Weighted Bearing-Compass Dynamics: Edge and Leader...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TNSE.2017.2754944, IEEETransactions on Network Science and Engineering

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Weighted Bearing-Compass Dynamics:Edge and Leader Selection

Eric Schoof, Airlie Chapman, and Mehran Mesbahi

Abstract—This paper considers the design and effective interfaces of a distributed robotic formation running planar weightedbearing-compass dynamics. We present results which support methodologies to construct formation topologies using submodularoptimization techniques. Further, a convex optimization framework is developed for the selection of edge weights which increaseperformance. We explore a method to select leader agents which can translate and scale the formation, and a corresponding controllerthat promotes the formation keeping its overall shape intact during manipulation. The results are supported with examples that illustratethe approach and their differing levels of performance.

Index Terms—Bearing rigidity, formation control, submodularity, network design, leader selection.

F

1 INTRODUCTION

The objective of formation control in multi-agent systems isto acquire and adapt the geometric configuration of agents,or formation shape, using distributed control protocols.The design of formation control algorithms can be brokeninto two main steps. The first is designing the distributedcontrol protocol which will achieve a desired formationshape. The second step is manipulating the formation byadapting the formation protocol or augmenting the protocolwith external signals. This paper builds on our previouswork on bearing-compass dynamics [1] which provides adistributed dynamic model able to almost globally acquirea desired formation shape. The focus of this work is ondeveloping theory and algorithms applicable to the bearing-dynamics dynamics to efficiently acquire and manipulatethe formation shape. This is achieved through the selectionof effective communication links and agents in the network- edges and leaders.

Bearing-compass dynamics fits under a broader class ofdynamics referred to as distributed protocols. Distributedprotocols are popular in the area of robotic formation controldue to the fact that an individual agent needs only localinformation about its neighbors for operation [2], [3], [4],[5], [6], [7]. One of the most popular local informationtypes is relative distance-based measurements [8], [9]. Theexamination of relative bearing-based measurements hasbecome of interest in the last decade, with this currentwork fitting into this subclass [10], [11], [12], [13]. Bothdistance-based and bearing-based protocols rely on graphrigidity theory to guarantee that the protocols will convergeto a unique formation shape based on the location of the

• Eric Schoof is with the department of Electrical and Electronics Engineer-ing, University of Melbourne, Australia. [email protected]

• Airlie Chapman is with the department of Mechanical Engineering,University of Melbourne, Australia. [email protected]

• Mehran Mesbahi is with the William E. Boeing department of Aeronauticsand Astronautics, University of Washington, USA. [email protected].

The research of the authors has been supported by ARO grant W911NF-13-1-0340 and AFOSR grant FA9550-16-1-0022.

relative measurements within the communication graph.The ability to reason about and design the placement ofrelative measurements within a network is one of the topicsaddressed in this paper.2n. Similar to the selection under rank in §4.1 the minimumnumber of leader agents, namely two will be selected..Designing topologies to optimize for a given metric has beenaddressed in the literature. These methods typically fit intotwo broad classes. The first class examines the continuousdesign of variables on the graph topology, e.g., throughedge weights and input control weights. The second classis a combinatorial approach that examines node or edgeselection problems. For example, Wan et al. maximizedthe largest eigenvalue of the graph Laplacian through theselection of edge weights [14]. Chapman et al. designed edgeweights distributively and online to reject network-level dis-turbances [15]. Zelazo and Mesbahi leveraged convex opti-mization techniques to design edge weights for performanceimprovements under the H2 and H∞ norms [16]. With theobjective of maximizing the algebraic connectivity of thegraph Laplacian, Ghosh and Boyd examined techniques togrow graphs with improved connectivity [17].Augmenting distributed robotic algorithms with agents thatare not following to the same control law has also beenconsidered in recent years [18], [19]. These non-conformingagents are referred to as leaders, or anchors, while theothers are named followers. The overall setup is com-monly termed leader-follower dynamics. Zhao and Zelazohave considered the leader-follower setup for the bearing-compass dynamics as a method of controlling the translationand scale of the formation [13], [20]. Techniques from convexoptimization and suboptimal greedy algorithms have beenused to select influential access points into a network forleaders for the consensus dynamics, also referred to as theleader selection problem [21], [22], [23]. Examining leaderselection for the bearing-compass dynamics is one of theproblems addressed in this work.Submodular optimization has emerged as a popular tech-nique to solve the discrete topology design problem [24],



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[25], [26], [27]. An attraction of this theory is that greedyselection heuristics may be employed to achieve provablynear-optimal solutions. The current work will rely heavilyon this framework to design well-performing graphs underbearing rigidity performance measures. A related work byShames and Summers examined a similar approach to builddistance rigid formations and select influential leader loca-tions within a network using submodular optimality theory[28].The contribution of this paper is two-fold. First, we extendthe bearing-compass dynamics [1], [13] to allow differentweights on the agent interaction topology and include theaddition of fixed anchors. Second, we use the field of sub-modular optimization to build and manipulate formationsvia edge and node selection, respectively. This providesprovably near-optimal selection performance that would becomputationally infeasible to perform optimally. We showhow several possible submodular metrics perform underedge and node selection. We also use convex optimizationto select edge weights that improve the convergence offormation acquisition and manipulation protocols withoutincreasing the required control authority for any individualagent in the formation.We begin by introducing background concepts and notationin §2. In §3, we extend our previous work [1], [6] and thework of others [11], [12], [13] by examining properties ofa weighted version of the bearing-compass dynamics. Weproceed in §4 to discuss a process by which one can define aformation for favorable performance under the dynamics.Next, in §5, we explore how we can move or scale aformation running the weighted bearing-compass dynamicsso it can be placed in a desired orientation through leaderselection. A state-dependent weighting scheme is presentedwhich allows the formation to more easily retain its shapewhile it is being manipulated. Concluding remarks andnotes on future directions are provided in §6.

2 BACKGROUND

In this section, a brief background is provided on the nota-tion, dynamics and definitions that are used in this paper.Given a matrix A ∈ Rm×n, the notation [A]ij is used torepresent the element of A in row i ∈ 1, . . . ,m and columnj ∈ 1, . . . , n. The eigenvalues of a matrix A ∈ Rn×nare notated as λi(A) and ordered such that Reλ1(A) ≤Reλ2(A) ≤ · · · ≤ Reλn(A). The notation A � 0 and A � 0designates the matrix A as being positive definite and positivesemi-definite, respectively. We use the notation 0 and 1 torefer to the vectors of all zeros and all ones, respectively,with the dimension inferred from the context. The 2-normof a vector v is denoted ‖v‖. The normalized vector ofv is denoted v̂ = v/ ‖v‖. Given a vector v ∈ R2, theperpendicular of v = [ v1 v2 ]T is v⊥ = [ −v2 v1 ]T .Werepresent block diagonal matrices with a repeated structureas Di (Ai), where Ai appears as the ith block diagonalof D. Given vectors a,b ∈ R2, the projection of a ontob is defined as Projba = (a

