Decentralized Formation Control Part I: Geometric Aspects

Decen tralized formation con trol part I: Geometric asp ects M.-A. Belabbas ∗ Jan uary 14, 2011 Abstract In this pap er, we dev elop new metho ds for the analysis of decen tralized control systems and w e apply them to formation con trol problems. The basic set-up consists of a system with m ultiple agents represen ted by the no des of a directed graph whose edges represent an a v ailable communication channel for the agen ts. W e address the question of whether the information flo w defined by the graph is sufficien t for the agents to accomplish a given task. F ormation con trol is concerned with problems in which agents are required to sta- bilize at a giv en distance of other agents. In this con text, the graph of a formation enco des b oth the information flo w, as describ ed ab o ve, and the distance constrain ts, by fixing the lengths of the edges. A formation is said to be rigid if it cannot b e contin u- ously deformed with the distance constraints satisfied; a formation is min imally rigid if no distance constraint can b e omitted without the formation losing its rigidity . Hence, the graph underlying minimally rigid formation pro vides just enough constrain ts to yield a rigid formation. An op en question we will settle is whether the information flo w afforded by a minimally rigid graph is sufficient to define global stabilizing control la ws. W e sho w that the answ er is negative in the case of directed information flow. In this first part, we establish basic prop erties of formation control in the plane. F ormations and the asso ciated control problems are defined mo dulo rigid transforma- tions. This fact has strong implications on the geometry of the space of formations and on the feedback control laws, since they need to resp ect this inv ariance. W e study b oth asp ects here. In detail, we show that the space of formations of n agents is C P( n − 2) × (0 , ∞ ) where C P( n ) is the complex-pro jective space of complex dimen- sion n . W e subsequently illustrate how the non-trivial top ology of this space relates to the parametrization of the formation by in ter-agent distances. W e then establish conditions feedbac k con trol la ws need to satisfy in order to yield a closed-lo op system that resp ects b oth the inv ariance under the action of the Euclidean group and the constrain ts on the information flow. ∗ M.-A. Belabbas is with the School of Engineering and Applied Sciences, Harv ard Universit y , Cambridge, MA 02138 belabbas@seas.harvard.edu 1 Con ten ts 1 In tro duction 2 2 F ramew orks and rigidity 4 2.1 F ramew orks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Rigidit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Pro jective space, Euclidean group and LS-category 8 3.1 Euclidean group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Pro jective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.1 R P( n ) for n ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2.2 Complex pro jective spaces . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 The Lusternick-Sc hnirelmann category . . . . . . . . . . . . . . . . . . . . . 12 4 The Geometry of the space of formations 13 4.1 The space E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 P arametrization of E 3 b y edge lengths . . . . . . . . . . . . . . . . . . . . . 15 4.3 The space E n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 P arametrization b y edge lengths . . . . . . . . . . . . . . . . . . . . . . . . 18 4.5 The space E 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.6 Henneb erg sequences and group actions . . . . . . . . . . . . . . . . . . . . 23 5 Decen tralized control with directed graphs 25 5.1 F easibilit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Con trol problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2.1 Distributed formation control . . . . . . . . . . . . . . . . . . . . . . 27 5.2.2 The mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3 Linearization of the dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Conclusion 36 1 In tro duction Decen tralized and formation control problems hav e b een o ccup ying a central part of the efforts in control theory for the past decade. The reason for this growing interest stems in part from the large num b er of p oten tial applications—from the study of schooling and flo c king, to sensor netw orks, to formation fligh t [EAM + 04, SPL07]—and in part from the scien tific c hallenges they presen t. Giv en an ensemble of agents, a graph whose vertices are identified with the agents is used to describ e the information flow in the system: a directed edge from agent i to j means 2 that agen t i observes agen t j . The main ob jective b ehin d our work in formation con trol is to understand ho w the constrain ts on the information flow in a decentralized system affect its dynamics. Informally sp eaking, w e would like to answer questions suc h as “how m uch do es agent i need to know in order for a giv en ob jectiv e to b e reached?” Here, ob jectives can b e as v aried as trying to remain in a certain part of the phase space or making sure that a given configuration is stable. W e address such questions in the case of directed formation con trol problems. The systems inv olv ed, in addition to their in trinsic in terest and in spite of their rather simple description, can hav e a fairly complex b eha vior even in lo w dimensions and thus pro vide a rich test-bed for framew orks addressing the ab o v e-men tioned issues. In part I of this pap er, w e study topological and geometric prop erties of formations in the plane and establish a few general results ab out feedback control laws that respect b oth the in v ariance and the information flo w of the system. In part I I, we in tro duce an algebraic framew ork that captures the range of b eha viors that a decentralized system can ac hiev e and use it to study global stabilization of directed formations. In this pap er, we adopt the following p oin t of view: coop erativ e control systems are most often made of a n umber of similar or nearly similar subsystems, and eac h subsystem comes with its o wn co ordinate system. Think of a flo c k where each agen t sees its neighbors with resp ect to its o wn p osition. While the parametrization of the individual systems can b e made straigh tforward, it is often not the case for the ensem ble of systems. Beyond the well-kno wn and studied fact that interactions b et w een systems can bring additional constrain ts, the parametrization of the ensem ble requires more care in tw o asp ects: the first is that the straigh tforward direct sum of the co ordinates of eac h agen t often results in a redundant parametrization; w e will elab orate on this b elo w. The second asp ect originates from top ological considerations: the dynamics of the subsystems dep end on a subset of the v ariables necessary to describ e the ensemble, as defined b y the information flow in the system. W e call these the lo calized co ordinates. W e will sho w ho w the interpla y b et ween these lo calized co ordinates and the top ology of the state-space of the ensem ble affects the dynamics of all decentralized feedbac k laws for the ensemble. Consider ha ving, as an example, three autonomous agents in the plane called x 1 , x 2 and x 3 . Agent i only knows the r elative p osition of agent i + 1 (tak en mod 3 ) with resp ect to its own position. With these constraints, is it p ossible for the agen ts to reac h an arbitrary triangular formation in the plane? It has b een sho wn in [CMY + 07] that it is the case, up to the mirror symmetric of the formation, for almost all initial conditions for the agen ts. Ho wev er, as noted in [CAM + 10], results b ey ond the ones relating to three agents hav e b een particularly difficult to obtain. The pap er is organized as follows. In Section 2, w e introduce fundamentals of graph and rigidit y theory and establish most of the notation used in the paper. In particular, w e in tro duce the mixed-adjacency and the edge-adjacency matrices that ha v e not b een used b efore. These op erators are useful when expressing the dynamics in differen t co ordinate systems. In Section 3, we present the necessary geometric bac kground for the pap er. It consists mainly of a review of pro jective spaces, the Euclidean group and the Lusternick- 3 Sc hnirelmann category . The following tw o sections contain the main results of this pap er. W e establish in Section 4 geometric prop erties of the space of formations and relate the edge- lengths parametrization to the Lusternick-Sc hnirelmann category . In the last section, w e in tro duce a mo del for formation con trol that respects the in v ariance prop erties in tro duced in Section 4 and derive its basic characteristics. 2 F ramew orks and rigidity F ormation con trol problems are intimately related to rigidity and graph theory . W e presen t in this section the relev ant aspects and refer the reader to [GSS93] for additional details. 