Closing Ranks in Vehicle Formations Based on Rigidity

Tolga Eren1 Peter N. Belhumeur2 A. Stephen Morse3

Abstract

In this paper, a systematic way of maintaining rigidityin case of vehicle removals in formations for coordi-nating mobile autonomous vehicles with limited com-munication/sensing links is presented. The main con-cern is the minimal rearrangement of links in such away that the links that have not been removed are pre-served as they are, and the minimum number of newlinks are created in the formation to maintain rigidity.The problem is solved by the tools used in the Hen-neberg method. This method gives us provably rigidformations with the minimum number of links. Theproposed framework meets this challenge for removedvehicles with any number of links in both 2 and 3-space.The same problem is also considered for the minimalDelaunay edge formations, which is a special class ofthe formations created by the Henneberg method withsome geometrical properties.

1 Introduction

This paper addresses the ”closing ranks” problem informations for coordinating large number of mobile au-tonomous vehicles with limited communication/sensingrequirements. By a formation is meant a group of ve-hicles moving in 2 or 3-space in such a way that thedistance between every pair of vehicles does not changeover time under ideal conditions. A framework to cre-ate formations with the minimum number of commu-nication/sensing links was presented in [3]. By closingranks, in an analogy to the army where each unit actsin a coordinated way especially to meet a challenge,is meant the minimal rearrangement of links betweenvehicles in case of vehicle removals (e.g. a vehicle isdestroyed or has a failure) such that the remaining ve-hicles will maintain the formation. In a formation withthe minimum number of communication/sensing linkssuch as the one shown in Figure 1a, for example, thevehicle pointed by an arrow is removed as shown inFigure 1b. One way to solve the problem would be tocreate a link between every pair of vehicles that had alink with the removed vehicle as shown in Figure 2a.

1Yale University, Department of Electrical Engineering, NewHaven, CT 06520 USA e-mail: tolga.eren@yale.edu

2Columbia University, Department of Computer Science, NewYork, NY 10027 USA e-mail: belhumeur@cs.columbia.edu

3Yale University, Department of Electrical Engineering, NewHaven, CT 06520 USA e-mail: as.morse@yale.edu

(a) (b)

Figure 1: (a)The initial formation with minimum numberof communication/sensing links. The vehicle tobe removed is pointed by an arrow. (b) Theremaining non-rigid formation after the vehicleis removed.

(a) (b)

Figure 2: (a)One way of maintaining rigidity is to connectevery pair of vehicles (dotted links) that wereconnected to the removed vehicle. (b) The fourlinks chosen as depicted (dotted links) will suf-fice to maintain the rigidity of the formation.

However, as it will be shown in the sequel, just four newlinks created in the way as depicted in Figure 2b wouldbe sufficient to maintain the formation. The main goalof this paper is to give a systematic way for creatingthe minimum number of new links to maintain a forma-tion while preserving the existing ones when a vehicleis removed.

The paper is organized as follows. In §2, an intro-duction to the terminology and notation that will beused in the paper will be presented. In §3, methods tomaintain rigidity in the closing ranks problem in theformations created by the Henneberg method in both2 and 3-space are proposed. In §4, a special class ofthe formations created by the Henneberg method isintroduced, namely the minimal Delaunay edge forma-tions. In §5, the closing ranks problem for the minimal

Delaunay edge formations is addressed in both 2 and3-space.

2 Point Formations and Rigidity

A point formation Fp = ({p1, p2, . . . , pn},L) provides away of modelling a set of n-vehicles moving in forma-tion in 2 or 3-space, where p , column {p1, p2, . . . , pn}is a set of n points in IRd together with a set L ofm maintenance links, labelled (i, j), where i and jare distinct integers in {1, 2, . . . , n}. The length oflink (i, j) is the Euclidean distance between points pi

and pj . The points represent the positions of vehi-cles and the links represent the specific distances tobe maintained between vehicles. By a trajectory of Fp

is meant a continuously parameterized, one-parameterfamily of points {q(t) : t ≥ 0} in IRnd which containsp. In practice, vehicles can not be expected to main-tain the specified distances exactly, because of sens-ing errors, vehicle modelling errors, etc. The idealbenchmark point formation against which the perfor-mance of an actual vehicle formation is to be mea-sured is called a reference formation. Such a forma-tion is said to undergo rigid motion along a trajectoryq([0,∞)) , {column {q1(t), q2(t), . . . , qn(t)} : t ≥ 0}if the Euclidean distance between each pair of pointsqi(t) and qj(t) remains constant all along the trajec-tory. A formation Fp is said to be rigid if rigid motionis the only kind of motion it can undergo along anytrajectory on which the lengths of all links in L remainconstant. The ideas of a point formation and a rigidpoint formation are essentially the same as the conceptsof a “framework” and a “rigid framework” studied inmathematics as well as within the theory of structuresin mechanical and civil engineering (see for example[5],[6]).