Tb/ ‖a‖ ‖b‖)b implying thata− Projba = Projb⊥a.We denote undirected weighted graphs as G = (V,E,w),where V is the set of nodes i ∈ V with cardinality |V | = n

and E is the set of edges {i, j} ∈ E, with i, j ∈ V andcardinality |E| = m. The incidence matrix H ∈ Rm×nencodes the edge set E such that for edge k representingedge {i, j} ∈ E, [H]ki = −1 and [H]kj = 1, and i < j. Ifedge {i, j} ∈ E is the kth row in the incidence matrix H ,then the relative state between agents i and j can be writtenrj − ri = rij = rk, which disambiguates the state of nodek and the relative state of the nodes incident on edge k. Wethen have the relation that H̃r =

[rT1̄ r

T2̄ · · · r

Tm̄

]T,

where r ∈ R2n, H̃ = (H ⊗ I2), and⊗ denotes the Kroneckerproduct. The vector w ∈ Rm is the weight vector withelements wk̄ for each edge k̄ ∈ E . If w is omitted, weassume that w = 1 and write G = (V,E). We define theset of neighbors of agent i as j ∈ N (i) if {i, j} ∈ E, whereN (i) is called the neighborhood of i.

2.1 Modular and Submodular Functions

We denote functions of the form f (S) : 2n → R, whereS ⊆ S and |S| = n, as set functions. The empty set is denoted∅. These functions can have many properties that we canexploit, some of which are reviewed next.

Definition 1. A set function f (S) : 2n → R is modular iffor all subsets S ⊆ S it can be expressed as

f(S) = w(∅) +∑s∈S

w(s),

for some weight function w : s ∈ S → R.

Definition 2. A set function f (S) : 2n → R is monotoneincreasing if, for all subsets A ⊆ B ⊆ S it holds that f(A) ≤f(B).

Definition 3. [24, Def. 2.1] A set function f (S) : 2n → R issubmodular if, for all subsets A ⊆ B ⊆ S and s /∈ B, then

f(A ∪ {s})− f(A) ≥ f(B ∪ {s})− f(B). (1)

Modular functions can be considered a special case of sub-modular functions where equality holds in (1). The intuitionbehind Def. 3 is that adding an element to a smaller set givesa greater gain than adding the same element to a largerset, sometimes called “diminishing returns”. Submodularfunctions bear similarities to convex functions [25]. Thisconnection can be used to provide guarantees on greedyalgorithms optimizing over submodular functions which isexplored in Thm 5 below.

Proposition 4. [24, Prop. 2.7] A positive linear combination ofsubmodular functions is submodular.

Theorem 5. [24, Thm 4.3] Consider the problem of selecting Kelements from a set S which optimizes the value of a monotonicallyincreasing submodular cost function f (S) for S ∈ S and |S| =K. If we make a greedy selection of R elements at each iteration,then the value of our greedy selection f (S) will satisfy

f (S)− f (∅)f (S∗)− f (∅)

≥ 1−(q − λq

)(q − 1q

)q−1, (2)

where S∗ is the globally optimal selection, K = qR− p for someq ∈ Z+, p an integer in the range 0 ≤ p ≤ R−1, and λ = R−pR .



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Figure 1: Optimality bound when performing a greedyselection of K elements over a submodular cost function,for different batch sizes R, assuming f (∅) = 0.

When K is an integer multiple of R, the bound in Eq. (2) becomestight. For the case when R = 1, Eq. (2) becomes

f (S)− f (∅)f (S∗)− f (∅)

≥ 1−(K − 1K

)K≥ 1− 1

e≈ 63%. (3)

The last approximation is obtained in the limit as K →∞.

A consequence of Thm 5 is that picking K items from aset one at a time using a greedy submodular heuristic, i.e.,adding an element to a set which maximizes the heuristic ateach step, will return a solution whose value is within ap-proximately 37% of the optimal solution. Different boundscan be obtained by varying K and R, as demonstrated inFig. 1.

2.2 Bearing Measurements on a Graph

In this paper, we use the term “node” to describe a particularelement of V . There is also a more loosely termed “agent”which refers to something (usually a vehicle) that can com-municate or act through the edges ofE, and can have its ownstate. We define the agent positions in 2-D space at time tby the vector r(t) = [ rT1 (t) r

T2 (t) . . . r

Tn (t) ]

T , whereri (t) = [ rix (t) riy (t) ]

T . Core to this paper, is under-standing agents formation shape when a family of bearingconstraints are placed on pairs of agents. To this end, weuse the term formation, denoted with f ∈ R2n, to represent adesired state for a collection of agents. Franchi and Giordano[12] explored the notion of coupling of a formation with agraph, using the pairwise bearing measurement set

Θ (G, f) ={f̂ij : {i, j} ∈ E

}. (4)

We will take an excursion from Franchi and Giordano ter-minology and refer to this as a framework. The collectionof agent formations r which are bearing equivalent, or justequivalent, to a given framework Θ is denoted by

χ (Θ) ={r : r̂Tij f̂

⊥ij = 0, ∀ f̂ij ∈ Θ

}. (5)

The set of bearing equivalent formations represents thoseagent locations that conform to the bearing constraints inΘ. A formation f and g are similar if g is a translatedand/or scaled version of f . If all equivalent formations aresimilar then the framework is termed parallel rigid or bearingrigid. Algebraic conditions that guarantee bearing rigidity

based on the notion of infinitesimal bearing rigidity willbe explored in §3. In order to more easily classify similarformations, we introduce the notion of centroid and scaleof a formation [1], [13]. Given a set of agents at positionsr ∈ R2n, the centroid of the agents is defined as

C (r) =1

n

∑i∈V

ri. (6)

The scale of the agents on the other hand is defined as

S (r) =

√1

n

∑i∈V‖qi‖2, (7)

where qi := ri − C (r). Given a formation f ∈ R2n, theunitless formation f̃ is defined as

f̃ = [f − (In ⊗ C (f))1n]S (f)−1 , (8)

with properties C(f̃) = 0, S(f̃) = 1, and∥∥∥f̃∥∥∥ = √n. We note

a unitless formation f̃ is similar to f .The unique pair of similar formations to formation f whichshare the same scale and centroid as r is defined as

ξ± (f , r) = ±f̃S (r) + (In ⊗ C (r))1 (9)= {ξ+ (f , r) , ξ− (f , r)} ,

where ξ+ (f , r) and ξ− (f , r) denote the positively and neg-atively scaled formations, respectively.

3 WEIGHTED BEARING-COMPASS DYNAMICSIn this paper, we extend the analysis of our previouslyintroduced bearing-compass particle dynamics [1] with aweighting term on each edge as

ṙi = ui (Θ) + ũi (10a)

ui (Θ) =∑

j∈N (i)

wij(r̂Tij f̂⊥ij )r̂

⊥ij :=

∑i∈N (i)

u(ij), (10b)

where Θ = (G, f), G = (V,E,w). Here, we use the notationb(ij) to represent the component of bi due to j. We notethe notation b(ij) is distinct from bij and that b(k̄) can beused in place of b(ij) when edge indexing is preferable. Ifũi = 0 for all i ∈ V , then we say the dynamics is unforced.Otherwise, we say the dynamics is forced.The dynamics is referred to as bearing-compass as it canbe implemented with only relative bearing information toneighboring agents and access to a compass to localize themeasurements in a frame. The former can be achieved via anon-board low-fidelity camera [7] where distance measure-ments can be unreliable compared to bearing measurements.Agents with ũi 6= 0 are referred to as leader agents, asthrough the networked dynamics, the non-leader agentsfollow the leaders.As we proceed through this section, we will look at variousfeatures of the bearing-compass dynamics. To this end, thefollowing proposition is a useful tool which will be usedthroughout this paper to simplify our proofs.