2.1 F rameworks Let G = ( V , E ) be a gr aph with n v ertices — that is V = { x 1 , x 2 , . . . , x n } is an ordered set of v ertices and E ⊂ V × V is a set of edges. The graph is said to be dir e cte d if ( i, j ) ∈ E do es not imply that ( j, i ) ∈ E . W e let | E | = m be the cardinality of E . W e call the outvalenc e of a v ertex the n umber of edges originating from this vertex and the invalenc e the num b er of incoming edges. Directed graphs are used to encode the information flo w in decen tralized control prob- lems. W e follo w the conv ention that an arrow lea ving v ertex i for vertex j means that agent i observes agent j . F or example, for the graph of Figure 1a, w e ha ve that x 1 sees b oth x 2 and x 4 , x 2 and x 4 see x 3 and so forth. The top ology of a graph can conv enien tly b e captured by some op erators, instead of the sets V and E . This is done using adjac ency matric es : Definition 1. Given a dir e cte d gr aph G = ( V , E ) , its adjac ency matrix A d ∈ R n × n is define d by A d,ij =  − 1 if ( i, j ) ∈ E 0 otherwise. Assume that the edges are ordered. The edge-adjacency matrix is a | E | × | E | matrix whose entry i, j is − 1 if edge i and edge j originate from the same v ertex i , 1 if edge i ends at v ertex where edge j starts and 0 otherwise. Notice that A e,ij is zero if edge i starts where edge j ends and that the diagonal entries are − 1 : Definition 2 (Edge-adjacency matrix) . Given a dir e cte d gr aph G = ( V , E ) , its e dge- adjac ency matrix A e ∈ R m × n is define d by A e,ij =    − 1 if e i = ( s, t ) , e j = ( s, t 0 ) , s, t, t 0 ∈ V 1 if e i = ( s, t ) , e j = ( t, s 0 ) , s, s 0 , t ∈ V 0 otherwise. 4 The mixed-adjacency matrix is a | E | × | V | matrix whose en try i, j is − 1 if edge i originates from vertex j , 1 if edge i ends at vertex j and 0 otherwise: Definition 3 (Mixed adjacency matrix) . Given a dir e cte d gr aph G = ( V , E ) , its mixe d adjac ency matrix A m ∈ R n × m is define d by A m,ij =    − 1 if e i = ( j, s ) , s ∈ V +1 if e i = ( k , j ) , k ∈ V 0 otherwise. The mixed- and edge-adjacency matrices are used to relate the dynamics of formation con trol systems in terms of the p osition of the agents to its expression in terms of edge lengths. W e will often hav e to consider the matrices A m ⊗ I and A e ⊗ I where ⊗ is the Kronec ker pro duct and I the tw o-b y-tw o identit y matrix. In order to keep the notation simple, we write A (2) m and A (2) e for these Kroneck er pro ducts. Example 1. The mixe d-adjac ency matrix of the 2-cycles of Figur e 1b is B =       − 1 1 0 0 0 − 1 1 0 1 0 − 1 0 0 0 1 − 1 − 1 0 0 1       . (1) and its e dge- adjac ency matrix is A e =       − 1 1 0 0 − 1 0 − 1 1 0 0 1 0 − 1 0 1 0 0 1 − 1 0 − 1 0 0 1 − 1       (2) wher e e dge 1 c orr esp onds to z 1 , etc.  W e call a fr amework an em b edding of a graph in R 2 endo wed with the usual Euclidean distance, i.e. given G = ( V , E ) a directed graph, a framework p is a mapping p : V → R 2 . 5 x 1 x 2 x 3 x 4 x 5 (a) x 1 x 2 x 3 x 4 z 1 z 2 z 3 z 4 z 5 (b) Figure 1: (a) A representation of the directed graph V = { x 1 , x 2 , x 3 , x 4 , x 5 } , E = { ( x 1 , x 4 ) , ( x 1 , x 2 ) , ( x 2 , x 3 ) , ( x 3 , x 1 ) , ( x 4 , x 3 ) , ( x 4 , x 5 ) } . (b) The 2-cycles formation. By abuse of notation, we will often write x i for p ( x i ) . W e define the distanc e function δ of a framew ork with n v ertices as δ ( p ) : R 2 n → R n ( n − 1) / 2 + : ( x 1 , . . . , x n ) → 1 2             k x 1 − x 2 k 2 k x 1 − x 3 k 2 . . . k x 1 − x n k 2 k x 2 − x 3 k 2 . . . k x n − 1 − x n k 2             , where R + = [0 , ∞ ) . W e denote b y δ ( p ) | E the restriction of the range of δ to edges in E . F or a graph G with m edges, w e define L =  d = ( d 1 , . . . , d m ) ∈ R m + for which ∃ p with δ ( p ( V )) | E = d  , and w e denote by L 0 the in terior of L . In other w ords, L is the set of feasible assignments of edge lengths of a giv en graph. F or example, for a triangle with edge lengths d 1 , d 2 and d 3 , L is given b y the inequalities d 1 + d 2 ≥ d 3 , d 2 + d 3 ≥ d 1 , d 1 + d 3 ≥ d 2 and d i ≥ 0 . When dealing with framew orks of a given graph, it is often the case that some prop erties are true except for a small set of framew orks, such as framew orks with all the v ertices aligned, or frameworks with t wo vertices sup erposed. In order to easily deal with these particular cases, we in tro duce the follo wing definition [Con05]: Definition 4 (Generic frameworks) . A fr amework is generic if the c o or dinates of al l its the vertic es ar e algebr aic al ly indep endent over the r ationals. The ab o v e definition means that the p ositions of the vertices cannot b e describ ed as the zeros of a polynomial with rational co efficien t. It is easy to see that generic framew orks are 6 dense in the space of frameworks, and that non-generic frameworks are of measure zero in this space. W e end this section with the following definition: Definition 5 (k-vertex connected graphs) . A gr aph is connected if for every p air of vertic es x i , x j ∈ V , ther e is a p ath in G that starts at x i and ends at x j . A gr aph is said to b e k- v ertex connected if every gr aph G 0 obtaine d by r emoving k − 1 vertic es fr om V , and the e dges incident to these vertic es, is c onne cte d. 2.2 Rigidit y Giv en a graph G with m edges, w e are giv en m positive n umbers d i and consider the framew orks of G whose distance function satisfies δ ( p ) | E = d i . Rigidit y theory is concerned with how man y edges are necessary so that a framew ork cannot b e con tinuously deformed, with the exception of translations and rotations. The rigidity matrix of the framework is the Jacobian ∂ δ ∂ x restricted to the edges in E . W e denote it by ∂ δ ∂ x | E . There are several notions of rigidit y that are relev an t for our w ork: 1. Static rigidity : A framew ork is said to b y statically rigid, or simply rigid, if there are no nearby framew orks p 0 , mo dulo rotation and translation, with δ ( p 0 ) | E = d . 2. Infinitesimal rigidity : A framework is said to b e infinitesimally rigid if there are no v anishingly small motions of the vertices that keep the edge-length constrain ts on the framew ork satisfied. This translates into [GSS93]: rank( ∂ δ ∂ x | E ) = 2 n − 3 . 3. Minimal rigidity : A framew ork is said to b e minimally rigid if none of the m frame- w orks with m − 1 edges obtained by remo ving one edge is rigid. 4. Glob al rigidity : A framework is said to b e globally rigid if the only other framew ork satisfying the same edge lengths is its mirror symmetric. While infinitesimally rigid frameworks are rigid, the conv erse is not true, see [GSS93] for an example. The framework in Figure 1a is not statically rigid since contin uous motions of x 5 are allow ed. The framework of Figure 1b is statically rigid but not globally rigid: indeed one could tak e the framew ork where x 2 is sent to its mirror symmetric along z 3 and satisfy the same distance constraints. W e will lo ok into global rigidity in more detail in later sections. 7 x 1 x 2 x 3 x 4 (a) x 1 x 2 x 3 x 4 x 5 (b) Figure 2: (a) is minimally and infinitesimally rigid, while (b) is infinitesimally rigid but not minimally rigid. In dimension tw o, rigid formations are completely characterized b y Laman’s theorem. Giv en a graph with n vertices and m edges, a subgraph is obtained b y keeping a subset of the vertices and the edges that link these v ertices. Precisely , a subgraph G 0 is given by a pair ( V 0 ⊂ V , E 0 ⊂ E ) with ( i, j ) ∈ E 0 if and only if i, j ∈ V 0 and ( i, j ) ∈ E . Theorem 1 (Laman) . A planar fr amework with n vertic es is generic al ly rigid if and only if - it has 2 n − 3 e dges - for every sub gr aph with n 0 vertic es and m 0 e dges we have m 0 ≤ 2 n 0 − 3 . The first part of the statement is a simple dimensionality argument: sp ecifying n agents in the plane can b e done b y giving their 2 n coordinates from whic h w e subtract 2 degrees of freedom due to inv ariance under translation and 1 degree of freedom due to inv ariance under rotation. Hence, the non-trivial part is the second part, which we understand as a densit y argument: the 2 n − 3 edges cannot b e concentrated around to o few vertices. The pro of is rather in volv ed, and we refer to [Lam70] for more information. 3 Pro jectiv e space, Euclidean group and LS-category W e now in tro duce the main geometric notions that we will need in this pap er. Giv en a framework p , we say that vertices are total ly c oincidental if they are all mapp ed to the same lo cation b y p : p ( x 1 ) − p ( x 2 ) = . . . = p ( x n − 1 ) − p ( x n ) = 0 . W e consider frameworks suc h that the v ertices are not totally coinciden tal for the rest of this paper, but we allow t wo or more v ertices to b e mapp ed to the same p oin t. W e study the set E n of configurations of n non-totally coinciden tal agents in the plane. 