocity, the locked system would abstract overall group maneuver. We also define the compatible decompositions of 共12兲 s.t. TL

共19兲

]

I ⌺2共q兲 − I ⌺3共q兲 − I ¯ ⌺m共q兲 − I

冥

共18兲

Journal of Dynamic Systems, Measurement, and Control

where we use the following definition

冋

CL CLE CEL CE

册

ª S−Tdiag关M1,M2, ¯ Mm兴

d −1 共S 兲 dt

+ S−Tdiag关C1,C2, ¯ Cm兴S−1

共24兲

with arguments being omitted for brevity. We call the n-DOF system in 共22兲 the locked system, which abstracts the overall group maneuver with vL in 共16兲 and ML共q兲 in 共21兲 as its velocity and inertia, respectively. Using 共17兲 and 共18兲, we can show that FL = F1 + F2 + ¯ + Fm, i.e., the effects of external disturbances on the locked system is given by the sum of those of the individual agents. We also call the 共m − 1兲n-DOF system in 共23兲 the shape system, which explicitly describes internal group formation by having the formation variable qE 共11兲 as its configuration. Due to the inertia block-diagonalizing property 共20兲, the couplings between the locked and shape systems are only through the terms CLE共q , q˙ 兲q˙ E and CEL共q , q˙ 兲vL in 共22兲 and 共23兲, that are functions of q and q˙ . Thus, the locked and shape systems can be decoupled without utilizing acceleration feedback. SEPTEMBER 2007, Vol. 129 / 665


PROPOSITION 1. The partially decoupled dynamics 共22兲 and 共23兲 have the following properties: 1. ML共q兲 and ME共q兲 are symmetric and positive definite. Moreover, total kinetic energy of the group (10) is decomposed into the sum of those of the shape and locked systems, s.t. m

␬共t兲 ª

1

兺 2 q˙ M 共q 兲q˙ T i

i

i

i=1

T i

1 1 = vLTML共q兲vL + q˙ ETME共q兲q˙ E 2 2

共25兲 ˙ ˙ 2. ML共q兲 − 2CL共q , q˙ 兲 and ME共q兲 − 2CE共q , q˙ 兲 are skew symmetric; T 3. CLE共q , q˙ 兲 + CEL 共q , q˙ 兲 = 0; 4. Total environmental and control supply rates are decomposed into the sum of those of the locked and shape systems, i.e.,

Fig. 2 A circuit-network representation of the passive decom␬L„q , vL… ª 21 vTLML„q…vL and ␬E„q˙ , vE… position, where 1 T ª 2 q˙EME„q…q˙E

m

s p共t兲 ª

兺 F q˙ = F v T i i

T L L

+ FETq˙ E

and

i=1

m

sc共t兲 ª

兺 T q˙ = T v T i i

T L L

+ TETq˙ E

共26兲

共27兲

i=1

Proof. Item 1 is a direct consequence of 共20兲. In order to prove items 2 and 3, let us define M共q兲 and C共q , q˙ 兲 ª diag关M1共q1兲 , M2共q2兲 , ¯ Mm共qm兲兴 ª diag关C1共q1 , q˙ 1兲 , C2共q2 , q˙ 2兲 , ¯ Cm共qm , q˙ m兲兴. Using 共20兲 and ˙ − 2C = −关M ˙ − 2C兴T兲, we ˙ = C + CT 共from M 共24兲 with the fact that M then have

˙ 共q兲 where we omit the arguments to avoid cluster. Thus, M L ˙ − 2CL共q , q˙ 兲 and ME共q兲 − 2CE共q , q˙ 兲 are skew symmetric and T CLE共q , q˙ 兲 = −CEL 共q , q˙ 兲. Item 4 can also be easily shown by using 共12兲 and 共17兲. 䊏 Following Proposition 1, ML共q兲 , ME共q兲 and CL共q , q˙ 兲 , CE共q , q˙ 兲 in 共22兲 and 共23兲 can be thought of as inertia and Coriolis matrices of the locked and shape systems. Thus, with the cancelation of the coupling terms via the controls TL and TE, the original mn-DOF group dynamics 共10兲 can be decomposed into the n-DOF locked and 共m − 1兲n-DOF shape systems, whose dynamics are decoupled and similar to that of the usual mechanical systems. A variety of control techniques developed for general mechanical systems can then be utilized 共e.g., passivity-based control兲. One remarkable property of the passive decomposition is that the decoupled system has the same passivity property as the 共passive兲 open-loop system with the kinetic energy ␬共t兲 in 共25兲 and the environmental power as the storage function and the supply rate 关24兴, i.e., with the decoupling control TL = CLE共q , q˙ 兲q˙ E and TE = CEL共q , q˙ 兲vL, 666 / Vol. 129, SEPTEMBER 2007

where the last equality is from item 4 of Proposition 1. Note that this equality 共27兲 is also satisfied by the open-loop group dynamics 共10兲 with Ti = 0. In the orbital formation flying, with the environmental power supplied/extracted by the gravitation field, this equality 共27兲 will determine the periodic group motion. Therefore, we would be able to decouple the formation and maneuver from each other without affecting periodicity and orbits of the group motion. This enforced passivity 共27兲 would also enhance coupled stability of the group, when it forms a large interconnected system with other 共passive兲 agents/groups, or it is physically coupled with 共passive兲 external systems, as a feedback interconnection of passive systems is necessarily stable 关25兴. This passivity property has been successfully exploited in some robotic applications 关20–23,27兴, and its application for multi-agent formation control is a future research topic. A circuit network representation of the decomposed dynamics 共22兲 and 共23兲 is given in Fig. 2, where 共1兲 the locked and shape systems are both passive; 共2兲 the coupling is energetically conservative 共as shown in 共27兲兲; and 共3兲 the total group kinetic energy and control/environmental supply rates are decomposed. In 关22,23兴, the passive decomposition is developed in the geometric setting 共i.e., coordinate invariant兲 for multiple mechanical systems under general holonomic constraints. Thus, using the results of 关22,23兴, any formulations of mechanical system dynamics 共e.g., w.r.t. a rotating frame or unit quaternion for motion in SO共3兲兲 and holonomic constraints 共e.g., distances among the agents兲 can be used as the group dynamics and the formation variable. However, in this paper, to make the presentation clearer with a simple decomposition 共12兲共and 共18兲兲, we confine our attention to the dynamics and the formation variable given by 共10兲 and 共11兲. Geometrically, the locked system velocity vL defines the projection of the group velocity to the level set Hc共t兲 ª 兵p 苸 Remn 兩 qE共p兲 = qE共q共t兲兲其, while the shape system velocity vE defines its orthogonal complement w.r.t. the inertia matrix 共Riemannian metric兲 as shown by the inertia block-diagonalizing property 共20兲. If vE = 0 共i.e., qE is constant兲, the shape system dynamics and the coupling term CLE共q , q˙ 兲q˙ E will vanish, and the locked system dynamics and the coupling term CEL共q , q˙ 兲vL will become the constrained dynamics 共Levi-Civita connection兲 on Transactions of the ASME


Hc共t兲 and the second fundamental form, respectively 关33兴. For more details on the geometric property of the passive decomposition, refer to 关22,23兴.