Proposition 6. If b(ij) = −b(ji), enumeration over nodes andneighborhoods is equivalent to enumeration over edges with∑

i∈VaTi

∑j∈N (i)

b(ij) = −∑k̄∈E

aTk̄ b(k̄).



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Figure 2: Vector definitions for dynamics (10).

Proof: Noting that LHS represents an enumeration of everyedge twice, once representing aTi b(ij) and once representingaTj b(ji), the result follows.

The unforced bearing-compass dynamics exhibits propertiessuch as invariance of the formation centroid and scale,which make it attractive for maintaining stable formations.The forced dynamics is then able to directly manipulatethe centroid and scale. The following proposition formerlypresents these properties.

Proposition 7. Under the weighted dynamics (10), the formationcentroid and scale evolve as

∂C (r)

∂t=

1

n

∑i∈V

ũi

∂S (r)

∂t=

2

nS (r)

∑i∈V

qTi (ũi −1

n

∑j∈V

ũj).

Additionally, if ũi = 0 for all i ∈ V , then ∂C (r) /∂t = 0 and∂S (r) /∂t = 0, and the centroid and scale are said to be invariantunder the unforced dynamics.

Proof: Differentiating Eq. (6), we have

∂C (r)

∂t=

1

n

∑i∈V

ṙ =1

n

∑i∈V

(ui + ũi)

=1

n

∑{i,j}∈E

(u(ij) + u(ji)

)+

1

n

∑i∈V

ũi

=1

n

∑i∈V

ũi.

Similarly, if we differentiate Eq. (7), we have

∂S (r)

∂t=

∂

∂t

√1

n

∑i∈V

qTi qi

=1√

n∑

i∈V qTi qi

∂

∂tqTi qi

=2

nS (r)

∑i∈V

qTi (ũi −1

n

∑j∈V

ũj),

where the final equality comes from a similar proof in [1].

In order to examine the dynamics (10) it is helpful to havea machinery to encode the bearing information embeddedwithin the dynamics. The bearing rigidity matrix has thisfunction and can be constructed from the formation f andthe underlying graph G.

Definition 8. The bearing rigidity matrix is denoted R (Θ),where Θ (G, f) is the framework, is defined as

R (Θ) = ∂ĝ∂f, (11)

where g = H̃f , and ĝ =[f̂T1̄ f̂

T2̄ · · · f̂

Tm̄

]T. An

alternative expression for (11) is1

R (Θ) = Dk̄[

1

‖fk̄‖Id

]Dk̄ [Pk̄] H̃

:= Dk̄[

1

‖fk̄‖Id

]R̃ (Θ) ,

where Pk̄ := I2 − f̂k̄ f̂Tk̄ is a projection onto the orthogonalcomplement of fk̄, and R̃ (Θ) is called the unit rigidity matrix.

The rigidity matrix exhibits the property that r ∈ χ(Θ) ifand only if R (Θ) r = 0, or equivalently R̃ (Θ) r = 0.

Theorem 9. Under the unforced dynamics (10), r (t) will asymp-totically converge to χ(Θ).

Proof: Without loss of generality, let the centroid of r (0) bethe origin. Let f = ξ+ (f , r(0)). Consequently, ‖r (0)‖ = ‖f‖and C(f) = 0. Applying the change of variable z = r − f ,where rij = zij + fij , then for each i ∈ V ,

żi =∑

j∈N (i)

wij r̂Tij f̂⊥ij r̂⊥ij

=∑

j∈N (i)

wij

‖zij + fij‖2(zij + fij)

Tf̂⊥ij (zij + fij)

⊥

=∑

j∈N (i)

wijzTij f̂⊥ij

‖zij + fij‖2(zij + fij)

⊥

:=∑

j∈N (i)

g(ij)(z, f). (12)

Consider the Lyapunov function V (z) = 12zT z, we can

expand its time derivative using Eq. (12) and apply Prop.6 as

V̇ (z) =∑i∈V

zTi∑

j∈N (i)

g(ij)(z, f)

= −∑k̄∈E

zTk̄ g(k̄)

= −∑k̄∈E

wk̄zTk̄zTk̄f̂⊥k̄

‖zk̄ + fk̄‖2 (zk̄ + fk̄)

⊥

= −∑k̄∈E

wk̄zTk̄f̂⊥k̄

‖zk̄ + fk̄‖2

(zTk̄ z

⊥k̄ + z

Tk̄ f⊥k̄

)

= −∑k̄∈E

wk̄(zTk̄f⊥k̄

)2‖fk̄‖ ‖zk̄ + fk̄‖

2 . (13)

As V̇ (z) ≤ 0 and V̇ (z) = 0 if and only if z ∈ χ(Θ), it followsthat r ∈ χ(Θ) and r(t) converges to χ(Θ), establishingasymptotic stability by La Salle’s theorem [29].

1. It is assumed throughout that no two agents are coincident withina formation, and so ‖fk̄‖ 6= 0.



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Asymptotic convergence to χ(Θ) can be further strength-ened to exponential stability. This is explored in the follow-ing theorem.

Theorem 10. Under the unforced dynamics (10), the Lyapunovfunction V = 12 (r− f)

T(r− f) := 12z

T z satisfies

V̇ ≤ S (f)−1 zT R̃TDk̄[m2

kI2]R̃z := −S (f)−1 zTQz,

where f = ξ+ (f , r(0)), and we define the term m2k(r0) :=

wk̄

∥∥∥f̃k̄∥∥∥ /(S (f)−1 ‖z0‖+ ∥∥∥f̃k̄∥∥∥)2. Consequently, r (t) willglobally exponentially converge to χ(Θ).

Proof: If we expand Eq. (13), we have

V̇ (z) = −∑k̄∈E

wk̄

((rk̄ − fk̄)

Tf⊥k̄

)2‖fk̄‖ ‖rk̄ − fk̄ + fk̄‖

2

= −∑k̄∈E

wk̄(rTk̄f⊥k̄

)2‖fk̄‖ ‖rk̄‖

2

= −∑k̄∈E

wk̄ ‖fk̄‖‖rk̄‖

2 rTk̄

(f̂⊥Tk̄ rk̄

)f̂⊥k̄ .