8 3.1 Euclidean group The in v ariance of the framew orks under rotations and translations can b e formalized as an in v ariance under a group action. W e elab orate on this here. Recall the group S E (2) of affine rigid transformation of the plane, i.e. transformations consisting of a rotation and translation. Notice that w e do not consider inv ariance under mirror symmetry at this p oin t. This group is a three-dimensional connected Lie group. By introducing the so-called affine co ordinates [ x 11 , x 12 ] T → [ x 11 , x 12 , 1] T , w e can write a typical elemen t of this group as A ( θ , a, b ) =   cos( θ ) sin( θ ) a − sin( θ ) cos( θ ) b 0 0 1   , (3) with θ ∈ [0 , 2 π ] , and a, b ∈ R . First, observe that A ( θ , a, b ) A ( θ 0 , a 0 , b 0 ) = A ( θ + θ 0 , a + a 0 , b + b 0 ) , from whic h w e conclude that the matrices of the t yp e of Equation (3) indeed form a group (the inv erse of A ( θ , a, b ) is A ( − θ , − a, − b ) ). Consider the pro duct A ( θ , a, b )   x 11 x 12 1   =   a + cos( θ ) x 11 + sin( θ ) x 12 b − sin( θ ) x 11 + cos( θ ) x 12 1   ; the right-hand side of the ab o ve equation corresp onds to the translation b y ( a, b ) of the p oin t x 1 first rotated by an angle θ ab out the origin. In particular, the third coordinate is alw ays one. W e will write A · x =  a + cos( θ ) x 11 + sin( θ ) x 12 b − sin( θ ) x 11 + cos( θ ) x 12  to denote the group action of S E (2) on R 2 just describ ed. 3.2 Pro jective spaces In this subsection w e giv e a brief primer on pro jectiv e spaces. W e recommend to the reader who is not already familiar with these spaces to read Section 4 in parallel, as the construction done in that section illustrates man y of the definitions presented here. Let V b e a n − dimensional vector space; its pro jectiv e space V P ( n ) is the space of lines in V passing through the origin [Mun00]. Pro jective spaces are examples of quotien t spaces, i.e. 9 spaces obtained as the quotient of a manifold M by a group action [Hel78]. T o elab orate on this, we define S n − 1 to b e the unit sphere in R n cen tered at the origin: S n − 1 = { u ∈ R n | X i u 2 i = 1 } . Let V = R n ; ev ery line passing through the origin in R n in tersects S n − 1 in tw o antipo dal p oin ts. Using this observ ation, w e can define an action of the group Z 2 on S n − 1 whic h sends a p oin t ( x 1 , . . . , x n ) to its an tip o dal ( − x 1 , . . . , − x n ) . There are no p oin ts in S n − 1 suc h that x = − x . This action is thus free, or without fixed p oin t. The real pro jectiv e space R P( n ) is then defined as the set of equiv alence classes of p oin ts in S n − 1 under the ab o v e-defined Z 2 action, or similarly— since every equiv alence class consists of a p oin t in the sphere and the antipo dal p oin t— as the sphere with antipo dal points iden tified: R P( n ) = S n − 1 / Z 2 =  x ∈ S n − 1 with x ∼ − x  , where a ∼ b means that a and b are iden tified. Pro jective spaces app ear most commonly in control theory and signal processing as instances of Grassmanian manifolds, and there is large bo dy of literature dealing with how to p erform numerical computations and integration on such spaces. They are rather difficult to visualize, but there is a useful recursive construction for them whic h w e describ e here. x 2 x 2 x 1 x 1 x 3 x 3 x 4 x 4 x 2 x 2 x 1 x 3 x 3 x 4 x 4 x y Figure 3: An action of Z 2 send a p oin t x on the circle to its antipo dal − x . The quotien t S 1 / Z 2 = R P(1) is the set of equiv alence classes of p oin ts under this Z 2 action. If w e iden tify the tw o elemen ts in the equiv alence class of x 1 , we obtain tw o iden tical circles tangent at x 1 . These circles con tain the same p oin ts and each is th us a cop y of R P(1) Let us first consider the space R P(1) of lines in R 2 . By the previous paragraphs, it is obtained by iden tifying an tip odal points of the circle x 2 + y 2 = 1 in R 2 . W e can represen t ev ery equiv alence class by a p oin t in the upp er half plane (i.e. b y its representativ e with y > 0 .), with the exception of the equiv alence class of p oin ts with y = 0 , where the ab o v e rule do es not yield a unique c hoice; see Figure 3. Iden tifying the p oin ts x 1 and − x 1 turns 10 the circle into a figure 8, with its tw o circles tangent at x 1 . These are th us tw o copies of R P(1) , since they b oth con tain the same p oin ts. W e conclude that R P(1) ' S 1 . While the iden tification of pro jectiv e spaces with spheres do es not generalize to higher dimensions, the construction do es. 3.2.1 R P( n ) for n ≥ 2 Using the same approach for R P(2) , w e start with the sphere x 2 1 + x 2 2 + x 2 3 = 1 in R 3 and R P(2) is the space of equiv alence classes under the Z 2 action that sends a p oin t to its antipo dal. W e can similarly choose to represent equiv alence classes b y their representativ e with x 3 > 0 — this space is a disk of dimension tw o, or D 2 — which yields a unique representativ e except for p oin ts on the circle x 3 = 0 , since b oth p oin ts in the equiv alence class hav e the third co ordinate zero. But by the previous section, the iden tification of an tip o dal p oin ts on the circle yields R P(1) , and then w e can represent R P(2) as a disk D 2 with its b oundary S 1 ha ving its an tip odal p oin t iden tified to yield R P(1) . This construction, whic h is illustrated in Figure 4, generalizes in a straightforw ard manner to higher dimensions: R P( n ) can b e seen as a ball D n with an R P( n − 1) b oundary . Figure 4: The action of Z 2 on the sphere sends a p oin t x to its antipo dal − x . The quotient S n / Z 2 is the space of equiv alence classes under this action. W e k eep the represen tativ es of these equiv alence classes with x 3 > 0 , e.g. A o ver − A . They are in the upp er half-sphere. F or p oin ts on the equator, e.g. B and − B , b oth represen tativ es ha ve x 3 = 0 . Identifying these p oin t corresp onds to defining a copy of R P( n − 1) at the equator. 3.2.2 Complex pro jectiv e spaces 11 Complex pro jectiv e spaces are defined similarly to real pro jectiv e spaces, but the underlying v ector spaces are complex. Hence, we are looking at the space of complex lines (i.e. copies of C ) through the origin in the complex vector space C n . W e can express C P( n ) as a quotient space as follows: giv en 1 z ∈ C n +1 0 with z = ( z 0 , . . . , z n ) , z i ∈ C , the complex line passing through 0 and z is the set of p oin ts z 0 = ( λz 0 , . . . , λz n ) for λ ∈ C 0 . Th us C P ( n ) is given b y the quotien t C P( n ) ' C n +1 0 / C 0 where the action of C 0 is the m ultiplication b y λ giv en ab o ve. An elemen t in C P( n ) is often giv en by a point in C n +1 0 , with the understanding that this p oin t is defined up to multipli- cation by λ ∈ C 0 ; these are the so-called homo gene ous c o or dinates and they are denoted b y a brack et notation [ z 0 : . . . : z n ] to indicate that ( z 0 , z 1 , . . . , z n ) is a represen tative of an equiv alence class: [ z 0 : . . . : z n ] ' [ λz 0 : . . . : λz n ] , ∀ λ ∈ C 0 . Observ e that having all the z i = 0 is not allo wed since in that case λz do es not define a line. Using homogeneous co ordinates, w e can exhibit copie s of C P( n − 1) in C P( n ) , similarly to what was done in the real case. First, observ e that if z 0 6 = 0 , we can divide the homogeneous co ordinates b y z 0 to obtain [1 : z 1 /z 0 : . . . : z n /z 0 ] , and if z 0 6 = 0 , all the z i , i 6 = 0 can b e zero sim ultaneously since z 0 = 1 ensures that not all co ordinates are zero. Th us if z 0 6 = 0 , the coordinates z 1 , . . . z n are in C n without restriction. This space is top ologically equiv alent to a disk D 2 n of real dimension 2 n . When z 0 = 0 , the homogeneous co ordinates are of the form [0 : z 1 : . . . : z n ] : they are the co ordinates of a copy of C P( n − 1) . Hence, we ha ve a decomp osition, similar to the one of R P( n ) , of C P( n ) into a disk D 2 n and with C P( n − 1) at its b oundary . Using the ab o ve construction recursiv ely , we obtain C P( n ) = D 2 n ∪ D 2 n − 2 ∪ . . . ∪ D 0 where D 0 is a point. Eac h time a disk is added in the construction, some identifications of p oin ts, as describ ed in the ab o ve paragraph, ha ve to b e made. 3.3 The Lusternic k-Schnirelmann category The Lusternic h-Schnirelmann category of a manifold, or LS-category , is an imp ortan t top o- logical in v ariant that is related to the minimal n um b er of c harts con tained in any atlas for the manifold. W e describ e it in this section. A set S in R n is homotopic [Mun00] to a point if there exists a contin uous map F ( t, x ) : [0 , 1] × R n → R n suc h that F (0 , x ) is the iden tity on S and F (1 , x ) maps S to a p oin t. Let M be a smo oth manifold of dimension n and V = { V i , i = 1 . . . k } b e a collection of closed connected sets which are homotopic to a p oint , or null-homotopic. Conv ex sets, or 1 C n 0 = C n − { 0 } 12 (a) A null-homotopic set in R 2 (b) A non-null- homotopic set. Figure 5 sets that are homeomorphic to a conv ex set, are homotopic to a p oin t; sets that are not n ull-homotopic include sets which hav e “holes” in them. In dimension 1 , the n ull-homotopic sets are the connected interv als. In R 2 , n ull-homotopic sets are the simply c onne cte d sets : sets suc h that every closed path contained in them can b e con tin uously deformed to a p oin t (see Figure 5). Let I b e an index set and V I = { V i , i ∈ I | V i is a closed set } a collection of closed sets indexed b y I . W e sa y that the collection V I is a cov er for M if M ⊂ ∪ i ∈ I V i . The Lusternick-Sc hnirelmann category [CLOT03] of M , or LS-category , is the top ological in v arian t defined as follows: Definition 6 (Lusternic k-Schnirelmann Category) . Given a close d manifold M , the Lusternick- Schnir elmann c ate gory of M , written cat( M ) , is the the le ast c ar dinality | I | , over al l p ossible V I that ar e c overs for M . F or example, it is easy to see that one needs at least tw o connected interv als to cov er the circle S 1 . Hence cat( S 1 ) = 2 . A ctually , one can pro ve that cat( S n ) = 2 : t wo disks of dimension n are needed to cov er S n . Even though the original definition of the LS- category was in terms of closed cov ers as in tro duced ab o ve, it is no wada ys more common to encounter a definition in terms of op en cov ers, i.e. co vers where the sets V i are op en. W e refer to [CLOT03] for more information and relations b et ween the t wo quan tities. F or our purp oses, we will need the Lusternick-Sc hnirelmann category of complex pro- jectiv e spaces, whic h is kno wn to b e [CLOT03] cat( C P( n )) = n + 1 . 