4 Centralized Maneuver and Formation Control Design Once the group dynamics is decomposed as in 共22兲 and 共23兲, design of formation and maneuver controls becomes fairly straightforward, as the dynamics of the shape system 共i.e., formation兲 and the locked system 共maneuver兲 are similar to that of usual mechanical systems. These controls, which will be designed in this section, however, generally require the state information of all group agents 共e.g., via centralized communication兲. In Sec. 5, for applications where such communication requirement is not achievable, we will provide a controller decentralization which only requires undirected line communication graph topology among the agents. See Sec. 7.1 for a detailed procedure of applying the centralized control to a specific example 共formation flying of three rigid bodies兲. 4.1 Control Design for Group Translation. From 共16兲, the locked system velocity of the group translation is given by vL =

m1x˙ 1 + m2x˙ 2 + ¯ + mmx˙ n 苸 Re3 m1 + m2 + ¯ + mm

共28兲

thus, we can define the translation locked system configuration by xL ª

m 1x 1 + m 2x 2 + ¯ + m mx m 苸 Re3 m1 + m2 + ¯ + mm

共29兲

i.e., the position of the center of mass. We choose this xL as the maneuver variable for the group translation so that the maneuver and formation aspects are decoupled from each other. With this map xL : Re3m → Re3 共29兲, we can project the translation locked system dynamics on Re3. As shown in 关22,23兴, this projection is possible since the manifold of the group translation dynamics 共1兲 is flat 共i.e., Riemannian curvature vanishes 关31兴 as shown by the constant mass mi on its flat configuration space Re3兲, and the level sets of the formation variable 共8兲 are also given by flat planes. For more details, refer to 关22,23兴. Following 共22兲 and 共23兲 with the definition of xL in 共29兲, the group translation dynamics 共1兲 is decomposed s.t. MLx¨ L = tL + fL

共30兲

MEx¨ E = tE + fE

共31兲

where xL 苸 Re and xE 苸 Re are the locked and shape systems configurations given in 共29兲 and 共8兲, ML = 共m1 + m2 + ¯ + mm兲I3⫻3 and ME 苸 Re3共m−1兲⫻3共m−1兲 are the symmetric and positive definite mass matrices, tL 苸 Re3 and tE 苸 Re3共m−1兲 are the controls 共to be designed兲, and fL = f1 + f2 + ¯ + fm 苸 Re3, fE 苸 Re3共m−1兲 are the environmental disturbances on the translation locked and shape systems, respectively. To achieve the control objectives 共5兲 and 共6兲, we design the locked and shape system controls tL , tE to be 3

共32兲

tE ª MEx¨ Ed共t兲 − KvE关x˙ E − x˙ Ed共t兲兴 − KEp 关xE − xEd共t兲兴

共33兲

xdE共t兲苸 Re3共m−1兲

are the desired translation mawhere neuver and formation trajectories, KLv , KLp 苸 Re3⫻3 and KEv , KEp 苸 Re3共m−1兲⫻3共m−1兲 are symmetric and positive-definite proportional-derivative 共PD兲 control gains. The environmental disturbance fi in 共1兲 can be compensated for either by individual local controllers or by incorporating cancelation of fL , fE of 共30兲 and 共31兲 into the controls 共32兲 and 共33兲. Journal of Dynamic Systems, Measurement, and Control

˙ L + QL共␨, ␻兲␻L + QLE共␨, ␻兲␨˙ E = ␶L + ␦L H L共 ␨ 兲 ␻

共34兲

HE共␨兲␨¨ E + QE共␨, ␻兲␨˙ E + QEL共␨, ␻兲␻L = ␶E + ␦E

共35兲

where each term is defined according to its counterparts in 共22兲 T T and 共23兲 with ␨ ª 关␨T1 , ␨T2 , … , ␨m 兴苸 Re3m and ␻共t兲 ª d / dt␨共t兲 3m 苸 Re . Here, from 共16兲, the locked system velocity ␻L共t兲苸 Re3 is given by

冋兺册 m

␻L ª

H j共␨ j兲

−1

关H1共␨1兲␻1 + H2共␨2兲␻2 + ¯ + Hm共␨m兲␻m兴

j=1

共36兲 In contrast to the translational locked system 共30兲, as shown in 关22兴, the attitude locked system 共34兲 does not have a well-defined configuration on Re3 in the sense that there does not exist a function ␨L共t兲苸 Re3 s.t. 共d / dt兲␨L = ␻L 共i.e., ␻L is a pseudo-velocity 关34兴兲. Therefore, the maneuver control objective described by a 共position兲 trajectory ␨Ld共t兲苸 Re3 would not be achievable by the locked system 共34兲. The lack of this projection is due to the fact that the distribution, which is orthogonal to the level sets of the formation variable 共9兲 w.r.t. the inertia metric is not integrable. For more detail, see 关22,23兴. This maneuver trajectory tracking, however, can still be achieved if the group dynamics 共2兲 is confined in a single level set of the formation variable 共9兲 and the desired maneuver trajectory ␨Ld共t兲 is defined on this level set. Thus, when the position of the locked system 共34兲 needs to be controlled, we restrict the desired formation ␨dE to be constant and define ␨Ld共t兲 on the target level set Hd ª 兵共␨1 , ␨2 , . . . , ␨m兲苸 Re3m 兩 ␨E共␨1 , ␨2 , . . . , ␨m兲 = ␨dE其. Since the relative attitude between m-agents is rigidly fixed on Hd 共i.e., ␨E is constant兲, the total group maneuver on Hd can be specified by the attitude of any agent on Hd. Due to this property, we can design ␨Ld共t兲 as a desired trajectory for any one of the agents. For example, if the first agent is chosen, ␨Ld共t兲 will be designed s.t. the desired maneuver achieved by ␨1共t兲 → ␨Ld共t兲 on Hd. While the desired maneuver trajectory ␨Ld共t兲 is defined on Hd, the control for it must operate even when ␨E共t兲 = ␨dE is not perfectly achieved. To do this, we define the following map ˜␨ : 共␨ , ␨ , ¯ ␨ 兲哫 Re3 s.t. L 1 2 m ˜␨ 共t兲 ª A ␨ 共t兲 + A ␨ 共t兲 ¯ + A ␨ 共t兲 + b 苸 Re3 L 1 1 2 2 m m

3共m−1兲

tL ª MLx¨ Ld共t兲 − KvL关x˙ L − x˙ Ld共t兲兴 − KLp 关xL − xLd共t兲兴

xLd共t兲苸 Re3,

4.2 Control Design for Group Attitude. Following 共22兲 and 共23兲, the group attitude dynamics 共2兲 with the formation variable ␨E共t兲苸 Re3共m−1兲共9兲 is decomposed s.t.

共37兲

m 兺k=1 Ak = I

where Ak 苸 Re3⫻3 are full-rank matrices s.t. to ensure ˜ 共d / dt兲␨L = ␻L, when 共␨1 , ␨2 , . . . , ␨m兲苸 Hd 共i.e., on Hd where ␻1 = ␻ ¯ = ␻ , from 共36兲 and 共37兲 with 兺m A = I, 共d / dt兲˜␨ = ␻ 2

m

k=1

k

L

L

= ␻i, ∀i 苸兵1 , . . . , m其兲. Also, the vector b 苸 Re3 in 共37兲 is designed to enforce consistency between the map ˜␨ and the designed traL

jectory ␨Ld共t兲 on Hd 共e.g., if ␨Ld共t兲 is designed for the first agent, b will be chosen s.t. ˜␨L共t兲 = ␨1共t兲 when 共␨i , ␨2 , . . . , ␨m兲苸 Hd兲. We choose this map ˜␨L共t兲共37兲 as the attitude maneuver variable. We can then rewrite the maneuver control objective 共5兲 for the attitude dynamics s.t. ˜␨ 共t兲 → ␨d 共t兲 L L

on Hd

共38兲

Geometrically, with this map ˜␨L in 共37兲, the desired maneuver ␨Ld共t兲 is lifted from Hd to the ambient configuration space 共Re3m兲 of the group attitude dynamics 共2兲. Therefore, control action can be defined in the ambient manifold for this lifted desired maneuver 共see 共39兲兲. Since the map ˜␨L is smooth, this defined control SEPTEMBER 2007, Vol. 129 / 667


will smoothly converge to the desired one on Hd as ␨E共t兲 → ␨dE. Using the maneuver variable ˜␨L共t兲 in 共37兲 as a “pseudoconfiguration” for the locked system 共34兲, we design the attitude locked and shape system controls to be

␶L = QLE共␨, ␻兲␨˙ E共t兲 + HL共␨兲␨¨ Ld共t兲 + QL共␨, ␻兲␨˙ Ld共t兲 − ␭vL共␻L共t兲 − ␨˙ Ld共t兲兲 − ␭Lp 共˜␨L共t兲 − ␨Ld共t兲兲