We now apply the relationship

(f̂⊥Tk̄ rk̄)f̂⊥k̄ = Projf̂⊥

k̄

rk̄ = rk̄ − Projf̂k̄rk̄ = Pk̄rk̄

to see that

V̇ (z) = −∑k̄∈E

rTk̄wk̄‖fk̄‖‖rk̄‖

2Pk̄rk̄

= −rT H̃TDk̄

[wk̄‖fk̄‖‖rk̄‖

2Pk̄

]H̃r. (14)

Now, we observe that

Dk̄


2Pk̄

]H̃f = 0 (15)

as Pk̄fk̄ = 0 due to Pk̄ being a projection onto the orthogonalcomplement of fk̄. Thus, we can add Eq. (15) into Eq. (14) as

V̇ (z) = − (r− f)T H̃TDk̄


2Pk̄

]H̃ (r− f)

= −zT H̃TDk̄


2Pk̄

]H̃z.

Similarly, we can also introduce an additional Dk̄ [Pk̄] term,using the fact that Pk̄ is a projection matrix and so P

2k̄

= Pk̄.Thus,

V̇ (z) = −zT H̃TDk̄ [Pk̄]T Dk̄

[wk̄ ‖fk̄‖‖rk̄‖

2 I2

]Dk̄ [Pk̄] H̃z.

However,

‖fk̄‖‖rk̄‖

2 =‖fk̄‖

‖zk̄ + fk̄‖2 ≥

‖fk̄‖(‖zk̄‖+ ‖fk̄‖)

2 .

Applying Thm 9, ‖zk̄‖ ≤ ‖z‖ ≤ ‖z0‖. Thus, we can write

wk̄ ‖fk̄‖‖rk̄‖

2 ≥wk̄S (f)

∥∥∥f̃k̄∥∥∥(‖z0‖+ S (f)

∥∥∥f̃k̄∥∥∥)2= S (f)

−1wk̄

∥∥∥f̃k̄∥∥∥(S (f)

−1 ‖z0‖+∥∥∥f̃k̄∥∥∥)2 := S (f)

−1m2

k.

From the definition of m2k̄

and the unit rigidity matrix R̃then

V̇ ≤ −S (f)−1 zT R̃TDk̄[m2k̄I2

]R̃z.

The worst case initial error is ‖z0‖ ≤ ‖−2f‖ = 2√nS (f),

which follows from the fact that the trajectory of the errorz under the dynamics (10) travels along a sphere of radius

‖f‖ [1], [13]. Noting that m2k̄≥ wk̄

∥∥∥f̃k̄∥∥∥ /(2√n+ ∥∥∥f̃k̄∥∥∥)2,let β = mink̄mk > 0, then

V̇ (z) ≤ −βS (f)−1 zT R̃T R̃z.

Isolating the components of z that are orthogonal to thenullspace of R̃, denoted z⊥, then

V̇ (z⊥) ≤ −βS (f)−1 λq(R̃T R̃

)zT⊥z⊥

= −βS (f)−1 λq(R̃T R̃

)V (z⊥) ,

where λq(R̃T R̃

)> 0 is the smallest non-zero eigenvalue

of R̃T R̃. Hence the dynamics will converge exponentiallyto the nullspace of R̃, or equivalently, to χ (Θ) by La Salle’stheorem [29].

Theorem 10 describes convergence to the set χ (Θ) . Toguarantee that the steady state formation is parallel to fthen the framework must be bearing rigid. With this is mind,Zhao and Zelazo [13] examined the effect of a small changein agent positions 4r ∈ R2n with respect to the bearingmeasurements, where if R (Θ)4r = 0, in this case 4r iscalled an infinitesimal bearing motion. An infinitesimal bear-ing motion corresponding to a translation or a scaling of thewhole framework is called trivial. The framework Θ (G, f)is called infinitesimally bearing rigid if all infinitesimal bear-ing motions are trivial. Intuitively, this means that suchframeworks will appear to maintain their shape, regardlessof scaling or position, under the dynamics (10b). We nowstate a rank condition on the bearing rigidity matrix forchecking if a given framework exhibits infinitesimal bearingrigidity. Throughout this paper, we will say that a formationis bearing rigid implying that it exhibits infinitesimal bearingrigidity.

Theorem 11. [30] A framework Θ (G, f) is infinitesimally bear-ing rigid, if and only if

rank [R (Θ)] = 2n− 3,

with the nullspace of R (Θ) spanned by orthogonal vectors v1 =FT

[1T 0T

]T, v2 = F

T[0T 1T

]T, and v3 = f̃ .

Here, F =[In ⊗

[1 0

]In ⊗

[0 1

] ] and f̃ is the underlying unitformation.



6

The vectors v1, v2 and v3 in Thm 11 capture the modesrepresenting the invariance of the bearing measurement totranslation in x, translation in y, and scale, respectively.A consequence of Thm 11 is that the bearing dynamicswill converge to a formation parallel to f if the underlyingframework is bearing rigid. Further, as the dynamics areinvariant to scale and centroid by Prop. 7 then the dynamicswill converge to one of the two formations in ξ± (f , r). Asdiscussed in our previous work [1], and summarized inthe following corollary of Thm 11, this will almost alwaysbe the positively scaled formation ξ+ (f , r), with ξ− (f , r)corresponding to an unstable equilibrium point.

Corollary 12. Let the framework Θ (G, f) be infinitesimallybearing rigid. Under the unforced dynamics (10), r (t) will almostglobally exponentially converge to ξ+ (f , r0), with the worst-caseconvergence rate proportional to λ4

[R̃ (Θ)T R̃ (Θ)

].

4 FORMATION ACQUISITION

In this section, we discuss a method to form a desiredformation shape on n agents over which to apply theweighted bearing-compass dynamics (10). A human oper-ator may provide such a shape, for example from an imageas in Fig. 3a, for the purposes of a distributed sensingor coverage problem [31]. Recreating the image using theavailable n agents, the image can be partitioned into nequal area cells using a Voronoi partitioning algorithm[32], [33]. In doing so, a set of desired agent positionsf =

[fT1 f

T2 · · · fTn

]T, centered on each cell, is con-

structed. From Cor. 12, if agent positions f are coupled withan infinitesimally bearing rigid framework Θ(G, f ), thenagents running the weighted bearing-compass dynamicswill converge to f . To this end, this section will focus onselecting links E from a set of available edges, such as thosein Fig. 3b, to acquire a favorable graph G, visualized in Fig.3c from a set of available edges.It is assumed that the links are selected such that whenagents are at f the graph G will have short edges to re-duce communication costs. This is accomplished by select-ing links from a distance graph [9]. This graph, denotedGD = (V,ED), is composed of edges

ED = {{i, j} : ‖fj − fi‖ ≤ dmax} ,

where dmax is chosen to be sympathetic to communicationrange limits of the agents (i.e., wireless signal strength ora distance such that a monocular camera can still resolveneighboring agents).We will use the desired agent positions f and the feasibleedge set ED to select some E ⊆ ED such that the dynamics(10) will both converge and be optimal over various metrics,such as maximizing convergence.In §4.1, we will explore a necessary and sufficient conditionon the bearing rigidity matrix R (Θ) for the dynamics toconverge. More specifically, we will select an edge set E ⊆ED such that the conditions on infinitesimal bearing rigidity,outlined in Thm 11, are met, which means the formationsatisfies rank [R (Θ)] = 2n − 3. It is assumed that dmax issufficiently large so that such a selection E with this criteriaexists.