4 The Geometry of the space of formations W e define E n to be the space of equiv alence classes, under rotations and translations, of formations of n agents in the plane. W e use the notions introduced in the previous 13 section to characterize these spaces. F or a differen t point of view with an eye on statistical applications, we refer to the excellent surv ey [Ken89]. W e start by the space E 3 of three agen ts in the plane. W e w ork out this case in details as it sheds light on the more abstract constructions needed for n agents. W e recall here that the agen ts in the formation are lab el le d . 4.1 The space E 3 Consider three agen ts x 1 , x 2 and x 3 in the plane R 2 . W e can describe their p osition with a vector x ∈ R 6 with the first tw o co ordinates of x giving the p osition of x 1 , the next t wo co ordinates the p osition of x 2 and so forth. W e will write for the rest of this section x = [ x 1 , x 2 , x 3 ] T for x ∈ R 6 . Since formations are defined up to a translation and rotation, it is clear that the ab o v e co ordinates on the set of formations with 3 agents is redundan t. This redundancy takes the form of a group action of S E (2) on R 6 giv en b y A · x =   Ax 1 Ax 2 Ax 3   (4) where the action of A ∈ S E (2) on x i ∈ R 2 is the one given in the previous section. The space of totally coincidental formations is the subspace of R 6 defined as N =  x ∈ R 6 | x 1 = x 2 = x 3  . F act 1 : N is inv ariant under the action of S E (2) . This fact is obvious since x i = x j implies Ax i = Ax j . F act 2 : E 3 is connected. Since N is a linear subspace of co dimension four, it do es not separate R 6 in tw o regions and E 3 has a single connected comp onen t. W e now sho w that the space E 3 is actually isomorphic to R 3 − { 0 } , but the mapping that sends a p oin t in R 3 to a triangle is not obvious. W e will later give a more abstract pro of of this fact. Prop osition 1. The sp ac e E 3 of triangular formations in the plane is R 3 0 . Pr o of. W e work out explicitly the quotien t ( R 6 − N ) /S E (2) . Given an equiv alence class of triangles, we can alwa ys choose a representativ e with x 1 = (0 , 0) , which takes care of the translational redundancy . If we assume that x 1 6 = x 2 , w e can take a representativ e suc h that the x 1 − x 2 edge is aligned with the x − axis in R 2 . Hence, we can describ e a non-degenerate triangle b y a point u ∈ R 3 0 with x 1 = (0 , 0) , x 22 = 0 and x 21 = u 1 , x 31 = u 2 and x 32 = u 3 . This represen tation is still redundan t: a p oin t u yields a triangle whic h is 14 equiv alent by a rotation of 180 degrees to the formation given by − u . The quotien t of R 3 0 b y u ' − u , i.e. the space of formations with x 1 not equal to x 2 , is R P(2) × R + as describ ed in the previous section. W e now show ho w the degenerate formations are attac hed to the space of non-degenerate formation just describ ed. If x 1 = x 2 , then x 2 6 = x 3 —otherwise the formation is totally coinciden tal. Assume w e hav e u 1 = 0 and ( u 2 , u 3 ) 6 = (0 , 0) . Geometrically , this corresponds to c ho osing the p oin t x 3 in the plane minus the origin. This p oin t can b e describ ed in p olar co ordinates as x 3 = r e iθ . T o simplify matters, we assume that the formation is normalized in the sense that the sum of the edge lengths is fixed. It is easy to see that this fixes a v alue in R + for co ordinates in the pro duct R P(2) × R + . Hence, the space of (normalized) formations with x 2 6 = x 1 is R P(2) and the formations with x 2 = x 1 , which are describ ed by z 3 = r e iθ , are a circle on its b oundary . The situation is similar to the one depicted in Figure 4, where the upp er half-sphere corresp onds to formations with x 2 6 = x 1 and the equator to formations with x 2 = x 1 . Since the formations are defined up to a rotation, all formations with x 2 = x 1 are iden tified. This corresp onds to shrinking the circle in Figure 4 to a point, whic h yields tw o iden tical spheres S 2 touc hing at a p oin t similarly to the situation of Figure 3. Rep eating this construction for every normalization co efficien t in R + , we conclude that the space of formations is S 2 × R + ' R 3 0 .  4.2 P arametrization of E 3 b y edge lengths W e now lo ok at the relation b et ween the lo calized co ordinates L and the geometry of E 3 . Let us write d 1 = k x 2 − x 1 k , d 2 = k x 3 − x 2 k and d 3 = k x 3 − x 1 k . Using the result of Prop osition 1, we know w e can consider the space of formations where, sa y , the sum of edge lengths d 1 + d 2 + d 3 = 1 , and the space of suc h formations is then the 2 − sphere S 2 . The edge lengths hav e to satisfy the triangle inequalities d 1 ≤ d 2 + d 3 , d 2 ≤ d 3 + d 1 , d 3 ≤ d 1 + d 2 . (5) Th us the set of admissible d i forms a con vex subset of R 3 , whose edges are such that the ab o v e inequalities are equalities (see Figure 6). Prop osition 2. Ther e is an op en set U ⊂ L such that ther e ar e at le ast two fr ameworks that c orr esp ond to ( d 1 , d 2 , d 3 ) ∈ U . The proof establishes a link b et w een the num b er of non-congruen t formations and the top ology of S 2 . W e will generalize this result b elo w. Pr o of. The set describ ed by the inequalities (5) is a cone in R 3 . It is easy to see that an in tersection of this cone with a plane orthogonal to the main diagonal 2 is a conv ex set which 2 w e call main diagonal in R n the subspace spanned by the vector (1 , 1 , . . . , 1) 15 Figure 6: The set L for triangular formations is a cone in R 3 + . W e depict its intersection with a plane with normal [1 , 1 , 1] at [ a, a, a ] . W e call this in tersection L a . corresp onds to having the sum of the edge lengths normalized. Precisely , if the orthogonal plane intersects the main diagonal at ( a, a, a ) , then d 1 + d 2 + d 3 = a ; w e call this intersection L a . Fixing the v alue of a corresp onds to fixing the v alue of the v ariable corresp onding to R + in the description of E 3 . Without loss of generality , we consider the space of normalize d formations in three agents, whic h is S 2 b y Proposition 1. Consider the map ν : S 2 → L a : p ( V ) → δ ( p ( V )) that maps frameworks to edge lengths. Because ν is contin uous, it is enough to sho w that it is not injective to pro ve the existence of U . By definition of L , ν is onto. Reasoning b y a con trap ositiv e argumen t, assume that ν is injectiv e; in other w ords, there is only one framework for every giv en v ector ( d 1 , d 2 , d 3 ) ∈ L a . This means that w e can uniquely assign a p oin t in S 2 to ev ery ( d 1 , d 2 , d 3 ) and vice-v ersa. But this is in contradiction with the fact that the Lusternic k-Schnirelmann category of S 2 is 2 : closed conv ex sets are n ull-homotopic and the ab o ve claim is equiv alent to cov ering S 2 with a single such set.  It is easy to see that in the case of triangles, there are exactly tw o suc h formations; they corresp ond to a triangle and its mirror symmetric, as illustrated in Figure 7. 16 Figure 7: The space of normalized triangles in the plane is S 2 . The set of normalized edge lengths corresp onds to a triangle, depicted in Figure 6, whic h w e ha v e deformed abov e in to a disk L a . T o each p oin t in the disk corresp ond tw o triangles with the same edge lengths but different orien tations. The map ν of Prop osition 4.2 is a pro jection of the sphere onto the disk L a . The kno wledge of the lo calized co ordinates (a p oin t in the disk) is thus not sufficien t to kno w the configuration of the ensemble (a p oin t in S 2 ). 4.3 The space E n W e describ e the space of configurations of n − agents in the plane. Theorem 2. The sp ac e of e quivalenc e classes of formations of n agents in the plane E n is E n ' C P( n − 2) × R + . Pr o of. Recall that a general formation in the plane has 2 n − 3 degrees of freedom, tw o for eac h of the n agents and three degrees of freedom are lost to translational and rotational in v ariance. The pro jectiv e space C P( n − 2) has a complex dimension of n − 2 and real dimension of 2 n − 4 , and thus dim( C P( n − 2) × R + ) = 2 n − 3 . Without loss of generality , w e iden tit y the plane R 2 with C , hence the p ositions of the agen ts are giv en b y n complex num b ers z 1 = x 11 + j x 12 , . . . , z n = x n 1 + j x n 2 . In complex co ordinates, the rotation of a formation b y an angle θ corresp onds to m ultiplying the co ordinates z i b y e j θ . Similarly to what w as done in the case of n = 3 agents, we use the translational in v ariance to put the first agen t at 0 + j 0 . Thus a formation is describ ed b y n − 1 complex n umbers ( z 2 , z 3 , . . . , z n ) with the identification ( z 2 , z 3 , . . . , z n ) ' e j θ ( z 2 , z 3 , . . . , z n ) , ∀ θ . 17 Recall that the set obtained via the identification ( z 2 , z 3 , . . . , z n ) ' r e j θ ( z 2 , z 3 , . . . , z n ) , ∀ θ ∈ [0 , 2 π ] , r ∈ R + is C P( n − 2) . Hence, we ha ve E n = C P( n − 2) × R + as announced.  