兩␻ik共t兲 − ␻kj 共t兲兩艋 ␣e−␥t where

␻ki 共t兲

共42兲

is the kth component of ␻i共t兲苸 Re . Thus, using 共42兲 3

with the definitions of ␻L and ˜␨L in 共36兲 and 共37兲 and the boundedness of Ai and Hi共␨i兲, we can show that there exists a finite scalar ␤ ⬎ 0 s.t. 共with arguments omitted兲共43兲

共39兲

␶E = QEL共␨, ␻兲␻L共t兲 + HE共␨兲␨¨ Ed共t兲 + QE共␨, ␻兲␨˙ Ed共t兲 − ␭vE共␨˙ E共t兲 − ␨˙ Ed共t兲兲 − ␭Ep 共␨E共t兲 − ␨Ed共t兲兲

共40兲

where QLE共␨ , ␻兲␨˙ E and QEL共␨ , ␻兲␻L are the decoupling controls, ␨dE共t兲苸 Re3共m−1兲 is the desired formation trajectory, ␨Ld共t兲苸 Re3 is the desired maneuver profile defined on the target level set Hd, and ␭Lv , ␭Lp 苸 Re3⫻3 and ␭Ev , ␭Ep 苸 Re3共m−1兲⫻3共m−1兲 are symmetric and positive-definite PD-control gains. The environmental disturbances ␦i in 共2兲 can be compensated for either by individual local controllers or by incorporating cancellation of ␦L , ␦E of 共34兲 and 共35兲 into the attitude locked and shape controls 共39兲 and 共40兲. When the maneuver tracking 共38兲 is desired, we can utilize large PD-control gains ␭Ev , ␭Ep in 共40兲共with a constant ␨dE兲 so that the group attitude quickly converges to Hd, and, from there, the locked control 共39兲 will enforce the desired maneuver as specified by 共38兲. The desired formation ␨dE共t兲 can still be time-varying when we do not need to explicitly control the position of the locked system 共e.g., velocity tracking ␻L共t兲 → ␻Ld共t兲兲. The properties of the 共stabilizing兲 centralized controls 共32兲, 共33兲, 共39兲, and 共40兲 are summarized in the following theorem. THEOREM 1. 共1兲 Consider the group translation dynamics 共1兲 under the centralized control 共32兲 and 共33兲. Suppose that the disturbances fi are negligible (e.g., by local cancelation). Then, 共x˙ L共t兲 − x˙ Ld共t兲 , xL共t兲 − xLd共t兲兲 → 0 and 共x˙ E共t兲 − x˙ dE共t兲 , xE共t兲 − xdE共t兲兲 → 0 exponentially. 共2兲 Consider the group attitude dynamics 共2兲 under the centralized control 共39兲 and 共40兲. Suppose that the inertia matrix Hi共␨i兲 and the Coriolis matrix Qi共␨i , ␻i兲 are bounded. Suppose further that the disturbances ␦i are negligible (e.g., by local cancelation) and the target formation ␨dE is constant. ˙ Then, 共˜␨ 共t兲 − ␨˙ d 共t兲 ,˜␨ 共t兲 − ␨d 共t兲兲 → 0 and 共␨˙ 共t兲 , ␨ 共t兲 − ␨d 兲 L

L

→ 0 exponentially.

L

E

L

E

E

E

E

nentially, since 共␨˙ E , ␨E − ␨dE兲 → 0 exponentially, QL共␨ , ␨˙ 兲 is bounded, and HL共␨兲 is bounded and positive definite. Thus, simi˙ ki 共t兲 − ␻ ˙ kj 共t兲兩 lar to 共42兲, there exists a finite scalar ¯␣ ⬎ 0 s.t. 兩␻ − ␥ t 艋 ¯␣e for any i , j = 1 , . . . , m. Therefore, similar to 共43兲, we can find a finite scalar ¯␤ ⬎ 0 s.t. for any i 苸兵1 , . . . , m其:

共44兲 where we use the fact that 共1兲 ∀i , j 苸兵1 , . . . , m其, ␻i − ␻ j → 0, ex˙ j = 0, as 兺m ␾ j = I; and 共3兲 ponentially 共from 共42兲兲; 共2兲兺mj=1␾ j=1 ˙ − H−1H ˙ H−1H is bounded, since H ˙ = Q + QT ˙␾ j共␨兲 = H−1H j L L j 쐓 쐓 쐓 L L T ˙ − Q = −关H ˙ − Q 兴兲 is bounded with bounded Q 共쐓 共from H 쐓

쐓

쐓

쐓

쐓

苸兵1 , . . . , m , L其兲. Thus, with 共43兲 and 共44兲 and the bounded H쐓 , Q쐓 共쐓 苸兵1 , . . . , m , L其兲, the locked system dynamics 共34兲 under the control 共39兲 can be written as: ¨ ˙ ˙ HL共␨兲共˜␨L − ␨¨ Ld兲 + QL共␨, ␻兲共˜␨L − ␨˙ Ld兲 + ␭vL共˜␨L − ␨˙ Ld兲 + ␭Lp 共˜␨L − ␨Ld兲 = ␦L + d共t兲

共45兲

where d共t兲苸 Re is an exponentially decaying vector s.t. 兩d共t兲兩艋 De␥t with a finite scalar D 艌 0. Thus, if ␦i = 0, the locked system dynamics 共45兲 is the exponentially stable dynamics 关35兴 with the ˙ exponentially decaying disturbance d共t兲. Thus, 共˜␨ − ␨˙ d ,˜␨ − ␨d 兲 3

Proof. 共1兲 This is a direct consequence of the fact that the translation locked and shape systems 共30兲 and 共31兲 under the centralized control 共32兲 and 共33兲 are given by the linear time invariant 共LTI兲 mass-spring-damper dynamics. 共2兲 With a constant ␨dE, the shape system dynamics 共35兲 under the centralized control 共40兲 is given by: HE共␨兲␨¨ E + QE共␨, ␻兲␨˙ E − ␭vE␨˙ E共t兲 − ␭Ep 共␨E共t兲 − ␨Ed共t兲兲 = ␦E 共41兲 The exponential convergence of 共␨˙ E , ␨E − ␨dE兲 → 0 with bounded HE共␨兲 and QE共␨ , ␻兲 and ␦i = 0 can then be proved by using the 1 1 Lyapunov function V 共t兲 = ␨˙ T H ␨˙ + ␨T ␭E␨ + ⑀␨T H ␨˙ , where ␨

where i is any integer in 兵1 , . . . , m其, and Akj and ␾kj 共␨兲 are the kth row vector 共k = 1 , . . . , 3共m − 1兲兲 of A j and ␾ j共␨兲 ª HL−1共␨兲H j共␨ j兲. Here 关兺mj=1共Akj − ␾kj 共␨兲兲兴 = 0, since 兺mj=1A j = 兺mj=1␾ j共␨兲 = I. From 共41兲 with ␦ = 0, we can also show that ␨¨ 共t兲 → 0 expo-

2 E

E E

2 E p E

E

E E

⑀ ⬎ 0 is a small scalar s.t. V␨共t兲 is positive definite. For more details, refer to p. 194 of 关35兴. Let us denote the exponential convergence rate of 共41兲 by ␥ ⬎ 0. Since ␨˙ E → 0 exponentially, from the definition 共9兲, there then exists a finite scalar ␣ ⬎ 0 s.t. for all i , j = 1 , . . . , m: 668 / Vol. 129, SEPTEMBER 2007