(a) (b) (c)

Figure 3: a) A human operator draws the formation shapeon a map, and a Voronoi partitioning of the agents is com-puted. b) A distance graph is constructed which providesthe edges we will select from. c) Edges are selected toensure the formation is bearing rigid, and is as performantas possible given different performance metrics. Dottededges represent additional edges past what is needed for in-finitesimal bearing rigidity, as extra edges typically improveperformance.

(a) (b)

(c) (d)

Figure 4: Greedy edge selection under metrics: a) fr (S),b) fr (S) + αft (S) and c) fr (S) + αft (S) until the graphis infinitesimally rigid and then f`(S). (d) Edge weightselection using Alg. 2 with the edge set from Fig. 4c. Solidlines indicate edge required for infinitesimal rigidity usingAlg. 1 and dashed indicate additional edges to improve con-vergence using greedy selection. Edge thickness indicatesthe relative edge weights.

While our formation edge set must be bearing rigid, wewould also like the set to promote good performance of thedynamics. We can add additional terms to the edge selectionto promote this, which we will explore in §4.2. Further, wemight be given a set of edges that we must use, in whichcase we can optimize the weights of those edges to furtherimprove performance. This is explored in §4.3.An example of such a refinement can be seen in Fig. 4, wherethe edges have been selected from those in the distancegraph in Fig. 3b.

Submodularity is key to our approach to the edge selectionproblem. In particular, the theorem below serves as theworkhorse for the results that follow.

Theorem 13. Let S ⊆ E and

Q (S) = U (S)TU (S) ,



7

where U (S) = (I (S ∪ S0)⊗ Id)U (E), I (T ) represents the Tcolumns of the identity matrix and S0 is the columns of U(E)associated with S = {∅}. Then the following are submodularfunctions:a) fr (S) = rank [Q (S)]b) ft (S) = trace [Q (S)]c) f` (S) = logdet [Q (S)] for Q (S) � 0.Furthermore, ft (S) is, in fact, modular.

Proof: (a) We observe that

rank [Q (S)] = rank[U (S)

TU (S)

]= rank

[U (S)

T].

The matrix U (S) corresponds to the S ∪ S0 column pairsof U (E). Maximizing fr (S) over S is then equivalent toselecting the 2-pair columns of each s ∈ S in U (E) thatmaximizes the rank. This is the standard matroid rankproblem and fr is submodular [24].

In support of (b) and (c), observe that

Q (S) = U (S)TU (S)

= U (E)T

(I (S ∪ S0)⊗ I2)T (I (S ∪ S0)⊗ I2)U (E)

= U(E)T∑

s={∅},s∈S

[(I ({s})⊗ I2)T (I ({s})⊗ I2)T

]U (E)

= Q(∅) +∑s∈S

Q ({s}) . (16)

(b) As trace [Q (S)] =∑

s∈S trace [Q ({s})] it follows thattrace [Q (S)] is modular.(c) For M � 0 and N,P � 0 thendet (M) det (M +N + P ) ≤ det(M + N)det(M + P )[34]. As log(·) is an increasing function then

0 ≤ log(det(M +N)det(M + P ))− log(detMdet(M +N + P ))≤ logdet(M +N) + logdet(M + P )− (logdetM + logdet(M +N + P ))≤ logdet(M + P ))− logdetM)− (logdet(M +N + P )− logdet(M +N)). (17)

Consider now A ⊆ B ⊆ S and s /∈ B. Assume that Q(A) �0 as Q(S) is formed by the sum of positive semidefinitematrices, then Q(B) − Q(A), Q({s}) � 0 . Let M = Q(A),N = Q(B)−Q(A), P = Q({s}) and applying (16) and (17)as

0 ≤ logdet(Q(A) +Q({s})− logdetQ(A)− (logdet(Q(B) +Q({s})− logdetQ(B))≤ logdetQ(A ∪ {s})− logdetQ(A)− (logdetQ(B ∪ {s})− logdetQ(B)) .

Hence, by Def. 3, logdet [Q (S)] is submodular.

4.1 Edge Selection for Bearing Rigidity

From Cor. 12, a necessary condition for dynamics (10) toconverge to a formation similar to f is that the frameworkΘ is infinitesimally bearing rigid. From Thm 11, it is thendesirable to select a subset of edges S ⊆ ED so that

Algorithm 1 Select edges which guarantee infinitesimalbearing rigidity.Input: Graph to refine: GD = (V,ED)Ensure: rank[R(ED)] = 2n− 3Input: Submodular set function: f(S), S ⊆ ED

1: S ← ∅2: while rank[R(S)] < 2n− 3 do3: e∗ = argmax

e∈ED\S[f(S ∪ {e})− f(S)] ‡

4: S ← (S ∪ {e∗})5: end while6: return S

rank [R] = 2n − 3. Representing the rigidity matrix as amatrix valued function with respect to the edges S then

R(S) = ∂ĝ(S)∂f

where ĝ(S) =[f̂Ts̄1 f̂

Ts̄2 · · ·

]T, for S = {s1, s2, . . . } ⊆

ED . The unitless rigidity matrix R̃(S) is similarly defined.We leverage this rank property in the following proposition.

Proposition 14. The set function

fr (S) = rank [R (S)] (18)

for S ⊆ ED is submodular.

Proof: This result follows directly from Thm 13.a by lettingS0 = {∅} and

U(S) = R(S)= (I(S ∪ S0)⊗ I2)R(ED).

Since we have shown that the rank of the bearing rigiditymatrixR is submodular over the underlying edge set E, wecan apply Thm 5 to select edges in a greedy fashion, leadingto an infinitesimally bearing rigid formation. This processis described in Alg. 1 and demonstrated in Fig. 4a withthe selection of the solid edges to acquire a bearing rigidformation. In fact, as the rank increases by at most one withthe addition of an edge, the greedy method corresponds tothe check and selection of linear independent columns ofR(ED). Hence, Alg. 1 will attain an infinitesimally bearingrigid formation with the minimal number of edges, namely2n− 3.

4.2 Edge Selection for Convergence

In §4.1, we showed how to select edges such that the result-ing formation is bearing rigid. However, in large formationswith many possible edges, it is likely that many such selec-tions will meet this minimum rank requirement, and someselections might offer better performance than others. In thissection, we explore ways to promote formation convergenceas a part of the edge selection process. The underlyingLyapunov function and the associated Lyapunov rate fromThm 10 is V = 12z

T z and V̇ ≤ −S (f)−1 zTQz, respectively.

‡. If, as in Thm 5, R > 1, then this line will select the subset e ⊆ED \S such that |e| = R if |ED \ S| ≥ R and |e| = |ED \ S| otherwise.