4.4 P arametrization by edge lengths F ormation control is concerned with finding feedback laws that stabilize agents at a giv en distance from a subset of the other agents, doing so based only on partial information ab out the formation. It is thus of in terest to kno w ho w man y formations satisfy the distance constrain ts specified. The first requiremen t that w as encountered w as the one of rigidit y: if the formation is not rigid, there exists a con tinuous set of formations that satisfy the sp ecified edge lengths. The study of these manifolds has a long history that can b e traced back to at least Lippman Lipkin and P eaucelier. They were inv estigating the design of non-rigid frameworks that w ould transform a circular motion to a straight-line motion. In this con text, the framework is understo od as an input/output system, and some of the vertices hav e fixed p ositions. More recen tly , William Th urston prov ed that using a complex enough framework, an y curve in the plane can b e approximately obtained as an output of a non-rigid framework [Thu97], or that one “could sign one’s name” with a planar framew ork. Recall that if a framework is rigid, it do es not in general imply that only this framework and its mirror symmetric satisfy the giv en edge length constraints. W e called frameworks for which this is true glob al ly rigid. Before discussing this p oin t an y further, w e formally define the mirror symmetry of a framew ork x to b e the op eration that sends x i =  x i 1 x i 2  →  x i 1 − x i 2  . Notice that the mirror symmetric of a framework is not equiv alen t to the original formation via a rigid transformation since it rev erses the orien tation. Unlik e minimal rigidity , global rigidit y is a function of the framework and not of the underlying graph G . Moreo ver, deciding whether a given framew ork is globally rigid is an NP-hard problem [EBM79], ev en for one dimensional frameworks. In order to av oid com- plications inheren t to this dependence on a particular framew ork, one considers generic al ly glob al ly rigid formations, as they can b e characterized in terms of their underlying graphs: Theorem 3 (Hendric kson, Connelly) . A gr aph G with n ≥ 4 vertic es is generically globally rigid in R 2 if and only if G is generic al ly r e dundantly rigid and vertex 3-c onne cte d. 18 x 1 x 2 x 3 x 4 (a) x 1 x 2 x 3 x 4 x 5 x 6 (b) Figure 8: (a): A generically globally rigid framework. (b): A globally rigid framew ork that is not generically globally rigid [Con05]. It is globally rigid if and only if the v ertices lie on a conic in R 2 . Observe that it is also minimally rigid. A graph is generically redundantly rigid if, for an y generic framework with underlying graph G , deleting one edge leav es the framework rigid. Hence, the minimal ly rigid forma- tions, which are the main ob ject of study in co operative control, ar e not generic al ly glob al ly rigid. This yields a lo wer b ound of 4 on the num b er of equilibria of formation con trol system with n ≥ 4 agents solely relying on the inter-agen t distance to stabilize a minimally rigid formation. Indeed, the mirror symmetry insures that there alwa ys is an even num b er of framew orks with a giv en edge lengths vector, and this num b er is tw o only for globally rigid formations. A finer b ound on the num b er of generic non-congruent formations in the plane for a giv en graph can b e obtained b y relating top ological c haracteristics of L and C P( n ) , as w e sho w below. Theorem 4. L et G b e a rigid gr aph on n ≥ 4 vertic es. If the set of lo c alize d c o or dinates L is nul l-homotopic, then ther e ar e at le ast 2 d ( n − 1) 2 e fr ameworks for a generic e dge length ve ctor and, in p articular, G is not glob al ly rigid. Recall that d α e is the smallest integer with α ≤ d α e . This result giv es a low er b ound on the num b er of non-congruent formations and an obstruction to global rigidit y: a formation c annot b e glob al ly rigid if L is nul l-homotopic. Pr o of. W e first observe that for any feasible d ∈ L ⊂ R m + and α ∈ R + , w e ha v e αd ∈ L . W e can thus normalize the sum of the d i b y taking the intersection of L with the hyperplane orthogonal to the unit diagonal in R m and passing through α > 0 , similarly to what was done in previous sections. W e call this in tersection L α . W e hav e that L α is non-empty if L is non-empty and, b y definition, P m i =1 d i = α for d ∈ L α . W e no w pro ve that L α is closed. Consider the map Φ : L α → C P( n − 2) 19 x 1 x 2 x 3 x 4 x 5 x 6 (a) x 1 x 2 x 3 x 4 x 5 x 6 (b) x 1 x 2 x 3 x 4 x 5 x 6 (c) x 1 x 2 x 3 x 4 x 5 x 6 (d) Figure 9: F our non-congruen t frameworks underlying the same minimally rigid graph on 6 v ertices in R 2 . 20 x 1 x 2 x 3 x 4 x 5 (a) x 1 x 2 x 3 x 4 x 5 (b) Figure 10: The t wo frameworks are minimally rigid in the plane, but are not isomorphic as graphs since x 3 has a different degree in (a) and (b). that tak es d to a framework in the plane with edge lengths giv en b y d . Because G is rigid, there is a finite num b er of suc h frameworks. The map Φ is m ulti-v alued and con tinuous on ev ery branc h of the image. This map is also on to: indeed, for every p oin t in C P( n − 2) , w e can pick a formation in the plane as in Theorem 2 with an arbitrary orientation. The pre-image of this p oin t under Φ is obtained by reading the relev ant edge lengths. Because Φ is contin uous, the preimage of a closed set under Φ is closed, and thus L α is closed. A dditionally , if L is null-homotopic, then clearly L α is null-homotopic for all α . Hence, if for a generic d ∈ L α there are k distinct formations in the plane, then we ha ve a closed co ver of C P( n − 2) b y k closed sets. Recall that the LS-category is a low er b ound on the cardinality of closed co vers on a space and th us cat( C P( n − 2)) = n − 1 ⇒ k ≥ ( n − 1) . Because the n umber of framew orks for a giv en d is alwa ys even, w e hav e that k ≥ 2 d n − 1 2 e .  4.5 The space E 4 W e explicitly describ e the space E 4 of framew orks with four v ertices in the plane. Observ e that all minimally rigid frameworks with 4 agents are of the t yp e of the (undirected) 2-cycles depicted in Figure 1b. This uniqueness prop ert y is lost for minimally rigid frameworks with n ≥ 5 vertices, as one can easily exhibit non-equiv alent minimally rigid framew orks with n = 5 (see Figure 10). The constraints on the edge lengths for the 2-cycles are giv en by d 3 ≤ d 1 + d 2 d 1 ≤ d 2 + d 3 d 2 ≤ d 3 + d 1 d 3 ≤ d 4 + d 5 d 4 ≤ d 5 + d 3 d 5 ≤ d 3 + d 4 d i ≥ 0 21 F rom these relations, we see that L is a con vex set. In fact, it is a cone in the positive orthan t of R 5 . Summing all the inequalities ab o ve, we ha ve P 5 i =1 d i + d 3 ≤ 2 P 5 i =1 d i + 2 d 3 , whic h simplifies to P 5 i =1 d i ≥ 0 . Hence, there are only five indep enden t inequalities. Moreo ver, some b ecome equalities on the hyperplanes that define the p ositiv e orthant in R 5 : for example, setting d 1 = 0 , the first and third inequalities yield d 2 = d 3 , etc. W e conclude that the picture is similar to the one for the three agent frameworks of Figure 6, alb eit in dimension fiv e. A ccording to Theorem 4, there are at least 2 d 4 − 1 2 e = 4 framew orks in the plane satisfying a generic edge lengths v ector d ∈ L . By insp ection, it is easy to see that there are exactly four suc h frameworks. It is informative to relate these frameworks to a discrete group action on E 4 . Giv en a framework in the plane with v ertices x 1 , . . . x 4 and such that d 3 > 0 , w e define: z 1 = x 2 − x 1 , z 2 = x 3 − x 2 , z 3 = x 1 − x 3 , z 4 = x 3 − x 3 , z 5 = x 4 − x 1 , (6) as in Figure 1b, and let z ⊥ 3 b e the orthogonal unit vector to z 3 . W e define the r efle ction symmetry along z 3 as R z 3 ( x ) = x − 2 h x, z ⊥ 3 i z ⊥ 3 , x ∈ R 2 , where h· , ·i is the usual inner pro duct in R 2 . W e then define the t wo operations R 1 ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 , R z 3 ( x 2 ) , x 3 , x 4 ) R 2 ( x 1 , x 2 , x 3 , x 4 ) = ( x 1 , x 2 , x 3 , R z 3 x 4 ) . W e clearly ha v e that R 2 i = 1 and R 1 R 2 = R 2 R 1 . (7) Consider the discrete group Z 2 × Z 2 with elements (1 , 0) , (0 , 1) , (1 , 1) , (0 , 0) and the group op eration is addition mo dulo 2. W e can identify R 1 and R 2 with the elements (1 , 0) and (0 , 1) : if w e understand composition of symmetries as addition of elements in Z 2 , then Equation (7) tells us that the symmetries R 1 and R 2 ob ey the same m ultiplication table as (1 , 0) and (0 , 1) . In this context, the identit y for framewo rks corresp onds to the additive iden tity (0 , 0) ∈ Z 2 × Z 2 . Indeed R 1 R 1 = 1 ↔ (1 , 0)+ (1 , 0) = (0 , 0) , R 1 R 2 ↔ (1 , 0)+ (0 , 1) = (0 , 1) + (1 , 0) ↔ R 2 R 1 , etc. W e also observ e that the elements (0 , 0) , (1 , 1) ∈ Z 2 × Z 2 form a prop er subgroup which corresp onds geometrically to the mirror symmetric of the formation. This is summarized in the table b elo w, where I is the iden tity: R 1 = (1 , 0) R 2 = (0 , 1) R 1 I=(0,0) R 1 R 2 = (1 , 1) R 2 R 1 R 2 = (1 , 1) I=(0,0) The section of the cone L obtained b y fixing the sum of the edge lengths, similarly to Figure 6, is a simplex of dimension 4. This simplex is a closed null-homotopic set, and four 22 copies of it are needed to cov er C P(2) as was shown in Theorem 4. W e represen t the four 4-simplices that cov er C P(2) in Figure 11. This figure is the equiv alent of Figure 7 for the case of n = 4 agents. Eac h of the 4 rectangles in the figure corresp onds to a 4-simplex and con tains frame- w orks related to frameworks in the other 4-simplices via the symmetries R 1 and R 2 . W e call LR the simplex corresp onding to frameworks where 2 is on the left of the 1 - 3 axis and 4 on the righ t, LL the simplex where b oth 2 and 4 are on the left, etc. W e illustrate ho w the simplices intersect on some co dimension 1 faces, corresp onding to the in v ariant space of R 1 and R 2 —i.e. the space of framew orks whic h are their own symmetric under R 1 or R 2 . The in tersection of these inv ariant spaces is a co dimension 2 space whic h corresp onds to frame- w orks inv ariant under both R 1 and R 2 . This inv ariance is tantamoun t to inv ariance under mirror symmetry . Since the sum of the edge lengths has b een normalized, each d i has an upp er b ound. Observ e that b y taking the limit as d 2 gro ws to its upp er b ound, we find that the left boundary of the figure contains framew orks whic h hav e three v ertices coinciden tal. But these frameworks are also inv arian t under R 1 and R 2 and thus there is an identification b et w een parts of this b oundary and the cen ter corresp onding to the framew orks inv ariant under mirror symmetry . In general, since we are representing a five dimensional space in t wo dimensions, we hav e to make some choices as to whic h c haracteristics of the space to represen t. In Figure 11, we will not describ e the boundaries into detail, except to mention that they contain degenerate framew orks and that in tricate iden tifications happ en there. 