One approach to determine whether a given formationis rigid or not starts by examining what happens toa given formation Fp = ({p1, p2, . . . , pn},L), along thetrajectory {q1(t), q2(t), . . . , qn(t)} : t ≥ 0} on whichthe Euclidean distances dij , ||pi − pj || between pairsof points (qi, qj) for which (i, j) is a link, are constant.Thus along such a trajectory

(qi − qj)′(qi − qj) = d2ij , (i, j) ∈ L, t ≥ 0 (1)

Solving the system of equations in (1) is very difficult ingeneral. One approach is to look for the first derivativesof trajectories instead of looking them directly.

(qi − qj)′(q̇i − q̇j) = 0, (i, j) ∈ L, t ≥ 0 (2)

These equations can be evaluated at p and rewritten inmatrix form as

R(q)q̇ = 0 (3)

where q̇ = column {q̇1, q̇2, . . . , q̇n}, and R is a speciallystructured m×dn array called a rigidity matrix. If theformation contains at least d+1 points which are notcontained in any proper hyperplane within IRd, thenthere are always d(d+1)

2 solutions of (2) correspondingthe derivatives of trajectories of rigid motion [5]. If thesystem of equations in (2) has no other solutions, i.e.

rank R(p) =

{3n− 6 if d=3 ,

2n− 3 if d=2 .

then the rigidity of formation would be implied.Clearly analyzing the rigidity requires one to study notonly the incidences of the links but also the positionsof points.

It is possible to associate with any point formation Fp agraph GF , {V,L} whose vertex set is the set of labelsof the points in Fp and whose edge set is L. The defini-tion of rigid formation implies that every formation Fp

whose graph GF is complete is automatically rigid. Theconverse however is not generally true. Combinatorialrigidity is concerned with what extent the rigidity offormations can be judged by knowing the vertices andtheir incidences, in other words, by knowing the under-lying graph. A formation Fp is said to be genericallyrigid if it is rigid and if there is an open neighborhoodof points about p in IRdn at which Fp is also rigid. Theconcept of generic rigidity does not depend on the dis-tances between the points of Fp and for this reason, it isa desirable specialization of the definition of a rigid for-mation for our purposes. The generic rigidity questioncan be posed solely in terms of the graph GFp withoutany reference to Fp’s actual points. In order for a graphto be generically rigid in the plane, there must be a sub-set L′ ⊆ L satisfying the following two conditions: (1)|L′| = 2n−3, (2) For all L′′ ⊆ L′,L′′ 6= ∅, |L′′| ≤ 2k−3,where k is the number of vertices which are endpointsof edges in L′′. These two conditions check whetherthere are at least 2n-3 independent edges and whetherthe edges are distributed in such a way that there areno redundancies. This combinatorial characterizationof generic rigidity was proved by Laman in 1970 [4]. Al-though the extension of this characterization to 3-spaceis a necessary condition, it fails to be sufficient.

Given this difficulty of characterization of rigidity in 3-space, we turn our attention to the problem of creatingprovably rigid formations in both 2 and 3-space insteadof checking whether a given formation is rigid or not.

We begin by reviewing an inductive method that ap-plies to isostatic graphs. By isostatic graph is meanta generically rigid graph such that removing any edgegives a non-rigid graph. Generically d-isostatic meansisostatic condition in d -space. The degree of a vertex iof a graph is the number of incident vertices to i. Givena generically isostatic graph G, an edge (i, j) which isnot in E is called an implicit edge for G, i.e. an implicitedge is redundant for the rigidity of G. Incident edgesshare a common vertex. We denote the set of adjacentvertices to vertex i by Vi. The following theorems (see[6]) give an inductive approach that creates genericallyd -isostatic graphs both in 2 and 3-space while main-taining rigidity.

Theorem 1. (Vertex Addition Theorem) Let G ,{V,L} be a graph with a vertex i of degree d; let G∗ ,{V∗,L∗} denote the subgraph obtained by deleting i andthe edges incident with it. Then G is generically d-isostatic if and only if G∗ is generically d-isostatic.