L

L

L

L

→ 0 exponentially, with the converging speed determined by either the exponential rate of the left-hand side of 共41兲共i.e., ␥兲 or that of 共45兲, whichever is slower. 䊏 The boundedness assumption on Hi共␨i兲 and Qi共␨i , ␻i兲 in item 2 of Theorem 1 can be achieved if the desired maneuver velocity 共d / dt兲␨Ld共t兲 and the initial angular rates ␻i共0兲共i = 1 , . . . , m兲 are bounded. This is because 共1兲 for the usual rigid-body attitude dynamics 共2兲, the inertia matrix Hi共␨i兲 is bounded w.r.t ␨i, and the Coriolis matrix Qi共␨i , ␻i兲 is bounded w.r.t. ␨i and linear w.r.t. ␻i 关36兴; 共2兲 the configuration space of the attitude dynamics 共2兲 is bounded, thus, with ␨i being bounded, Hi共␨i兲 is bounded; and 共3兲 from item 2 of Theorem 1, once 共d / dt兲␨Ld共t兲 and ␻i共0兲 are bounded, ␻i共t兲 will be bounded, thus, with bounded ␨i, Qi共␨i , ␻i兲 will also be bounded. Note from Theorem 1 that, under the controls 共32兲, 共33兲, 共39兲, and 共40兲, the translation and attitude of each agent 共1兲 and 共2兲 are converging exponentially to the desired position and attitude. Transactions of the ASME


Therefore, this stability would still be preserved under some small perturbation 共e.g., sensor and actuator noises, model uncertainty, etc.兲, and the trajectory of each agent will be close to the desired one, as long as this perturbation is small enough. 4.3 Hierarchy by Abstraction. Since the locked systems 共30兲 and 共34兲 have the dynamics similar to those of the individual agent 共共1兲 and 共2兲兲, abstracting each group by its locked system, we can generate a hierarchy among multiple agents/groups in a very natural way as follows. Consider N agents. From these N agents, let us designate q-agents and, among the rest, make p-groups o⌺ j共j = 1 , . . . , p兲, each consisting of m j agents 共i.e., N = q + 兺 pj=1m j兲. Suppose that we want to control the formation among these q-agents and p-groups, each considered as one entity. For the translation, following 共30兲 and 共31兲, we can then decompose each group o⌺ j into its 共1兲 locked system 1L j : 1jML 1jx¨ L = 1jtL + 1jfL; and 共2兲 shape system 1S j : 1jME 1jx¨ E = 1jtE + 1jfE, where 1j xL 苸 Re3 and 1jxE 苸 Re3共m j−1兲. Here, we use notation 1j쐓 in the place of 쐓 in 共30兲 and 共31兲 to show that jL j 共and jS j兲 is one level higher than their group agents. Similarly, each agent’s dynamics in o⌺ j may be written as ojmk ojx¨ k = ojtk + ojfk 共k = 1 , . . . , m j兲 to show that they are a level lower. Then, a desired formation in the group o ⌺ j can be achieved by controlling 1S j. Since the dynamics of each 1L j has the same structure as that of a single agent 共1兲, we can also make another group 1⌺ by collecting the q-agents and the p-locked systems 1L j , 1L j , . . . , 1L p and we can define the locked and shape systems 2L and 2S for this group 1⌺. By controlling 2S, we can then control the formation shape among the q-agents and the p-groups o⌺1 , o⌺2 , . . . , o⌺ p, whereas, by controlling 2L, we can control the overall behavior of the total collection of the q-agents and the p-groups. We can also put this locked system 2L into another group with other agents/groups. By doing so, we can

generate a hierarchy as shown in Fig. 1. A similar procedure can also be obtained for the attitude case, if each shape system 共35兲 converges to its 共constant兲 set point ␨dE much faster 共e.g., with large gains ␭Ev , ␭Ep 兲 than its locked system dynamics 共34兲, so that, as stated after 共37兲, its locked system 共34兲 will have the same structure as that of a single agent 共2兲 with a well-defined configuration ˜␨L 共i.e., 共d / dt兲˜␨L = ␻L on Hd兲. Just as in the translation case, treating the locked system as a single agent, we can then make a mixed group consisting of agents and locked systems, thus, we can generate the hierarchy in Fig. 1. This concept of hierarchy by abstraction is useful for controlling a mixed collection of many agents and groups. For example, see Sec. 7.3.

5

Controller Decentralization

As shown below, the centralized controls 共共32兲, 共33兲, 共39兲, and 共40兲兲 require state information of all group agents. This communication requirement may be too expensive or infeasible for some applications. In this section, we provide a controller decentralization that only requires each agent to communicate with 共or sense of兲 its two 共or one depending on the agent numbering兲 neighbors. With the help of the passive decomposition, we also show that the performance of this decentralized controller would still be satisfactory, if the desired group behavior is slow. We first choose the gain matrices KEv and KEp in 共33兲 to be block E共m−1兲 diagonal s.t. K쐓E = diag关K쐓E1 , K쐓E2 , . . . , K쐓 兴共쐓 = 兵v , p其兲 where Ej 3⫻3 K쐓 苸 Re 共j = 1 , . . . , m − 1兲 are symmetric and positive-definite matrices. This is always possible, since, in 共33兲, the only condition imposed on KEv and KEp is that they are symmetric and positive definite. Using 共17兲 with the definition of S共q兲 in 共12兲, the centralized translation controls 共32兲 and 共33兲 can then be decoded into the individual control ti 苸 Re3 of the ith agent 共1兲 s.t. for i = 1 , . . . , m,

共46兲

dT dT T 3共m−1兲 , and MEj 苸 Re3⫻3共m−1兲 is the horizontal partition where xdEj 苸 Re3 is the jth entity of xE in 共8兲 s.t. xdE = 关xdT E1 , xE2 , . . . , xE共m−1兲兴苸 Re 共m−1兲T

2T of the 共constant兲 shape system mass matrix ME 苸 Re3共m−1兲⫻3共m−1兲 in 共31兲 s.t. ME = 关M1T 兴. In 共46兲, any terms having E , ME , . . . , ME an index less than 1 or larger than m are to be zeros. Similar to 共46兲, with block-diagonal gain matrices ␭Ev , ␭Ep for the shape system control 共40兲, the centralized attitude controls 共39兲 and 共40兲 are decoded into individual control action s.t.

共47兲


SEPTEMBER 2007, Vol. 129 / 669


and

␶i ª − ␭vL␨˙ i − ␭Lp 共␨i − ␨id兲 + ␭vE共i−1兲共␨˙ i−1 − ␨˙ i兲 + ␭Ep 共i−1兲共␨i−1 − ␨i − ␨Ed 共i−1兲兲 − ␭vE共i兲共␨˙ i − ␨˙ i+1兲 − ␭Ep 共i兲共␨i − ␨i+1 − ␨Ed 共i兲兲

Fig. 3 Communication topology of centralized control: Decentralized control uses only decentralizable communication „black solid lines…

for i = 1 , . . ., m, where arguments are omitted for brevity. Some terms in 共46兲 and 共47兲 require state information of all agents, while the “decentralizable” terms require only those of its own and two neighboring agents 共or one neighbor for the first and mth agents兲. In other words, implementation of 共46兲 and 共47兲 requires centralized communication, as shown in Fig. 3, where the black solid arrows represent information flow required for those “decentralizable” terms in 共46兲 and 共47兲, which can be computed/ implemented by using just a local neighboring sensing or communication 共i.e., distributed control implementation兲. For the controller decentralization, we first decode the desired group formation and maneuver into desired trajectory of each agent. We assume that this desired trajectory is computed beforehand and stored in each agent. This desired trajectory is computed as follows: for the ith agent 共i = 1 , . . . , m兲, 共1兲 desired translation trajectory xdi 共t兲苸 Re3 is 共uniquely兲 obtained by solving d m1xd1 + m2xd2 + ¯ + mmxm = xLd m1 + m2 + ¯ + mm

d and xdj − x共dj+1兲 = xEj 共48兲

T T for j = 1 , . . ., m − 1, where xLd共t兲 and xdE共t兲 = 关xdE共1兲 , . . . , xdE共m−1兲兴T are the desired maneuver and formation of the group translation; and 共2兲 similarly, desired attitude trajectory ␨di 苸 Re3 for the ith agent 共i = 1 , . . . , m兲 is 共uniquely兲 determined by