8

Explicitly encoding the edges S ⊆ ED that contribute to Qthen

Q(S) = R̃(S)TDk̄(S)[m2k̄I2

]R̃(S), (19)

where R̃(S) := (I(S)⊗ I2) R̃, m2k̄ is defined as in Thm 10,Dk̄(S)

[m2

k̄I2]

:= (I(S)⊗ I2)Dk̄[m2

k̄I2]

(I(S)⊗ I2)T , andR̃ and Dk̄

[m2

k̄I2]

are constructed from the edge set ED .

The worst-case convergence rate to the null-space of Qis dictated by the smallest nonzero eigenvalue of Q. Thisalso dictates the exponential convergence envelope of thedynamics. The other non-zero eigenvalues of Q also play arole in describing the convergence of the dynamics. Edgeselection based on measures of these eigenvalues has theeffect of improving the overall dynamics performance andis the edge optimization criteria for this section.

The first considered measure is trace [Q], which is propor-tional to the instantaneous average convergence envelopeof the dynamics at time t when the initial error is ‖z (t)‖ issampled evenly on the unit circle. This follows from the factthat for a fixed scale S (f)−1,

d

dtE‖z(t)‖=1V (z (t))

≤ −S (f)−1 E‖z(t)‖=1zT (t)Qz (t)

= −S (f)−1 12n

n∑i=1

λi(Q)E‖z(t)‖=1zT (t) z (t)

=

( −1nS (f)

trace [Q])E‖z(t)‖=1V (z (t)).

Consequently, a larger trace [Q] will speed up the averageconvergence rate envelope.

An attraction of this measure is that it is modular withrespect to the edge selection problem on the graph. Thisfeature is formalized in the following proposition.

Proposition 15. Let Q(S) be defined as in (19); then the setfunction

ft (S) = trace [Q(S)] (20)

for S ⊆ ED is modular.

Proof: The proof follows from Thm 13.b by letting S0 = {∅}and

U (S) = Dk̄(S) [mk̄I2] R̃ (S)= (I(S)⊗ I2)Dk̄

[m2k̄I2

](I(S)⊗ I2)T (I(S)⊗ I2) R̃

= (I(S)⊗ I2)Dk̄[m2k̄I2

]R̃

= (I(S)⊗ I2)Dk̄(ED) [mk̄I2] R̃(ED).

A boon of the modularity of trace [Q] is that the greedyselection of edges under this measure provides the optimaledge selection. Interestingly, trace [Q] is independent of theunitless rigidity matrix R̃ (S). This becomes apparent by

(a) (b)

Figure 5: a) Curve for m2k̄

versus∥∥∥f̃k̄∥∥∥ for an unweighted

edge, i.e., wk̄ = 1, given desired formation f with initialerror z0. b) Average potential over 100 random initial con-ditions, where the rank curve is the formation from Fig. 4a,trace curve is the formation from Fig. 4b, log det curve isthe formation from Fig. 4c, the weighted log det curve is theformation from Fig. 4d and the inclusion of all edges ED inthe formation from Fig. 3b.

manipulating the trace of a single edge {s} ∈ ED , indicatingthe sth row of H by Hs; then

ft({s}) = trace[R̃ ({s})T Dk̄({s})[m2k̄I2

]R̃ ({s})]

= trace[(Hs ⊗ I2)Dk̄({s})[m2k̄I2

](HTs ⊗ I2)]

= trace[m2s([

1−1

]⊗ I2)P 2s (

[1 −1

]⊗ I2)]

= trace[m2s([

1 −1] [ 1−1

]⊗ I2)Ps]

= trace[2m2sPs] = 2m2s, (21)

and using the property that ft(S) =∑

s∈S ft ({s}) =2∑

s∈S m2s, as ft (S) is a modular function. The measure is

therefore solely dependent on the lengths∥∥∥f̃k̄∥∥∥ and weights

wk̄ of the selected edges, and the initial condition r0. Figure5a plots the function m2

kfor an unweighted edge. For the

regime∥∥∥f̃k̄∥∥∥ ≥ ‖z0‖S (f)−1 for all edges k̄, or equivalently

‖fk̄‖ ≥ ‖r0 − ξ+(f , r0)‖, then smaller∥∥∥f̃k̄∥∥∥corresponds to

a large m2k

and therefore a larger trace [Q], i.e., smalleredges in the final formation shape are preferable. For theregime

∥∥∥f̃k̄∥∥∥ ≤ ‖z0‖S (f)−1, the reverse is true. Examiningconvergence rate close to the equilibrium, namely when

‖r0 − f‖ ≤ mink‖fk̄‖ , (22)

then the optimal q-edge selection to maximize trace [Q]would be to select the q shortest edges

∥∥∥f̃k̄∥∥∥.As ft(S) is independent of R̃ (S), it will not necessarilyfavor a maximum rank rigidity matrix which is necessary toacquire the desired formation shape. An alternative is to usethis result in conjunction with the submodular rank measuredescribed in Prop. 14. This can be achieved by applying theadditive submodular property of Prop. 4 to form the newsubmodular function

f(S) = fr(S) + αft(S), (23)

where α > 0.



9

So as to encourage the minimal number of required edgesfor convergence to the desired formation associated withfr(S) = 2n − 3, then it may be desired to preferentiallyselect the rigidity rank measure over the trace measure. Tothis end, α can be selected to ensure that fr(S), an integer,dominates f(S). Specifically, when α < 1/ft(ED) then asft(S) is monotonically increasing αft(S) ∈ [0, 1) for all S ⊆ED . Hence, an edge e ∈ ED\S will be selected if it increasesrank in preference to if it increases trace as 2

fr(S ∪ {e})− fr(S) ≥ 1 ≥ α (ft(S ∪ {e})− ft(S)) .

The effect of this choice results in the trace measure “break-ing ties” if there is more than one choice of edge whichincreases rank when using the greedy selection heuristic forrank in Alg. 1. After sufficient edges SR have been addedto the graph such that the rigidity matrix is maximum rank,i.e., rank [R(SR)] = rank [Q(SR)] = 2n− 3, then to furtherimprove performance it may be desired to add additionaledges S ∈ ED \ SR. The trace [Q(SR ∪ S)] is one measureto accomplish this result.The solid edges in Fig. 4b indicate an infinitesimally rigidgraph formed under greedy selection for fr(S) + αft(S)with α sufficiently small such that α < 1/ft(ED). Com-paring the solid edges in Fig. 4a, i.e., selection only underrank, to Fig. 4b a formation with shorter edges is acquired.Additional edges to improve performance under fr(S) andfr(S)+αft(S) are indicated in Figs. 4a and 4b, respectively.Figure 5b shows the improved convergence rate of the edgeselection under both the trace and rank measure comparedto the rank measure alone.A limitation of the measure in Eq. (23) is that, as it is basedon the average convergence, it does not favor improvingthe worst case convergence. The worst case convergence isdictated by the smallest non-zero eigenvalue of Q, namelyλ4(Q). The set function f(S) = λ4(Q(SR ∪ S)) is not asubmodular function, however, so does not provide greedyguarantees when Alg. 1 is applied. An alternative measure,which promotes increasing the smallest eigenvalues of amatrix, is

∑2ni=4 logλi(Q(SR ∪ S)). From the discussion

on the nullspace of R in Thm 11, note that v̂1, v̂2 andv̂3 span the nullspace of Q(SR) with unitary eigenvalues,i.e., Q(SR) +

∑3j=1 v̂jv̂

Tj is full rank and logλn(v̂jv̂

Tj ) =

log(1) = 0. Hence, for rank [Q(SR)] = 2n− 3 it follows that2n∑i=4

logλi(Q(SR ∪ S))

=2n∑i=4

logλi(Q(SR ∪ S)) +3∑

j=1

logλn(v̂jv̂Tj )

=2n∑i=1

logλi[Q(SR ∪ S) +3∑

j=1

v̂jv̂Tj ]

= tracelog[Q(SR ∪ S) +3∑

j=1

v̂jv̂Tj ]††

= logdet[Q(SR ∪ S) +3∑

j=1

v̂jv̂Tj ].