Theorem 2. (Edge Split Theorem) Let G , {V,L}be a graph with a vertex i of degree d + 1, let Vi be theset of vertices incident to i, and let G∗ , {V∗,L∗} bethe subgraph obtained by deleting i and its d+1 incidentedges. Then G is generically d-isostatic if and only ifthere is a pair j, k of vertices of Vi such that the edge(j, k) is not in L∗ and the graph G′ = (V∗,L∗⋃

(j, k))is generically d-isostatic.

This method starts from Gd in d-space at the first step(Gd is a single edge in 2-space and a triangle in 3-space)and then it either adds a new vertex with d edges tothe existing graph, or it removes an existing edge byadding a new vertex connecting it to the end pointsof the removed edge and to other d− 1 vertices in theexisting graph. The resulting graphs that we have aftereach step are generically isostatic. A sample graph in3-space created by this method is depicted in Figure 3.

3 The Closing Ranks Problem

One way to solve the closing ranks problem would bedetermining links for the remaining vehicles from thevery beginning. However generating a whole new for-mation with the remaining vehicles is not practical ifa minimal rearrangement of links between vehicles suf-fices to keep the formation rigid. By the minimal re-arrangement, we mean the following in graph theoreticterms. Assume that a generically isostatic graph Gwith a vertex i is given. Let {1, 2, . . . , n} denote theset of vertices in G. Given that the vertex i is re-moved from the graph, we want to maintain rigiditywith the set of vertices {1, 2, . . . , n} \ {i}. Minimalrearrangement is preserving all the edges except theincident edges to i and creating new edges such thatthe new graph G∗ is isostatic. To maintain rigidity,

Figure 3: The Henneberg method in 3-space. V.A. standsfor the vertex addition, E.S. stands for the edgesplit. Double-lined edges indicate the edges cre-ated for the new vertex. Dashed edges indicatethe removed edges in the edge splitting.

one might connect each pair of vertices in Vi. Howeverthis strategy would not give us a generically isostaticgraph. In fact, for the removal of a vertex of degree k,we need to create k(k−1)/2 new edges with this strat-egy. The number of new edges required for isostasywould be k− 2 and k− 3 in 2 and 3-space respectively.Maintaining generic isostasy in case of vertex removalsin generically isostatic frameworks was considered inrigidity literature in [6]. One of the aims of this paperis to draw attention to their results for applications invehicle formations.

3.1 Formations in 2-spaceRemoving a Vehicle with Two and Three Links: Agenerically isostatic graph G is shown in Figure 4a.The vertex 8 is removed with the two edges (8, 5) and(8, 7) in G (Figure 4b). For maintaining rigidity, nochange is made with the remaining vertices and edges.The vertex 1 is removed with the three edges (1, 2),(1, 3) and (1, 5) in G. Then one of the edges, which isnon-implicit, (2, 3), (2, 5) or (3, 5) is created as shownin Figure 5a. The existence of a non-implicit link wasproved in [6]. The resulting graph is generically iso-static. The proofs are based on the vertex addition andedge split theorems. The existence of a non-implicitlink in the subsequent analogous proofs is based on theproofs in [6].

Removing a Vehicle with Four and Higher Number ofLinks:

Lemma 3. Let G = (V, E) be a generically isostaticgraph with a vertex i of degree four, which has a setof neighboring vertices, Vi. Let G∗ denote the resulting

(a) (b)

Figure 4: (a)The initial graph representing a point forma-tion. (b) The vertex 8 of degree two is removed.

non-rigid graph after the vertex is removed. Then forsome choice of two non-implicit edges among the ver-tices in Vi, the graph G∗∗ formed from G∗ by creatingthese edges is generically isostatic.

Proof: We choose an isostatic graph G with thevertex i and the vertices in Vi. We remove one of theedges of i, and call it G′. To maintain isostasy wecreate an edge between two points in Vi which is notimplicitly present. If we add this edge, we create G′′which is isostatic and has a vertex i with degree three.We can replace i with a non-implicit edge between twovertices in Vi. Hence a vertex with degree 4 can bereplaced by two edges between the vertices in Vi.

Figure 5b shows an example where the vertex 4 of de-gree four is removed and the edges are created to main-tain rigidity. Now, we remove a vertex with five orhigher number of edges in a rigid graph. We maintainrigidity using the following theorem.