where any terms having an index less than 1 or larger than m are to be zeros. Here, we restrict the control objective for the decentralized attitude control 共51兲 to be set-point regulation 共i.e., both ␨Ld and ␨dE are constant vectors兲. This is because trajectory tracking control requires information on the inertia and Coriolis terms of the decomposed dynamics 共i.e., HL共␨兲, HE共␨兲,QL共␨ , ␻兲, QE共␨ , ␻兲, QEL共␨ , ␻兲 and QLE共␨ , ␻兲 in 共47兲兲, which are functions of the states of all agents, thus, generally not decentralizable. In contrast, such a restriction is not necessary for the decentralized translation control 共50兲, since the translation dynamics 共1兲 has only constant inertia matrices ML, ME with no Coriolis terms. How to estimate such inertia matrices and Coriolis terms in a decentralized manner will be a topic for future work. The following theorem summarizes properties of the 共stabilizing兲 decentralized controls 共50兲 and 共51兲. THEOREM 2. 共1兲 Consider the group translation dynamics 共1兲 under the decentralized control 共50兲. Suppose that the target accelerations x¨ di 共t兲 is bounded and the disturbances fi共t兲 are negligible (e.g., by local cancelation). Then, 共x˙ L共t兲 − x˙ Ld共t兲 , xL共t兲 − xLd共t兲兲 → 0 exponentially. Also, 共x˙ E共t兲 − x˙ dE共t兲 , xE共t兲 − xdE共t兲兲 is ultimately bounded, whose bound can be made arbitrarily small by large formation gains KEv , KEp in 共50兲; 共2兲 Consider the group attitude dynamics 共2兲 under the decentralized controls 共51兲. Suppose that the inertia matrix Hi共␨i兲 and the Coriolis matrix Qi共␨i , ␻i兲 in 共2兲 are bounded. Suppose further that the disturbances ␦i in 共2兲 are negligible (e.g., by local cancelation), and target maneuver and formation ␨Ld and ␨dE are all constant. Then, ˙ and 共␨˙ E共t兲 , ␨E共t兲 − ␨dE兲 → 0 共˜␨L共t兲 ,˜␨L共t兲 − ␨Ld兲 → 0 exponentially. Proof. 共1兲 Let us define the following Lyapunov function Vx共t兲 s.t.

m

兺A␨

d i i

+ b = ␨Ld

and ␨dj − ␨dj+1 = ␨Ed 共 j兲

共49兲

i=1

m

␨Ld 苸 Re3

dT dT ␨dE = 关␨E共1兲 , . . . , ␨E共m−1兲兴T

for j = 1 , . . ., m − 1, where and are desired maneuver and formation of the group attitude, with Ai, b being defined in 共37兲. With these precomputed desired trajectories xdi 共t兲 and ␨di in 共48兲 and 共49兲 for each agent, we decentralize the translation and attitude controls 共46兲 and 共47兲 s.t. for i = 1 , . . ., m, ti ª

mix¨ id共t兲

共51兲

mi − 关KL共x˙ i − x˙ id共t兲兲 + KLp 共xi − xid共t兲兲兴 m1 + m2 + ¯ + mm v

− MEi−1x¨ Ed共t兲 + KvE共i−1兲共x˙ i−1 − x˙ i − x˙ Ed 共i−1兲共t兲兲 + KEp 共i−1兲共xi−1 − xi − xEd 共i−1兲共t兲兲 + MEix¨ Ed共t兲 − KvE共i兲共x˙ i − x˙ i+1 − x˙ Ed 共i兲共t兲兲 − KEp 共i兲共xi − xi+1 − xEd 共i兲共t兲兲 670 / Vol. 129, SEPTEMBER 2007

共50兲

Vx共t兲 ª

兺 i=1

冋

1 ␾i mi共x˙ i − x˙ id兲T共x˙ i − x˙ id兲 + 共xi − xid兲T共KLp 2 2

+ ⑀KvL兲 ⫻共xi − xid兲 + ⑀mi共x˙ i − x˙ id兲T共xi − xid兲

册

1 + 共xE − xEd兲T共KEp + ⑀KvE兲共xE − xEd兲 2

共52兲

where ␾i ª 共mi兲 / 共m1 + ¯ mm兲 and ⑀ ⬎ 0 is a small constant s.t. Vx共t兲 is positive definite. Using the dynamics 共1兲 with the decentralized control 共50兲 and the condition 共48兲, we then have Transactions of the ASME


共53兲

where we use the “telescoping” relation in the Appendix with 共a , b , G兲苸兵共x˙ , x˙ , KEv 兲 , 共x˙ , x˙ , KEv 兲 , 共x˙ , x˙ , KEv 兲 , 共x˙ , x˙ , KEv 兲其, and the definition ML = 共m1 + m2 + ¯ mm兲I3⫻3 from 共30兲. Thus, by choosing small enough ⑀ ⬎ 0 s.t. Vx共t兲 in 共52兲 and KLv − ⑀ML in 共53兲 are all positive definite, 共xi − xdi , x˙ i − x˙ di , xE − xdE兲 is bounded ∀t 艌 0. Therefore, from 共8兲, 共29兲, and 共48兲, 共xE − xdE , x˙ E − x˙ dE兲 and 共xL − xLd , x˙ L − x˙ Ld兲 are bounded ∀t 艌 0. Using the definitions of x˙ L共t兲 and xL共t兲 in 共28兲 and 共29兲 and the condition 共48兲, we can show that the translation locked system dynamics 共30兲 under the decentralized control 共50兲 is given by: ML共x¨ L共t兲 − x¨ Ld共t兲兲 + KvL共x˙ L共t兲 − x˙ Ld共t兲兲 + KLp 共xL共t兲 − xLd共t兲兲 = fL 共54兲 Therefore, if the disturbances fi in 共1兲 are negligible or properly canceled out, 共xL , x˙ L兲 → 共xLd , x˙ Ld兲 exponentially. We can also show that the shape system dynamics 共31兲 with the decentralized control 共50兲 is given by ME共x¨ E − x¨ Ed兲 + KvE共x˙ E共t兲 − x˙ Ed共t兲兲 + KEp 共xE共t兲 − xEd共t兲兲 m

= fE +

兺 g 共x¨ 共t兲,x˙ 共t兲 − x˙ 共t兲,x 共t兲 − x 共t兲兲 d i

i

i

d i

共55兲

d i

i

i=1

where the functions gi 苸 Re3共m−1兲, i = 1 , . . ., m, are linear w.r.t.

where, similar to 共53兲, we use the “telescoping” relation in the Appendix having 共a , b , G兲苸兵共␨˙ , ␨˙ , ␭Ev 兲 , 共␨˙ , ␨ , ␭Ep 兲 , 共␨ , ␨˙ , ␭Ev 兲 , 共␨ , ␨ , ␭Ep 兲其 with the fact that 共d / dt兲␨di = 0 and 共d / dt兲␨dE = 0. Thus, if the inertia matrices Hi and the Coriolis matrices Qi are bounded, there always exists a small enough ⑀ ⬎ 0 so that Ve␨共t兲 in the above equality is positive definite. Therefore, using this small ⑀ ⬎ 0 with ␦ = 0, we have 共␨˙ 共t兲 , ␨ 共t兲 − ␨d兲 → 0 i

i

i

i

˙ exponentially. Thus, from the condition 共49兲, 共˜␨L共t兲 ,˜␨L共t兲 − ␨d 兲 → 0 and 共␨˙ 共t兲 , ␨ 共t兲 − ␨d 兲 → 0 exponentially. 䊏 L