2. The same result follows if batches of edges are added, i.e., R > 1inThm 5.

The following proposition describes the submodularity of∑2ni=4 logλi(Q) providing the necessary features to guaran-

tee the suboptimal greedy selection performance of Thm 5.

Proposition 16. Let Q(S) be defined as in (19); the set function

f` (S) = logdet[Q(SR ∪ S) +3∑

i=1

v̂iv̂Ti ], (24)

for S ⊆ ED \ SR and where rank [f` (∅)] = rank [Q(SR)] =2n− 3, is submodular.

Proof: Following similarly to the proof of Prop. 15, then

U (S) =

[Dk̄(SR∪S) [mk̄I2] R̃ (SR ∪ S)∑3

i=1 v̂iv̂Ti

]

= I(S ∪ SR ∪ T )⊗ I2

[Dk̄(SR∪ED) [mk̄I2] R̃ (SR ∪ ED)∑3

i=1 v̂iv̂Ti

]= (I(S ∪ S0)⊗ I2)U (ED) ,

where T = {|ED| + 1, |ED| + 2, |ED| + 3} and S0 =SR ∪T . Further as

∑3i=1 v̂iv̂

Ti is an idempotent matrix then

UT (S)U (S) = Q(SR ∪ S) +∑3

i=1 v̂iv̂Ti . Therefore, U(S) is

in the necessary form to apply Thm 13.c and so the resultfollows.Consider the infinitesimally rigid formation acquired underthe selection measure fr(S) + αft(S) indicated with solidedges in Fig. 4a. As rank [Q(SR)] = 2n − 3 then additionaledges can be selected under the measure f` (S) to improveperformance. Fig. 4c indicates the greedy selection of anadditional 15 edges with dashed edges. The relative per-formance between the three measures is shown in Fig. 5bwith f` (S) outperforming the other two.

4.3 Edge Weight Selection

Once an edge set E has been selected to achieve infinites-imal bearing rigidity, additional performance improvementcan also be garnered if edge weights w can be designed non-uniformly. Assuming the weights are positive continuousvariables for certain selection criteria these can be optimizedusing convex optimization. Examining dynamics (10b), as∥∥∥(r̂Tij f̂⊥ij )r̂⊥ij∥∥∥ ≤ 1, then ∥∥u(ij)∥∥ ≤ wij and so the edgeweight wij is akin to the maximum available control effortavailable to correct for errors induced by the edge {i, j}.Not surprisingly, increasing edge weights has the effect ofincreasing the convergence rate of the dynamics. This iscaptured more explicitly through Cor. 12 where the worstcase convergence is dictated by the smallest non-zero eigen-value of Q(w), namely λ4(Q(w)), and the edge weightsappear linearly in the term m2

k̄(wk̄) within the definition of

Q. Balancing the maximum control effort available at eachnode, defined as umax, and maximizing λ4(Q(w)) then anequivalent optimization problem is:

maximizew

λ4(Q(w))

s.t. wk̄ ≥ 0,∑

j∈N(i)

wij ≤ umax. (25)

††. For matrix M , log[M ] denotes to the matrix logarithm of M .



10

The control effort cap in (25) can be represented in terms ofthe incidence matrix of the graph as

∣∣HT ∣∣w ≤ umax1, where|·| for a matrix indicates the element-wise absolute value.Applying a similar approach to Prop. 16 to remove thenullspace of Q then

λ4(Q) = λ1(Q+ β3∑

i=1

v̂iv̂Ti ). (26)

It is assumed that β is large enough to shift the eigenvaluesassociated with

∑3i=1 v̂iv̂

Ti above λ4(Q), i.e., β ≥ λ4(Q).

A sufficiently large β can be found by examining an up-per bound of 12n−3 trace [Q] which is the average over thenonzero eigenvalues of Q with

λ4(Q) ≤1

2n− 3

2n∑i=4

λi(Q) =1

2n− 3trace [Q]

=1

2n− 32

m∑k̄=1

m2k̄ (from equality (21))

≤ 22n− 3

m∑k̄=1

wk̄S(f)

4 ‖z0‖(from upper bound in Fig. 5a)

≤ S(f)2 ‖z0‖

1

2n− 3n

2umax =

numax4 (2n− 3)

S(f)

‖z0‖:= β,

where the final inequality follows from the maximum con-trol cap as

m∑k̄=1

wk̄ =1

2

n∑i=1

∑j∈N(i)

wij ≤1

2

n∑i=1

umax ≤n

2umax.

From the additive property (16) over the edges of Q then

Q =∑s∈E

Q({s})

=∑s∈ER̃ ({s})T Dk̄({s})

[m2k̄I2

]R̃ ({s})

=∑s∈E

ws̄R̃ ({s})T

∥∥∥f̃s̄∥∥∥(

S (f)−1 ‖z0‖+

∥∥∥f̃s̄∥∥∥)2 I2 R̃ ({s})

=∑s∈E

ws̄Qu({s}),

where Qu ({s}) is the matrix representing the unweightedcontribution of edge s ∈ E. Hence, the maximization ofλ4(Q) using (26) can be represented as a linear matrixinequality and solved as a convex optimization presentedin Alg. 2.We note that one reasonable strategy using what we havedeveloped is to do an initial edge selection based on themethods in §4.1 and §4.2, then apply Alg. 2 to refine thechosen edge weights. An example of the edge weight selec-tion is shown in Fig. 4d applied to the edges of Fig. 4c. Theimproved convergence rate can be seen in Fig. 5b.