Theorem 4. (replacing a vertex of degree k, k ≥ 5)Let G = (V, E) be a generically isostatic graph witha vertex i of degree k and with a set of neighboringvertices Vi. Then, for some choice of k−2 non-implicitedges among the vertices in Vi, the graph formed byremoving i and the edges whose one end point is i, andadding these k − 2 edges is generically isostatic.

Proof: The case for k = 4 has been shown in Lemma3. Now we are going to show for k ≥ 5. Assume thatthe theorem holds for k = n− 1. For k = n, we choosean isostatic graph G with vertex i and the set of neigh-boring vertices Vi. We remove one of the edges of i, andcall it G′. To maintain isostasy we create an edge be-tween two vertices in Vi which is not implicitly present.If we add this edge, we create G′′ which is isostatic andhas a vertex i with degree n−1. From our assumption,we can remove the vertex i by creating n−3 new edges.Hence, we can maintain isostasy by creating a total ofn − 2 edges in case of removing a vertex with degreen.

(a) (b)

Figure 5: (a) The vertex 1 of degree three is removed andthe edge (2,5) is created to maintain rigidity.(b) The vertex 4 of degree four is removed andthe edges (2,6),(5,7) are created to maintainrigidity.

3.2 Formations in 3-spaceRemoving a Vehicle with Three and Four Links: Givena generically isostatic graph G as shown in Figure 6a,consider the case when the vertex 3 with the threeedges (3, 1), (3, 2) and (3, 5) is removed as shown inFigure 6b. For maintaining rigidity, no changes withthe remaining vertices and edges need to be done.Consider the case when the vertex 8 with the fouredges (8, 2), (8, 5), (8, 7) and (8, 11) is removed inG (Figure 7a). Rigidity is maintained as follows.One of the edges, which is non-implicit in the set{(2, 5),(2, 7),(2, 11),(5, 7),(5, 11),(7, 11)} (Figure 7b) iscreated. The proofs are analogous to 2-space cases.

Removing a Vehicle with Five and Higher Number ofLinks: The vertex 7 with five edges is removed as shownin Figure 8a. To maintain rigidity with the remainingvertices, the following theorem is used. The proof isanalogous to Lemma 3.

Lemma 5. Let G = (V, E) be a generically isostaticgraph with a vertex i of degree five and with a set ofneighboring vertices Vi. Then, one of the followingis true: (i) for some choice of two non-incident edgesamong the vertices in Vi, the graph G1 formed from Gby creating these edges is generically isostatic; or (ii)for two choices of incident pairs of edges among thevertices in Vi, (not all incident with a single vertex),the two graphs G2 and G3 formed from G by creatingthese pairs of edges are both generically isostatic.

In Figure 8b, the edges (5,10),(6,8) are created to main-tain rigidity. Now, a vehicle with six or higher numberof links is removed in a rigid formation. The follow-ing theorem is used to maintain rigidity, which has ananalogous proof of Theorem 4.

Theorem 6. (replacing a vertex of degree k, k ≥ 6) LetG = (V, E) be a generically isostatic graph with a vertexi of degree k and with a set of neighboring vertices,Vi. Then, for some choice of k − 3 non-implicit edgesamong the vertices in Vi, the graph formed by removing

Figure 6: (a) A generically isostatic graph in 3-space. (b)The vertex 3 of degree three is removed, andthe resulting formation is still rigid.

Figure 7: (a) The vertex 8 of degree four is removed. (b)The link (7,11) is created to maintain rigidity.

i and the edges whose end point is i, and adding thesek − 3 edges is generically isostatic.

4 The Delaunay Formations

Here, some geometric flavor is added to the formationscreated by the Henneberg method. While adding newvertices by the vertex addition or the edge split, thenew edges will be picked in such a way that the mini-mum angle between two incident edges in a formationis as large as possible. The large angle property playsa role in avoiding obstructions between vehicles. Bythe Delaunay formations, three types of formations aremeant, namely the Delaunay triangulation, the mini-mal Delaunay triangulation and the minimal Delaunayedge formation (see Figure 9). The Delaunay triangu-lation, well known in computer graphics and in com-putational geometry literature, was proposed in [3] forcreating rigid point formations. A triangulation in theplane is a partition of a point set into triangles thatmeet only at shared edges. The Delaunay triangula-tion, although rigid but not necessarily isostatic, has anumber of properties that makes it attractive [1], [2].It maximizes the minimum angle of all triangulationsof a given vertex set. It is composed of the Delau-nay triangles. A triangle is Delaunay if and only if