E

E

E

With the decentralized controls 共50兲 and 共51兲, the communication requirement is now relaxed from the centralized communication to the neighboring communications 共i.e., black solid arrows in Journal of Dynamic Systems, Measurement, and Control

their respective arguments with no bias in the sense that 兩gi共t兲兩艋 ai1 兩 x¨ di 共t兲兩 + ai2 兩 x˙ i共t兲 − x˙ di 共t兲兩 + ai3 兩 xi共t兲 − xdi 共t兲兩, i = 1 , . . ., m, ∀t 艌 0 with ai1, ai2, ai3 ⬎ 0 being all finite scalars. Thus, following 共53兲 with the bounded x¨ di 共t兲, gi共t兲 is bounded ∀t 艌 0. Therefore, with the negligible fi, the shape system dynamics 共55兲 will be ultimately bounded and its bound can be made arbitrarily small by large enough gains KEp , KEv in 共50兲. 共2兲 Similar to 共52兲, let us define the following Lyapunov function:

m

V␨共t兲 ª

兺 i=1

冋

1 1 ˙T ␨ Hi共␨i兲␨˙ i + 共␨i − ␨id兲T共⑀␭vL + ␭Lp 兲共␨i − ␨id兲 2 i 2

册

1 + ⑀␨˙ iTHi共␨i − ␨id兲 + 共␨E − ␨Ed兲T共␭Ep + ⑀␭vE兲共␨E − ␨Ed兲 2 where Hi共␨i兲 is the attitude inertia in 共2兲 and ⑀ ⬎ 0 is a small constant so that V␨共t兲 is positive definite. Differentiating V␨共t兲 w.r.t. time with the dynamics 共2兲, the decentralized control 共51兲, the condition 共49兲, and the ˙ − 2Q 兲T 共or H ˙ =Q ˙ − 2Q = −共H passivity property that H i i i i i i T + Qi 兲, we can show that

Fig. 3兲. However, comparing Theorem 2 with Theorem 1, we can see that this controller decentralization has the following adverse effects/limitations: 共1兲 For the group translation, the formation aspect would be perturbed by the desired maneuver acceleration 共see 共55兲兲. In contrast, its maneuver is not affected by the decentralization at all 共i.e., the locked system dynamics 共54兲 is the same as that under the centralized control 共30兲兲; and 共2兲 for the group attitude, we need to restrict the desired maneuver ␨Ld to be constant due to those “nondecentralizable terms” as explained in the paragraph before Theorem 2. For the group attitude, since the decentralized control 共51兲 does not cancel out the coupling terms Q 共␨ , ␻兲␻ and Q 共␨ , ␻兲␨˙ EL

L

LE

E

in 共34兲 and 共35兲, there would be crosstalk between the formation and maneuver, which are quadratic w.r.t. the operating speed. For



the group translation, such crosstalk would affect only the formation 共shape system兲 when the desired maneuver has nonzero acceleration 共see 共54兲 and 共55兲兲. Thus, both for the translation and attitude, if desired group behaviors are slow enough 共i.e., small accelerations and velocities兲 so that such crosstalks become insignificant, precision of the formation and maneuver would still be satisfactory under the decentralized control. As in the centralized control case 共see the last paragraph of Sec. 4兲, due to the exponential convergence property of the decentralized control, the agent’s state will be close to the desired one even under some perturbations 共e.g., sensor and actuator noise, and model uncertainty兲, as long as their magnitude is small enough. Along the same reasoning in the paragraph after Theorem 1, for item 2 of Theorem 2, boundedness assumption on Hi共␨i兲 and Qi共␨i , ␻i兲 will also be ensured, if ␻i共0兲 is bounded.

6 Extension to an Equivalence Class of Formation Variables In this section, we extend the results to the case where the formation variable 共11兲 is given by an element in the following equivalence class of qE in 共11兲: EqE ª 兵qE⬘ 苸 Re共m−1兲n兩 ∃ E 苸 Re共m−1兲n⫻共m−1兲ns.t.qE⬘ = EqE其 where E is a full-rank constant matrix. Here, to unify the translation and attitude cases, we use the notations of Sec. 3 with q T T 兴共i.e., 苸兵x , ␨其. For example, consider q⬘E ª 关qT1 − qT2 , . . . , qT1 − qm agent 1 is the leader兲. This q⬘E 苸 EqE, since there is E s.t. its ith row is given by 关I , . . . , I , 0 , . . . , 0兴苸 Ren⫻共m−1兲n with i-copies of I 苸 Ren⫻n and 共m − i − 1兲-copies of 0 苸 Ren⫻n. With the extension to this class of formation variables, as those in 关9,37–39兴, our framework is not bound any more to a specific definition of the formation variables 共e.g. 共8兲 and 共9兲兲. Now, suppose that, as the formation variable 共11兲, we choose q⬘E = EqE 苸 EqE. Similar to 共12兲, we can then define the decomposition matrix S⬘共q兲 s.t. 共56兲 with S⬘共q兲 ª US共q兲, where S共q兲苸 Remn⫻mn and vL 苸 Ren are the decomposition matrix and the locked system velocities with qE in 共11兲 as the formation variable, and those with ⬘ are for q⬘E. From the block-diagonal structure and being constant of U in 共56兲, we can then decompose the group dynamics 共10兲 similar to 共22兲 and 共23兲 s.t.

⬘ 共q,q˙ 兲q˙ E⬘ = TL⬘ + FL⬘ ML⬘ 共q兲v˙ L⬘ + CL⬘ 共q,q˙ 兲vL⬘ + CLE

共57兲

⬘ 共q,q˙ 兲vL⬘ = TE⬘ + FE⬘ ME⬘ 共q兲q¨ E⬘ + CE⬘ 共q,q˙ 兲q˙ E⬘ + CEL

共58兲

where

冋冋

ML⬘

0

0

ME⬘

⬘ CL⬘ CLE

⬘ CE⬘ CEL

册冋册冋 = U−T

= U−T

ML

0

0

ME

册册

U−1

CL CLE CEL CE

共59兲

U−1

and, similar to 共17兲, using 共S⬘兲−T = U−TS−T,

冉冊 TL⬘

TE⬘

= 共S⬘兲−TT = U−T共S−TT兲 =

冋册冉冊 I

0

TL

−T

TE

0 E

共60兲

T T with T = 关TT1 , . . . , Tm 兴 . A similar relation also holds for F. Here, note from 共56兲–共60兲 that the locked system is invariant w.r.t. the formation variable change 共i.e., 쐓L⬘ = 쐓L兲. These decomposed systems 共57兲 and 共58兲 have the same structure and satisfy the Proposition 1 as their counterparts 共22兲 and


共23兲 do. Therefore, exactly same centralized control in Sec. 4 can be applied for 共57兲 and 共58兲. Suppose that, for 共57兲 and 共58兲, we design the following centralized control: ˜ L − qLd兲 ⬘ q˙ E⬘ + ML⬘ q¨ Ld + CL⬘ q˙ Ld − BL⬘ 共vL⬘ − q˙ Ld兲 − KL⬘ 共q TL⬘ = CLE 共61兲