4.6 Henneb erg sequences and group actions In this section, w e elab orate on a p ossible relation b et w een Henneb erg sequences, defined b elo w, and discrete group actions on minimally rigid frameworks. W e hav e shown that in the case n = 4 , the four frameworks that satisfy a giv en set of edge lengths can b e characterized as the orbit of the group Z 2 × Z 2 acting on an y formation that satisfies these edge lengths. W e did so by assigning to each generator of the group a reflection in the plane (namely , R 1 w as asso ciated with (1 , 0) and R 2 with (0 , 1) ) and v erified that these t w o indeed ob ey ed the same multiplication table as Z 2 × Z 2 . A ma jor result ab out minimally rigid formations in the plane is the Henneb erg Theo- rem and the asso ciated Henneb er g sequence [GSS93]. They state that all minimally rigid framew ork in the plane can b e obtained inductiv ely , starting with a single line segment (t wo v ertices and one edge) and applying at eac h time step one of the tw o op erations describ ed b elo w. Giv en a framew ork at step n − 1 with a set of v ertices V n − 1 and a set of edges E n − 1 of cardinalities n − 1 and m − 1 respectively , w e define: - vertex add: add a vertex to the framew ork and link it to tw o distinct existing vertices. The choice of v ertices is arbitrary . Specifically , for i 6 = j , i, j ≤ n − 1 , we ha ve: V n = V n − 1 ∪ { x n } E n = E n − 1 ∪ { ( x n , x i ) , ( x n , x j ) } . 23 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 mirror symm., (1,1), R 1 symm., (1,0) R 2 symm., (0,1) LR RR LL RL Figure 11: A depiction of the cov er of C P(2) b y four 4-simplices, depicted here as rectangles. Eac h simplex contains framew orks whic h can b e iden tified b y the p osition of 2 and 4 with resp ect to the 1 - 3 axis. F or example, the top-right simplex contains framew orks with 2 on the left and 4 on the right. The vertical axis corresp onds to a facet (i.e. subspace of co dimension 1) of the simplices which con tains frameworks that are inv ariant under R 1 . Similarly , the horizontal axis corresp onds to frameworks that are inv ariant under R 2 . The in tersection is a subspace of co dimension 2 which corresp onds to formations in v ariant under mirror symmetry . - Edge-split: c ho ose an edge in E n − 1 and delete it from the framework. A dd a vertex and link it to the t wo v ertices to which the deleted edge w as inciden t and to a third, distinct vert ex. Sp ecifically , for ( x i , x j ) ∈ E n − 1 and k 6 = i, j , we ha ve: V n = V n − 1 ∪ { x n } E n = E n − 1 ∪ { ( x n , x i ) , ( x n , x j ) , ( x n , x k ) } − { ( x i , x j ) } . W e illustrate the operations in Figure 12. The sequence of op erations obtained is called Henneb er g se quenc e . It has the property that all framew orks in the sequence are minimally rigid. There is no unique Henneb erg sequence leading to a minimally rigid framew ork in general. Consider the Henneb erg sequence leading to the 2-cycles that is depicted in Figure 13 The first op eration is a vertex-add applied to a single segmen t. This op eration can b e 24 x 1 x 2 x 3 x 4 v ertex-add x 1 x 2 x 3 x 4 (a) x 1 x 2 x 3 z 1 z 1 edge-split x 1 x 2 x 3 x 4 z 1 (b) Figure 12: The t wo basic operations of the Henneb erg sequence are illustrated. realized in t wo different wa ys: either the vertex is added to the left of the segmen t or to its righ t. This choice results in the group of symmetry Z 2 acting on the resulting triangular framew ork. There is an iden tical choice for the second vertex-add op eration which yields Z 2 × Z 2 as the symmetry group of the resulting framework, since the choices ab o v e are indep enden t. In summary , constructing the 2-cycles can b e done by tw o vertex-add op erations and the choices with which one can p erform these vertex-adds can b e related to the addition of the groups Z 2 as symmetry groups of the resulting framew orks. A natural question is th us whether one can find a general sc heme to relate a symmetry group to the Henneb erg sequences defining the framew ork. Such a c haracterization, by allo wing to resort to group theoretic techniques and the known classification of discrete groups, w ould greatly deep en our understanding of b oth directed and undirected formation con trol. 5 Decen tralized con trol with directed graphs In this section, w e formally in tro duce the decen tralized control mo del used. W e consider kinematic mo dels of the form ˙ x i = u i 25 x 1 x 3 x 1 x 3 x 2 x 0 2 Z 2 x 1 x 3 x 2 x 4 x 0 4 Z 2 Figure 13: A Henneb erg sequence for the 2-cycles. Starting with tw o no des ( x 1 , x 3 ) joined b y an edge, we used tw o vertex-add op erations to in tro duce x 2 and x 4 . Eac h vertex-add op eration adds a copy of the group Z 2 as the symmetry groups of the formation. The t wo op erations being commutativ e, the final symmetry group is Z 2 × Z 2 . The formation control problem is then the following: F ormation c ontr ol pr oblem : Giv en a graph G = ( V , E ) , and a set of target distances d ∈ L , find controls u i , which resp ect the information flow describ ed by G , such that for almost all initial conditions in R 2 n , the system stabilizes to a framew ork p with δ ( p ) | E = d — in other words, as t → ∞ , the inter-agen t distances are giv en by d . In order for the set of framew orks with in ter-agent distances given by d to b e finite, the underlying graph has to b e rigid. If it is moreov er minimally rigid, no edges in the graph can b e spared without losing rigidity . 5.1 F easibility Determining which frameworks are achiev able by agents in a formation con trol problem can b e a somewhat delicate problem when the underlying information flow is given b y a dir e cte d graph. In particular, the fact that the graph is rigid, or even minimally rigid, is not enough to ensure that the agents will b e able to reach the desired formation, and rigidit y is not alw ays needed. W e will sho w some examples below that illustrate some of these issues and refer the reader to [BS03, HADB07] and references therein for further information. Let us in tro duce some of the terminology used: an agent is called a le ader if it is not follo wing any other agent. An agen t is a c ole ader if it follo ws some agents and is himself b eing follow ed b y agents. An agent is a fol lower if it just follows agen ts and is not b eing follo wed. Observ e that any formation which has a follo wer with a single leader will not b e rigid, but nev ertheless ma y be feasible as is the case with the formation depicted in Figure 14a. The formation of Figure 14b, whic h has a follo wer with three coleaders is not feasible. Indeed, agen t x 3 has no means to influence the p osition of agen ts x 1 , x 2 and x 4 and these three will settle generically at a p osition whic h is not compatible with the constraints that 26 x 3 has to satisfy . The problem with this formation is easily identifiable: x 3 is follo wing more than 2 agents and the underlying graph is minimal ly rigid . In contrast, in the formation of Figure 14c, x 3 similarly has an outv alence greater than 2 but nev ertheless can satisfy its three constraints since the graph without x 3 is rigid and the constraints are consisten t—a consequence of the fact that the underlying graph is r e dundantly rigid . x 1 x 2 x 3 x 4 (a) x 1 x 2 x 3 x 4 (b) x 1 x 2 x 3 x 4 (c) Figure 14 5.2 Con trol problem 5.2.1 Distributed formation con trol W e describ e a general dynamical mo del for formation con trol with directed information flo w. The mo del has to reflect the following three conditions: 1. Eac h agen t is only a ware of the target lengths it has to achiev e. 2. Eac h agen t is only a ware of the p osition of its (co)leaders. 3. The agents do not share a common reference frame and only know the r elative p osi- tions of their coleaders with respect to their own positions. The first condition states that agents are only a ware of th e ob jectiv e they hav e to ac hieve, and not the ob jective of the formation as a whole. In particular, agents do not kno w how many other agents are in the formation. The second condition says that the information flow in the system is giv en by the underlying graph. These first tw o conditions are at the core of decen tralized control problems. The third condition enforce that the resulting system is in v ariant under the S E (2) action described in Section 4. F or example, if x i is such that ( i, j ) , ( i, k ) ∈ E , then u i is allow ed to dep end on x j − x i and x k − x i . T o the b est of our knowledge, these assumptions are v erified by most mo dels that hav e app eared in the literature. An exception is the discussion in [Y ADF09], which separates the design stage from the dynamics stage and hence each agent is designed with a complete kno wledge of the desired final formation. The first condition is then not satisfied. This has yielded only results of lo cal nature. 