Figure 8: (a) The vertex 7 of degree five is removed. (b)The links (5,10), (6,8) are created to maintainrigidity.

the circumcircle of the triangle does not enclose anyother vertex. By circumcircle is meant the circle pass-ing through the vertices of a triangle. Secondly thedistance between any two vertices in the Delaunay tri-angulation is at most a constant times the Euclideandistance. Finally, empirical results suggest that in anydimension the edge skeleton of the Delaunay triangu-lation usually has small vertex degree and small totallength [2].

There are often implicit edges in the Delaunay triangu-lation and this is undesirable for formations with min-imum communications/sensing links. Here two meth-ods of avoiding implicit edges are proposed while stillusing the nice property of large angles between the De-launay edges. These methods can be extended to 3-space. These two types of formations are called theminimal Delaunay triangulation and the minimal De-launay edge formation. While the minimal Delaunaytriangulation is composed of triangles, the minimal De-launay edge formation is not necessarily. Let i and jbe two vertices in V. An edge (i, j) is called a De-launay edge if and only if there exists a circle passingthrough i and j that does not enclose any other vertexin V. An isostatic point formation with the Delaunayedges is called the minimal Delaunay edge formation.Note that the minimal Delaunay triangulations and theminimal Delaunay edge formations are created by theHenneberg method by using the Delaunay triangle andthe Delaunay edge properties. In the sequel we give away of creating these two types of formations.

Creating the Minimal Delaunay Triangulations:We add two edges for each vertex addition to the endpoints of an existing edge in the point formation suchthat the resulting triangle has the smallest circumcircle.To meet our goals in measure of angles, we apply edgeflipping in the resulting point formation.

Creating the Minimal Delaunay Edge Forma-tions: We add two non-implicit edges with minimumlengths for each vertex addition. To meet our goals in

Figure 9: (a) The Delaunay triangulation of a given vertexset. (b) The minimal Delaunay triangulation ofthe same vertex set. (c) The minimal Delaunayedge formation of the same vertex set.

measure of angles, we apply edge flipping in the result-ing point formation.

5 Closing Ranks in the Delaunay Formations

The Delaunay Triangulations: We state the follow-ing theorem for maintaining rigidity and the Delaunayproperty in case of vertex removals in a formation cre-ated by the Delaunay triangulation.

Theorem 7. Let Fp = (V,L) be a generically rigidpoint formation created by the Delaunay triangulation.When a point pi is removed from Fp, the polygonal re-gion consisting of all the triangles that are incident topi is retriangulated satisfying the Delaunay property.Let F∗p denote the resulting point formation. Then F∗pis a generically rigid point formation.

The Minimal Delaunay Triangulations: For closingranks, we use Theorem 7 as in the case of the Delau-nay triangulations with the condition of one triangleper vertex as explained in creating the minimal Delau-nay triangulations.

The Minimal Delaunay Edge Formations: For closingranks, we need to create a set of k − 2 and k − 3 non-implicit links for a vertex of degree k in 2 and 3-spacerespectively as in the case of isostatic graphs, whereeach edge satisfies the Delaunay property. An exampleis shown in Figure 10. A minimal Delaunay edge forma-tion showing the vertex and its edges (dotted lines) tobe removed is depicted in Figure 10a. The new edges(double lines) are created to maintain rigidity whereeach edge is a Delaunay edge (Figure 10b).

Remark 8. Proposed methods can be extended to 3-space.

Figure 10: (a) A minimal Delaunay edge formation show-ing the vertex and edges (with dotted lines)to be removed, (b) New edges (depicted bydouble lines) are created to maintain rigiditywhere each edge is a Delaunay edge.

6 Conclusions

A systematic way of maintaining rigidity in case of ve-hicle removals in formations for coordinating mobileautonomous vehicles with the minimum communica-tion/sensing links is presented. The method createsthe minimum number of new links while preserving theones that have not been removed. Hence, creating anew formation from the very beginning is not needed.The method can be applied to removed vehicles withany number of links in both 2 and 3-space. The min-imal Delaunay edge formations, which are formationscreated by the Henneberg method with some geometricproperties, are also introduced. Splitting-merging andreconfiguration of rigid formations will be presented ina separate paper.

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