⬘ vL⬘ + ME⬘ q¨ Ed⬘ + CE⬘ q˙ Ed⬘ − BE⬘ 共q˙ E⬘ − q˙ Ed⬘兲 − KE⬘ 共qE⬘ − qEd⬘兲 TE⬘ = CEL 共62兲 where qLd and qdE⬘ = EqdE are the desired trajectories and ˜qL⬘ is the pseudo-configuration as in 共37兲. Note that the controls presented in Sec. 4 are special examples of these 共61兲 and 共62兲. Here, we use qLd and ˜qL instead of qLd⬘ and ˜qL⬘ , since vL⬘ = vL from 共56兲. Then, using 共60兲, we can map 共TL⬘ , T⬘E兲 to the controls 共TL , TE兲 for the original decomposed systems 共22兲 and 共23兲. First, the locked control 共61兲 is mapped s.t. ˜ L − qLd兲 TL = TL⬘ = CLEq˙ E + MLq¨ Ld + CLq˙ Ld − BL⬘ 共vL − q˙ Ld兲 − KL⬘ 共q This is because the locked system is invariant w.r.t. the formation variables as stated after 共60兲共i.e., 쐓L⬘ = 쐓L兲 and CLE ⬘ q˙ ⬘E = CLEE−1Eq˙ E from 共59兲. The shape control 共62兲 is also mapped s.t. TE = ETTE⬘ = CELvL + MEq¨ Ed + CEq˙ Ed − BE共q˙ E − q˙ Ed兲 − KE共qE − qEd兲 where we use 共59兲 with CEL ⬘ = E−TCEL, vL⬘ = vL, E−1q⬘E = qE, and E−1qdE⬘ = qdE. Here, we use the definitions BE ª ETB⬘EE and KE ª ETK⬘EE, both of which are positive definite and symmetric, since E is full rank. This shows that a set of centralized controls designed such as 共61兲 and 共62兲 for any formation variable q⬘E 苸 EqE are equivalent across different formation variables in EqE. In this sense, the centralized controls 共61兲 and 共62兲 constitute a class of equivalent controls. Now, consider the decentralized controls 共50兲 and 共51兲 and denote them by the unified notation Ti 共T 苸兵t , ␶其兲. Using 共17兲 and 共18兲 as for 共55兲, we can then show that these Ti can be mapped to ˜L 共TL , TE兲 of 共22兲 and 共23兲 s.t. TL = MLq¨ Ld − BL共vL − q˙ Ld兲 − KL共q − qLd兲 and TE = MEq¨ dE − BE共q˙ E − q˙ dE兲 − KE共qE − qE兲 + G, where some terms are zeros for the attitude control 共51兲 and G 苸 Re共m−1兲n is corresponding to the sum of 共bounded兲 gi functions in 共55兲. Since they are in the same form as 共61兲 and 共62兲, the structure of these controls will be carried over to the control 共TL⬘ , T⬘E兲 for 共57兲 and 共58兲 with G replaced by G⬘ ª E−TG from 共17兲. This implies that, under the decentralized controls 共50兲 and 共51兲, Theorem 2 also holds for other formation variables in EqE, too. Conversely, by converting it first to 共TL , TE兲, and then decentralizing it as in Sec. 5, we can also decentralize a centralized control 共TL⬘ , T⬘E兲 for any formation variable q⬘E 苸 EqE into the form 共50兲 and 共51兲 requiring only the line communication topology, as long as we choose the gains B⬘E and K⬘E for T⬘E s.t. BE ª ETB⬘EE and KE ª ETK⬘EE are block diagonal as required in Sec. 5. Perhaps, an even more interesting question would be how to decentralize the control according to the communication pattern implied by a formation variable q⬘E 苸 EqE, which may not necessarily be the line topology of Sec. 5. For instance, if q⬘E = 关qT1 T T − qT2 , . . . , qT1 − qm 兴 , can we decentralize the control 共TL⬘ , T⬘E兲 s.t. each decoded control Ti is a function of q1 and qi? Simple extensions of the proposed framework for this seem not to work at this moment. This is mainly because: 共1兲 The decentralized controls 共50兲 and 共51兲 are derived w.r.t. the specific qE in 共11兲. Thus, sim⬘ in 共50兲 and 共51兲, there is ply by replacing the terms qE共i兲 by qE共i兲 no guarantee that it also achieves the desired formation/maneuver and complies with the communication pattern implied by q⬘E; and 共2兲 In Sec. 5, the matrix ST共q兲 produces very naturally the decentralized controls 共50兲 and 共51兲 according to the communication pattern implied by qE. However, in general, this is not the case Transactions of the ASME


Fig. 4 3D and 2D simulation snapshots: Position and attitude of agent represented by sphere and body-fixed frame

with other 共S⬘兲T and q⬘E, when T⬘E contains the kinematic feedback terms 共e.g., K⬘Eq⬘E兲. How to achieve the controller decentralization for a given arbitrary communication structure is a topic for future research. For this, we will investigate a possible use of other tools, such as graph or matrix theory. See 关40兴 for a result in this direction. Fig. 5 Detailed data of the snapshots in Fig. 4

7

Simulation

7.1 Formation and Maneuver Control. We consider three agents whose dynamics is given by 6-DOF rigid body dynamics. We simulate each agent’s dynamics using 6-DOF block of MatLab SimuLink Aerospace Blockset. Following 关13兴, masses and principal-body-axis inertias of the three spacecraft are set to be: m1 = 20 kg, I1 = diag关0.73, 0.55, 0.63兴 k gm2 for agent 1, m2 = 10 kg, I2 = diag关0.36, 0.27, 0.31兴 k gm2 for agent 2, and m3 = 11 kg, I3 = diag关0.38, 0.30, 0.33兴 k gm2 for agent 3. The centralized control in Sec. 4 is used and environmental disturbances are assumed to be negligible. Snapshots of the simulation are given in Fig. 4, where the position and attitude of each agent are represented by the sphere and the body-fixed 共x , y , z兲-frame. The bodyfixed z-axis will be coincident with the inertial z-axis, when the pitch and roll angles are zeros. Simulation is performed for 130 s and snapshots are taken in every 5 s. Detailed simulation data are shown in Fig. 5 and its movie is available at http:// www.me.umn.edu/⬃djlee/Formation/FF3DAng1.avi 共or http:// www.me.umn.edu/⬃djlee/Formation/FF3DAng2.avi for a different view angle兲. We first derive the agent’s dynamics w.r.t. the inertial frame. For the translation, each agent’s dynamics is simply given by 共1兲. For the attitude dynamics, since the inertia matrix is given w.r.t. the body-frame, we use the standard Jacobian map to get the dynamics w.r.t. the inertial frame in the form of 共2兲. The group translation and attitude dynamics can then be decomposed as 共30兲, Journal of Dynamic Systems, Measurement, and Control

共31兲, 共34兲, and 共35兲, respectively 共see Sec. 3 for more details on the decomposition procedure兲. Since we assume negligible external disturbances, fL, fE, ␦L, and ␦E in 共30兲, 共31兲, 共34兲, and 共35兲 are all zeros. By adjusting parameters in 共32兲, 共33兲, 共39兲, and 共40兲, we assign different controls 共tL , tE , ␶L , ␶E兲 on different time windows of the simulation, as briefly stated below. Once they are obtained, we decode them into individual agent’s control ti and ␶i by using 共17兲 for the translation and the attitude, separately. During the first 50 s, to make an equilateral triangular formation shape 共with 20冑3m side-length兲, we choose xE共t兲 = 关0 , −20冑3 , 0 , 30, 10冑3 , 0兴T 苸 Re6 with x˙ E = 0. We also set xL共t兲 = 关vdx ⫻ t , ay , 0兴T with KLp = diag关0 , kLp , kLp 兴共kLp ⬎ 0兲 so that the geometric center of the triangle floats along the x-axis with a constant speed vdx 共=0.9 m / s兲 on a line 共y , z兲 = 共0 , 0兲. Here, the 共generally nonzero兲 constant offset ay is necessary, since the center of the mass 共described by the locked system兲 and the geometric center of the triangle 共whose motion we want to control兲 are not the same. For the attitude controls 共39兲 and 共40兲, we choose ˜␨L = ␨1. We also set both ␨Ld共t兲 and ␨dE共t兲 to be zero so that each agent’s roll, pitch, and yaw angles are stabilized to zero 共see Fig. 4 or 5兲. From 50– 80 s, the goal is to maintain the center of the triangle at 共50, 0 , 0兲 m, while turning only the yaw angles of the agent 1 and agent 2 by 2␲ / 3 and −2␲ / 3, respectively, so that all the SEPTEMBER 2007, Vol. 129 / 673


Fig. 7 Hierarchy by abstraction: Snapshots of nine agents, three groups „triangles…, and their locked systems „solid lines…

trol兲, due to the exponential convergence property, the group behavior was still close to the desired one, as long as the noise level was small enough. Therefore, we omit here the simulation results with the decentralized control and the noise.