27 5.2.2 The mo del If edge l connects agents i and j , we write e l = k x i − x j k 2 − d l , the error in square edge length. W e follow this conv ention, without loss of generalit y , in order to hav e the d i en ter the dynamics linearly . W e hav e the following model: x i x j x k z r z s Figure 15 1. Agen t with outv alence of 1 : If agent i has a unique leader j and the target length for k x i − x j k is d s , then ˙ x i = u ( d s ; e s )( x j − x i ) (8) 2. Agen t with outv alence of 2 : Assume that agent i has a tw o leaders j, k and that the target edge lengths are d r and d s resp ectiv ely (see Figure 15). W e will consider t wo cases, according to whether agen t i is able to measure explicitly ho w agen t k and agen t l are positioned relativ e to eac h other: (a) Distance only: In this case, the mo del is an extension to tw o v ariables of the mo del for an agen t with a single leader: ˙ x i = u 1 ( d r , d s ; e r , e s )( x j − x i ) + u 2 ( d r , d s ; e r , e s )( x k − x i ) (9) (b) Distance and angle: Define z s = x j − x i and z r = x k − x i . The mo del is ˙ x i = u 1 ( d r , d s ; e r , e s , z s · z r )( x j − x i ) + u 2 ( d r , d s ; e r , e s , z s · z r )( x k − x i ) . (10) Kno wing d r , d s , e r , e s as well as the inner product z r · z s indeed allows agent i to reconstruct the relativ e p ositions of agents j and k using simple trigonometric rules. 28 There are similar laws for agen ts with a higher outv alence. F or the reasons men tioned, w e only consider in the pap er agen ts with an outv alence of at most 2. W e alw a ys ha ve the additional implicit condition on the u i ’s that the resulting differenti al equation admits a solution on a long enough time in terv al. Equations (9) and (10) allow agents with tw o leaders to treat them differently . W e will also consider here mo dels that preven t this differentiation: ˙ x i = u ( e r , e s , z r · z s ) z r + u ( e s , e r , z r · z s ) z s . (11) Indeed, observ e that substituting e r for e s and z r for z s lea ves equation (11) unchanged. W e conclude this section by verifying that the mo del is inv ariant under the S E (2) action in tro duced ab o ve. Lemma 1. The dynamic al system describ e d by Equations (8) , (9) and (10) defines a smo oth dynamic al system on E n . Pr o of. Since E n is a quotien t space, it is sufficien t to chec k that the dynamics is equiv arian t under the group action, or in other words that if ˙ x = F ( x ) and S ∈ S E (2) , then S ˙ x = F ( S x ) = S F ( x ) . The result is a simple consequence of the fact that the e i are clearly inv ariant under an action of S , and so are the inner products z i · z j .  The description of the system in terms of the z v ariables is redundant. Because the LS-category of the state space of the system is strictly greater than 1, we cannot write a unique set of differential equations in terms of non-redundant v ariables, such as the edge lengths, to describ e the ev olution of the system. W e sho w that the assumptions on the information flo w put constraints on the con trol terms. Giv en a rigid graph and d ∈ L , we call design e quilibria the framew orks such that e i = 0 . Theorem 5. The design e quilibria of a le aderless infinitesimal ly rigid formation ar e such that u i (0 , 0 , w ) = u j (0) = 0 . The theorem states that the system cannot b e at a design equilibrium while undergoing a rigid transformation; e.g. the system cannot b e such that e i = 0 while the formation undergo es a translation. W e need t wo preliminary lemmas. W e first derive a useful formula for the dynamics in terms of the z v ariables. Let z = [ z T 1 , z T 2 , . . . , z T M ] T . 29 Lemma 2. Given a gr aph G with e dge-adjac ency matrix A e , define the diagonal matrix D with D ii = u ( e i ) if e dge i originates fr om an agent with a single le ader and D ii = u i ( e i , e j , z T i z j ) if e dge i originates fr om an agent with two le aders and z j links the origin of i to the other le ader. W e have ˙ z = A (2) e D (2) z . (12) W e recall that A (2) e is a shorthand notation for A e ⊗ I where I 2 is the 2 × 2 identit y matrix and ⊗ the Kroneck er tensor product. Pr o of. Notice that a cycle in the underlying graph of a framew ork yields a linear relation that the z v ariables hav e to satisfy . Hence, the z v ariables are not indep enden t if the underlying graph is not a tree and there are many non-equiv alent wa ys of writing the dynamics in terms of these v ariables. W e will v erify form ula (12) ro w b y row. Consider the follo wing generic situation depicted in the figure b elo w: x 1 follo ws x 2 and x 3 , x 2 follo ws x 4 and x 5 and x 3 follo ws x 6 , with z 1 = x 2 − x 1 and x 2 = x 3 − x 2 . W e hav e ˙ z 1 = ˙ x 2 − ˙ x 1 = u 1 ( e 3 , e 4 , z T 3 z 4 ) z 3 + u 2 ( e 3 , e 4 , z T 3 z 4 ) z 4 − u 3 ( e 1 , e 2 , z T 1 z 2 ) z 1 − u 4 ( e 1 , e 2 , z T 1 z 2 ) z 2 . Observ e that the first ro w of the edge adjacency matrix corresp onding to this formation is  − 1 − 1 1 1 0 . . . 0  . Hence, this equation is indeed the first row of (12). x 1 x 2 x 3 x 4 x 5 x 6 z 1 z 2 z 3 z 4 z 5 W e ha ve similar relations for the other ro ws of (12).  Lemma 3. L et G = ( V , E ) r epr esent a le aderless formation with | V | = n and | E | = m . Then ther e exists a matrix K ∈ R n × R m of r ank n such that A e = A m K. 30 Pr o of. Fix a vertex s in V and let j be an edge originating from this v ertex. Since G is leaderless, w e know that such a j exists. F rom the definition of A e , w e hav e that A e,kj is 1 if edge k starts at s , − 1 if it ends at s and zero otherwise. Observ e that, from the definition of A m , we hav e the same relations for A m,ks : A m,ks is 1 if edge k starts at vertex s , − 1 if it ends at vertex s and zero otherwise. Hence if edge j starts at v ertex s , the corresp onding columns in A e and A m are similar. Because the ab o ve is true for any vertex s ∈ V , and b ecause ev ery vertex has at least one edge originating from it, we hav e prov ed that K exists and it contains the n × n iden tity matrix in its column span, which yields the result.  W e write u x for ∂ ∂ x u ( x, y , z ) where x, y , z are dummy real v ariables. W e can now pro v e Theorem 5: Pr o of of The or em 5. Let us write diag ( D ) for the vector whose en tries are the diagonal en tries of D , where D is as defined in Lemma 2. Define the m × 2 n matrix Z by Z =      z 1 0 . . . 0 0 z 2 . . . 0 0 . . . . . . . . . 0 0 . . . z m      T . (13) A t a design equilibrium, we hav e e i ≡ 0 by definition and thus the en tries of D are u i (0 , 0 , w ) or u j (0) dep ending on the num b er of coleaders of the agent. Since e i ≡ 0 , we ha ve d dt ( z i · z i ) = 0 . A short computation using Equation (12) yields d dt      z 1 · z 1 z 2 · z 2 . . . z m · z m      = Z A e D z = Z A e Z T diag( D ) = 0 Because the formation is leaderless, we hav e, using Lemma 3, A e = A m K with K of full rank. The rigidit y matrix of the framew ork can b e written as R = Z A (2) m where we recall that A m is the mixed-adjacency matrix of the underlying graph. W e thus ha ve Z A m K Z T diag( D ) = RK Z T diag( D ) = 0 . When the formation is infinitesimally rigid and leaderless, R , K and Z are of full rank. Hence, R K ∈ R m × 2 m is of full row rank while Z T ∈ R 2 m × m is of full column rank. Hence, RK Z T is of full rank and w e conclude that diag( D ) = 0 .  31 The first condition states that agen ts are only aw are of the target distances to their coleaders, and that they are unaw are of the rest of the formation. Hence, agents do not kno w what the relativ e positions of their coleaders with resp ect to eac h other is at an equilibrium. W e elab orate no w on the implications of this fact on the mo del. Lemma 4. The 2-cycles formation is infinitesimal ly rigid for e dge lengths d ∈ L 0 . Pr o of. The rigidity matrix of the 2-cycles is giv en by: R =       z 1 − z 1 0 0 0 z 2 − z 2 0 − z 3 0 z 3 0 0 0 z 4 − z 4 z 5 0 0 − z 5       T (14) where the z i are defined by Equation (6). Recall that the rigidit y matrix R of a framew ork can b e expressed as R = Z A (2) m . where we recall that A m is the mixed adjacency matrix, A (2) m = A m ⊗ I , I b eing the 2 × 2 iden tity matrix and Z is given in equation 13. In the case of the 2-cycles, the mixed adjacency matrix A m ∈ R 5 × 4 is of rank 3 . The cok ernel 3 of A m is spanned by [0 , 0 , 1 , 1 , 1] T and [1 , 1 , 1 , 0 , 0] . Hence, the cok ernel of A (2) m is four dimensional and spanned b y the vectors [0 , 0 , 1 , 1 , 1] T ⊗ [1 , 0] T , [0 , 0 , 1 , 1 , 1] T ⊗ [0 , 1] T , [1 , 1 , 1 , 0 , 0] T ⊗ [1 , 0] T and [1 , 1 , 1 , 0 , 0] T ⊗ [0 , 1] T The matrix Z is of full rank unless z i = 0 for some i , whic h corresp onds to tw o agents sup erposed. W e th us ha v e that Z is of full rank for formations in L 0 . The k ernel of Z is giv en by the relations z 1 + z 2 + z 3 = 0 and z 3 + z 4 + z 5 = 0 . Because Z is of full ro w rank, R is of full (row) rank if A (2) m maps onto the coimage of Z . It is readily verified to b e the case from the ab o ve relations describing the cokernel of A (2) m and the kernel of Z .  Prop osition 3. L et d ∈ L 0 . The fe e db ack functions u i ( d i , d j ; x, y , z ) have to b e such that ∂ u i ∂ z ( d i ; d j ; 0 , 0 , z ) = 0 for al l z in the domain of u i . W e can informally understand this prop osition as a consequence of the fact that an agen t with tw o leaders do es not kno w what the angle b et ween its t wo leaders is at a design equilibrium, since this angle is a function of information the agen t does not ha ve access to. 32 x 1 x 2 x 3 x 4 z 1 z 5 z 2 z 4 z 3 Figure 16 Pr o of. Let us assume that agent 1 has tw o coleaders. Without loss of generality , we consider the situation of Figure 16, as agent 1 ’s dynamics do es not dep end directly on agen ts b ey ond its coleaders. The formation depicted is the 2-cycles. The formation is at a design equilibrium with edge lengths k z i k 2 = d i . The differential equations for z 1 and z 5 are  ˙ z 1 = u 2 ( e 2 ) z 2 − u 1 ( e 1 , e 5 , z 1 · z 5 ) z 1 − u 5 ( e 1 , e 5 , z 1 · z 5 ) z 5 ˙ z 5 = u 4 ( e 4 ) z 4 − u 1 ( e 1 , e 5 , z 1 · z 5 ) z 1 − u 5 ( e 1 , e 5 , z 1 · z 5 ) z 5 Let us consider a one-parameter family in L 0 with the property that the associated framew orks all ha ve d 1 and d 5 fixed to a constant v alue while the angle betw een z 1 and z 5 v aries. Suc h a family exists b ecaus e d is in the interio r of L . V arying d 3 while keeping the other d i ’s constant yields such a family with the notation of Figure 16. W e parametrize this family with s ∈ ( − ε, ε ) , ε > 0 and denote b y γ ( s ) the angle b et ween z 1 and z 5 . Because the 2-cycles is infinitesimally rigid when d ∈ L 0 b y Lemma 4, w e can use Theorem 5 to deduce that u 1 ( d 1 , d 5 ; 0 , 0 , γ ( s ) d 1 d 5 ) = u 5 ( d 1 , d 5 ; 0 , 0 , γ ( s ) d 1 d 5 ) = 0 for almost all s ∈ ( − ε, ε ) . Hence, u 1 ,z = u 5 ,z = 0 . The same argument can be rep eated for ev ery agen t with t wo coleaders.  W e conclude by stating the c omp atibility conditions that u i ha ve to satisfy in order to define a v alid formation control system: Definition 7. A n set of fe e db ack c ontr ol laws u i is compatible with the formation c ontr ol pr oblem if 1. u i ( d j ; e j ) is such that u i ( d j ; 0) = 0 if agent i has one c o-le ader. 2. u i ( d j , d k ; e j , e k , z T j z k ) is such that u i ( d j , d k ; 0 , 0 , z ) = 0 for al l z if agent i has two c o-le aders. 3 The cok ernel of a linear map f : A → B is the quotient space B / im( f ) . Its coimage is A/ ker( f ) . 33 5.3 Linearization of the dynamics Giv en a formation con trol problem, the equations in z v ariables are clearly redundan t, as the quan tit y P z i is zero along any cycle in the underlying graph. They ha ve the adv antage, ho w ever, of b eing in v ariant under translation, whic h renders some pro ofs below more transparen t. In this section, w e will lo ok at the linearization of the dynamics in the these v ariables. W e let F ( z ) b e the righ t-hand side of the differential equation describing the system: ˙ z = F ( z ) . Prop osition 4. Set z 0 i = ( u 1 x z i + u 2 x z j ) and z 0 j = ( u 1 y z i + u 2 y z j ) if z i originates fr om an agent with two c o-le aders given by z i and z j , and z 0 i = u x z i if z i originates fr om an agent with a single c o-le ader. Define Z 0 =      z 0 1 0 0 . . . 0 0 z 0 2 0 . . . 0 0 . . . . . . . . . 0 0 . . . 0 z 0 m      . (15) The Jac obian at a design e quilibrium of a formation c ontr ol system is ∂ F ∂ z = A (2) e Z 0 T Z. (16) Pr o of. W e first observe that ∂ ∂ z i u 1 ( e i , e j , z T i z j ) z i = 2 u 1 x z i z T i + uI 2 + 2 u 1 z z i z T j ∂ ∂ z j u 1 ( e i , e j , z T i z j ) z i = 2 u 1 y z i z T j + 2 u 1 z z i z T j ∂ ∂ z i u 2 ( e i , e j , z T i z j ) z j = 2 u 2 x z j z T i + 2 u 2 z z j z T i ∂ ∂ z j u 2 ( e i , e j , z T i z j ) z j = 2 u 2 y z j z T j + 2 u 2 z z j z T j 34 where we omitted the arguments of the functions on the right-hand side in order to keep the notation simple. Recall from Proposition 3 that at a design equilibrium u z = 0 . Hence, if z i originates from an agent with t wo co-leaders with ˙ z i = F i = . . . + u 1 ( e i , e j , z T i z j ) z i + u j ( e i , e j , z T i z j ) z j then: ∂ F i ∂ z i = − 2 u 1 x z i z T i − 2 u 2 x z j z T i = z 0 i z T i . If z j originates from an agent with a single leader, we ha ve: ∂ F j ∂ z j = u x z j z T j = z 0 j z T j . Putting the equations ab o ve together, w e get the result after some simple algebraic manip- ulations.  Recall that the problem is inv ariant under an action of the Euclidean group S E (2) on R 2 and that, in addition, the z v ariables are not indep enden t. As a consequence, the Jacobian of the system expressed in the z v ariables at an equilibrium will alw ays ha ve m ultiple zero eigen v alues. This degeneracy of the Jacobian do es not rev eal an ything about the dynamics of the system b ey ond the fact that equilibria are part of manifold of equilibria. F or all practical purp oses, this connected set of equilibria can b e thought of as a single equilibrium b y taking the quotient b y the action of the group. Under this quotient, this degeneracy of the Jacobian disapp ears. T o address this issue, the follo wing result gives a computational to ol to ev aluate the eigen v alues of the Jacobian that do not corresp ond to the action of the Euclidean group while working in the more con venien t inter-agen t distance co ordinates. Observ e that in the Prop osition below J ∈ R m × m . Corollary 1. L et G b e the gr aph of a minimal ly rigid formation. The eigenvalues of the Jac obian at a design e quilibrium ar e the eigenvalue zer o with algebr aic multiplicity 2 n + 3 − m and the eigenvalues of J = Z A e Z 0 T Pr o of. The result is a consequence of Theorem 1.3.20 in [HJ90] when applied to Prop osi- tion 4.  35 6 Conclusion W e hav e inv estigated some geometric prop erties of formation control. The main contribu- tions of this pap er can b e summarized as follows: - W e ha ve described the geometry of the space E n of configurations of n agents in the plane and shown that this space w as E n = C P( n − 2) × R + - W e ha ve sho wn how a global top ological c haracteristic of E n , namely the LS-category , relates to the coordinates used by the agen ts in the ensemble, called localized co ordi- nates. By doing so, we hav e derived a lo wer b ound on the num b er of non-congruen t framew orks with similar edge lengths when L is n ull-homotopic. - W e hav e established a num b er of conditions that a feedback control law needs to satisfy in order to mo del a formation control problem appropriately . In particular, formations with 3 and 4 agen ts ha ve b een inv estigated in details as examples of the main results. W e also conjectured the existence of a general pro cedure relating the Henneb erg sequence to discrete symmetry groups acting on framew orks. The results of this pap er will b e used extensiv ely in part I I, whose fo cus is on dynamical prop erties of formation con trol. W e dev elop there a framework to analyze the range of b eha viors ac hiev able by the system while resp ecting the information flow of the underlying graph. W e then apply this framework to show that there are no compatible feedbacks, as describ ed in Definition 7, that satisfactorily stabilize the 2-cycles. References [BS03] J. Baillieul and A. Suri, Information p atterns and he dging br o ckettâĂŹs the or em in c ontr ol ling vehicle formations , Pro c. of the 42th IEEE Conference on Decision and Control, v ol. 42, 2003, pp. 194–203. [CAM + 10] M. Cao, B.D.O. Anderson, A.S. Morse, S. Dasgupta, and C. Y u, Contr ol of a thr e e c ole ader formation in the plane , F estc hrift for John Baillieul (2010). [CLOT03] O. Cornea, G. Lupton, J. Oprea, and D. T anré, Lusternik-schnir elmann c ate- gory, mathematic al surveys and mono gr aphs, 103 , AMS, 2003. [CMY + 07] M. Cao, A. S. Morse, C. Y u, B. D. O. Anderson, and S. Dasgupta, Contr ol ling a triangular formation of mobile autonomous agents , Pro c. of the 46th IEEE Conference on Decision and Control, v ol. 46, 2007, pp. 3603–3608. [Con05] R. Connelly , Generic glob al rigidity , Discrete Comput. Geom. 33 (2005), 549– 563. 36 [EAM + 04] T. Eren, B.D.O. Anderson, A.S. Morse, W. Whiteley , and Peter N. Belh umeur, Op er ations on rigid formations of autonomous agents , Communications in In- formation and Systems 3 (2004), no. 4, 223–258. [EBM79] T. E., P . N. Belh umeur, and A.S. Morse, Emb e ddability of weighte d gr aphs in k-sp ac e is str ongly np-har d , Pro c. 17th Allerton Conf. in Comm unications, Con trol, and Computing, v ol. 17, 1979, pp. 480–489. [GSS93] J. E. Gra v er, B. Serv atius, and H. Serv atius, Combinatorial rigidity , Americal Mathematical So ciet y , 1993. [HADB07] J. M. Hendrickx, B. D. O. Anderson, J.-C. Delvenne, and V. D. Blondel, Di- r e cte d gr aphs for the analysis of rigidity and p ersistenc e in autonomous agents systems , International Journal on Robust and Nonlinear Con trol 17 (2007), 960–981. [Hel78] S. Helgason, Differ ential ge ometry, lie gr oups and symmetric sp ac es , Americal Mathematical So ciet y , 1978. [HJ90] R. A. Horn and C. R. Johnson, Matrix analysis , Cambridge Universit y Press, 1990. [Ken89] D.G. Kendall, A survey of the statistic al the ory of shap e , Statistical Science 4 (1989), no. 2, 87–99. [Lam70] G. Laman, On gr aphs and rigidity of plane skeletal structur es , Journal of Engi- neering Mathematics 4 (1970), 331–340. [Mun00] J. R. Munkres, T op olo gy , Prentice-Hall, 2000. [SPL07] R. Sepulc hre, D. Paley , and N. E. Leonard, Stabilization of planar c ol le ctive motion: A l l to-al l c ommunic ation , IEEE T ransactions on Automatic Control 52 (2007), 811–824. [Th u97] W.P . Thurston, Thr e e-dimensional ge ometry and top olo gy, volume 1 , Princeton Univ ersity Press, 1997. [Y ADF09] C. Y u, B.D.O. Anderson, S. Dasgupta, and B. Fidan, Contr ol of minimal ly p ersistent formations in the plane , SIAM Journal on Control and Optimization 48 (2009), no. 1, 206–233. 37