Fig. 6 Effect of disturbance on agent 3: Agent position/pitch angle represented by vertex of triangle and bar stemming from it

agents point to the center of the triangle 共50, 0 , 0兲关m兴. Thus, from the previous control, we change the following: 共1兲 xL共t兲 = 关50 + ax , ay , 0兴T with KLp = diag关kLp , kLp , kLp 兴, where, similar to ay, the constant offset ax is used to compensate for the position difference between the geometric center of the triangle and the center of mass; and 共2兲 ␨Ld共t兲 = 关0 , 0 , 2␲ / 3兴T and ␨dE共t兲 = 关0 , 0 , 4␲ / 3 , 0 , 0 , −2␲ / 3兴T. This choice of ␨Ld共t兲 is because we set ˜␨L = ␨1 共i.e., parametrize the locked system’s attitude by that of the agent 1兲. See Sec. 4.2. From 80– 130 s, we aim to drive the equilateral triangle from z = 0 m to z = −20 m while shrinking the triangle size and maintaining the agents’ pointing to the triangle’s center. Thus, we make the following change only on the previous translation control: xL共t兲 = 关50+ ax , ay , −2ut兴T and xE共t兲 = 关0 , −冑3共20− ut兲 , 0 , 3共20 − ut兲 / 2 , 冑3共20− ut兲 / 2 , 0兴T, where u共=0.2 m / s兲 is the shrinking rate. We also performed the same simulation with the decentralized control in Sec. 5 and achieved similar results. We think that this is because the desired group behavior is slow enough so that the effect of the controller decentralization is not significant 共see Sec. 5兲. We also injected several levels of sensor/actuator noises. However, as expected in Sec. 4 共or Sec. 5 for the decentralized con674 / Vol. 129, SEPTEMBER 2007

7.2 Group-Agents Interaction. The purpose of this simulation is to show that the group agents behave as one coherent group. We choose the same agents in Sec. 7.1 and use the decentralized control in Sec. 5. We impose sinusoidal disturbance on the agent 3 from 10 s, while the regulation controls stabilize the group translation and attitude to their respective set points. The dominant effects of the disturbance are in the directions of z-axis translation and pitch angle rotation. Data are plotted in Fig. 6. A movie of the simulation is also available at http://web.utk.edu/⬃djlee/movie/push.avi As shown in Fig. 6, the perturbed state of the agent 3 affects the total group behavior and the states of other agents via the decentralized control. The disturbance propagates from the agent 3 to the agent 2 and the agent 1 through the control actions. Due to the damping actions in the controls, the disturbance effect is attenuated while propagating to other agents, as shown by the mitigated perturbed z-axis motion and pitch rotation of the agent 2 and the agent 1. Similar results were also achieved with the centralized control. 7.3 Hierarchy by Abstraction. For this simulation, we consider nine 2-DOF 共x , y兲-planar point masses. Utilizing the hierarchy with the centralized control 共see Sec. 4.3 for more details兲, we partition them into three groups as shown in Fig. 7, where each triangle represents one group with three agents represented by its vertexes. We decompose each group by its locked and shape systems 共Li , Si兲, i = 1 , 2 , 3. The trajectory of each locked system Li is shown by the solid line stemming from the center of each triangle.

Fig. 8 Hierarchy by abstraction: Agent 5’s actuation temporarily failed and recovered



Fig. 9 Formation-maneuver decoupling with the centralized control

Fig. 10 Formation-maneuver coupling with the decentralized control

¯ , ¯S兲 of 共L , L , L 兲. We also define the locked and shape systems 共L 1 2 3 The locked systems 共L1 , L2 , L3兲 then inherit their controls from ¯ , ¯S兲. those of 共L

unactuated agent 5. However, as soon as the agent 5’s actuation is activated again, all the nine agents are speeding up again to catch upto the desired trajectory, and eventually stabilized on the arc of the small circle. This simulation clearly shows that the multiple agents behave as one coherent group. A movie of this simulation is also available at http://web.utk.edu/⬃djlee/movie/hier_wfail.avi 共or http://web.utk.edu/⬃djlee/movie/hier_wfail1.avi with deactivation during a different time period兲.

¯ , ¯S , S , S , S 兲 to achieve three In Fig. 7, initially we stabilize 共L 1 2 3 equilateral triangles with its center on the arc of the large circle 共left兲. By controlling ¯L and ¯S, we then drive the three triangles to the small circle 共right兲, while shrinking the distance among the three equilateral triangles. At the same time, the shape systems 共S1 , S2 , S3兲 are controlled to shrink and rotate each equilateral triangle. Note from Fig. 7 that not only the group sizes are shrinking, but also the locations of the equilateral triangles are swapped on the circles. A movie of this simulation is also available at http://web.utk.edu/⬃djlee/movie/hier_nfail.avi In Fig. 8, we use the same controls used in Fig. 7, but the actuation of agent 5 共group 2兲 has been temporarily deactivated. As shown in Fig. 8, to maintain the desired formation w.r.t. the agent 5, the remaining eight agents start to drift following the Journal of Dynamic Systems, Measurement, and Control

7.4 Effect of Decentralization. As shown in Sec. 5, under the decentralized control, the formation and maneuver are not decoupled from each other any more. This simulation is to show the effect of this incomplete decoupling of the decentralized controls. We consider three point masses 共1 kg兲 forming an equilateral triangle confined in the circle of 2 m radius. See Fig. 9 for an illustration. We design the formation and maneuver control so that the center of the triangle is revolving along the arc of the large circle with 4 m radius counter-clockwise with 8 s periodicity. The triangle itself is also rotating at the same rate so that one agent SEPTEMBER 2007, Vol. 129 / 675


共marked with ⴰ兲 is always farthest from the center of the 4 m radius circle 共i.e., on the circle of 6 m radius兲. We assume 10% mass identification error. We first utilize the centralized control and Fig. 9 shows the simulation results. In the upper plot, the snapshots are taken with 5 s shooting speed during 10– 50 s, while the lower plot shows the 2 norm of the maneuver and formation error. Similarly, the simulation results of the decentralized control are shown in Fig. 10. For the translational dynamics, the desired maneuver acceleration perturbs the shape system 共see item 1 of Theorem 2兲. This coupling effect induces the phase lag of the triangle rotation as shown by that the triangles are tilted in Fig. 10. However, with the centralized control in Fig. 9, this phase lag disappears, and the triangles maintain their rotation as their center rotates along the large circle. Ripples in the lower plots of Figs. 9 and 10 are due to the mass identification error. With the perfect mass parameters, those ripples disappear, and, with the centralized control, the maneuver and formation errors converge to zero exponentially 共or, with the decentralized control, the maneuver error converges to zero exponentially, while the formation error still maintains a constant bias兲. It is noteworthy to point out that the locked system dynamics and the maneuver error are not affected by the controller decentralization 共see item 1 of Theorem 2兲. Such a preservation of the maneuver behavior would not be achieved for the attitude dynamics due to the couplings QEL, QLE in 共34兲 and 共35兲.

8

Conclusion

In this paper, we propose a novel control framework for the formation of multiple rigid bodies. The key idea is the passive decomposition, with which we can decompose the nonlinear group dynamics into two decoupled systems: Shape system representing 共internal group兲 formation aspect, and locked system abstracting 共overall group兲 maneuver. By controlling them separately and individually, we can then achieve precise maneuver and formation controls simultaneously without any crosstalk between them. Furthermore, as the locked system is directly derived from the real dynamics of all agents, it would better abstract the group maneuver than an artificially-defined virtual system. We also connect group agents via bidirectional controls so that they can behave as one single coherent group 共i.e., no runaway of slower agents兲. By abstracting a group by its locked system, we can also generate a hierarchy among multiple agents and groups. We also

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Acknowledgment Research partially supported by NSF CMS-9870013, the Doctoral Dissertation Fellowship of the University of Minnesota, and the College of Engineering at the University of Tennessee.

Appendix Consider ai , bi , adi , bdi 苸 Ren, i = 1 , . . . , m. Similar to 共8兲, 共9兲, 共48兲, and 共49兲, we can then define aE , bE , adE , bdE s.t. 쐓E T T T ª 关쐓E共1兲 , 쐓E共2兲 . . . , 쐓E共m−1兲兴T 苸 Re共m−1兲n, where 쐓 苸兵a , b , ad , bd其 n and 쐓E共j兲 = 쐓 j − 쐓 j+1 苸 Re , j = 1 , . . . , m − 1. Similar to the diagonal gain matrices in the controls 共46兲 and 共47兲, we also define a blockdiagonal matrix G = diag关G1 , G2 , . . . , Gm−1兴苸 Re共m−1兲n⫻共m−1兲n, where G j 苸 Ren⫻n. The following “telescoping” relation then holds: With any term having an index less than 0 or larger than m being zero:

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