5 FORMATION MANIPULATION

In our previous work [6] we studied the performance of theforced dynamics (10a). External signals were applied from

Algorithm 2 Algorithm to maximize λ4 (Q) with respect tothe weight vector w given a fixed edge set E.

maximizew

δ

s.t.∑s∈E

ws̄Qu({s}) + β3∑

i=1

v̂iv̂Ti � δI,

w ≥ 0,∣∣∣HT ∣∣∣w ≤ umax1

leader agents V` ⊆ V with ũi 6= 0, where the selection crite-ria of a “favorable” leader was based on the leader’s abilityto translate or scale a bearing-constrained formation. Theconsidered form of ũi was an impulse signal. In this section,we adapt the tools developed in the previous sections toselect leaders which have external signals that “ground” theformation to a certain centroid and scale given informationfrom a set of anchors which behave as stationary neighbors.These leader agents will manipulate the formation as a whole,so we can acquire the desired formation scale and centroid.Taking a similar approach to Shames and Summers [28], weassume there are some external anchors that act as additionalbearing constraints to any agent that can sense them. Thereare many examples of such signals like the VHF omni-directional range (VOR) radio navigation system used byaircraft, localizing from signals of opportunity like CDMAcellular towers [35], or simply letting a subset of agents haveaccess to their position using GPS so they can measure theirbearing offset from locations set a priori. This anchored setupcomplements the work by Zhao and Zelazo [13], [20] whichconsiders a set of anchor nodes that are native to the graph.Let Va be the set of na anchor nodes with correspondingpositions fa ∈ R2na . Let Ea (S) = {{i, j} : i ∈ S, j ∈ Va}be the set of edges between anchors and the leaders in setS ⊆ V . This set represents the possible bearing measure-ments a leader agent can make. In this section, we will stillbe selecting a subset of edges, but in contrast to previoussections, we will do so by selecting edges connected to asubset of the agents. This will result in selecting a subset ofthe anchor edges via selecting leaders resulting in a systemwhich is “full rank” with respect to an augmented rigiditymatrix and exhibits “good performance”.Specifically, leaders have access to a family of na anchorreference positions. The ith anchor’s position is denotedas ai ∈ R2. The graph G is extended to include anchornodes Va, or cardinality na, and ma anchored edges {i, j}with weights wij for all i ∈ Va and j ∈ V` with theextended neighborhood of agents V` denoted as Na. Theagent’s position vector r is augments with the anchor po-sitions with rn+i = ai for i = 1, . . . , na and the desiredformation f is similarly augmented with fn+i = ai. Theleader agents forced signal is then defined using the newbearing measurements to the anchors as

ũi =∑

j∈Na(i)

wij(r̂Tij f̂⊥ij )r̂

⊥ij . (27)

The attraction of this form of external leader signal is thatmuch of the machinery established in the previous sectioncan be adapted to examine the anchored properties of these



11

dynamics. The first notion is the anchored rigidity matrixRa ∈ R2(m+ma)×2(n+na), can be formed as Ra (Θ) := ∂ĝ∂f .Noting that

[∂ĝ∂f

]ij

= 0 for j ∈ Va, as the anchors nodes areeffectively grounded, then

Ra =[R 0∂ĝa∂f 0

]where R is the rigidity matrix of the unanchored formationand ĝa =

[f̂m+1 f̂m+2 · · · f̂m+ma

]. The anchored

unitless rigidity matrix R̃a is defined similarly to the unit-less rigidity matrix R̃. Consequently, the anchors can serveas a method to remove the nullspace associated with thecentroid and scale of R. When sufficient bearing measure-ments are added to the dynamics, through the addition ofanchors and leaders to define a unique formation f , thenthe framework Θ is called anchored infinitesimally bearingrigid. This property manifests itself as a rank propertysummarized in the following theorem.

Theorem 17. For a framework to be anchored infinitesimallybearing rigid, we must have that

rank [Ra (Θ)] = 2n.

As the addition of an anchor grounds at most two di-mensions in the null space of R, at least two anchors areneeded, with associated leader agent edges, for an infinites-imally bearing rigid framework to become an anchoredinfinitesimally bearing rigid framework. Not surprisingly,for a anchored infinitesimally bearing rigid formation thenthe dynamics will converge exponentially to the uniqueformation f . The proof of this property echoes that of Thm10 using Thm 17. The result is summarized in the followingwithout proof, leveraging the property rank [Ra] = 2n toshow that the dynamics converge to a unique f , rather thana formation dependent on the initial conditions as in Cor.12.

Theorem 18. Let the framework Θ (G, f) be anchored infinites-imally bearing rigid. Under the forced dynamics (10) with (27),a Lyapunov function is V = 12z

T z, where z = r − f , withassociated Lyapunov rate bounded as

V̇ ≤ S (f)−1 zT R̃TaDk̄[m2

kI2]R̃az := −S (f)−1 zTQaz,

where m2k(r0) is as defined in Thm 10. Consequently, r (t) will

globally exponentially converge to f .

5.1 Leader Selection for Anchor Rigidity

As for the unforced dynamics, it is important to reasonabout effective graph topologies that meet the requirementsfor convergence and improve the overall performance of theforced dynamics (10). The ability to adapt the graph topolo-gies in the forced dynamics is assumed to be restricted tothe selection of a set of leader agents given at least na ≥ 2anchors. The topology selection problem is then a nodeselection rather than an edge selection problem.

With the objective of selecting a set of leader agents S ⊆V \Va to acquire the infinitesimally bearing rigid framework

necessary for the convergence of the dynamics, the anchoredrigidity matrix is

Ra(S) =[ R 0

∂ĝa(S)∂f 0

]where ĝa(S) =

{f̂ij}

where i ∈ S and j ∈ Va. Unlikethe edge selection rigidity matrix problem, 2na rows areadded to Ra for each leader selected. The matrix-valued setfunction R̃a(S) is defined similarly.The following proposition establishes that the rank set func-tion for Ra is submodular. Hence, the leader agents can beselected using greedy Alg. 1 until Ra has rank 2n. Similarto the selection under rank in §4.1 the minimum number ofleader agents, namely two will be selected.

Proposition 19. The set function

fr (S) = rank [Ra(S)] (28)for S ⊆ V \ Va is submodular.

Proof: This result follows directly from Thm 13.a by lettingS0 = {1, 2, . . . , 2m− 1, 2m} and

U(S) = Ra(S)= (I(S ∪ S0)⊗ I2na)Ra(V \ Va),

where for V \ Va = {v1, v2, . . . , vn} then ĝa(V \ Va) =[ĝa({v1})T ĝa({v2})T · · · ĝa({vn})T

].

5.2 Leader Selection for Convergence

We now consider the selection of leaders under measuresthat promote the convergence rate of the dynamics. Thematrix Qa which appears in the Lyapunov function of Thm18 can be written with respect to the leader set S ⊆ V \ Vaas

Qa(S) = R̃a(S)T[Dk̄[m2

k̄I2]

00 Dk̄a(S)

[m2

k̄I2] ] R̃a(S),

(29)where Dk̄

[m2

k̄I2]

is constructed from the edges betweenagents in V \ Va,

Dk̄a(S)[m2k̄I2

]:= (I(S)⊗ I2na)Dk̄a

[m2k̄I2

](I(S)⊗ I2na)

T,

and Dk̄a[m2

k̄I2]

is constructed from the edges between theagent set V \ Va and the anchors Va. Following a similarapproach to §4.2, trace [Qa(S)] is a modular function andthe log of the determinant of a nullspace-removed Qa(S) isa submodular function with respect to the leader set S. Thisfeature is summarized in the following two propositions.The proofs of these propositions are omitted due to theirsimilarities to Props. 15 and 21.

Proposition 20. Let Qa(S) be defined as in (29). The setfunction

ft (S) = trace [Qa(S)] (30)

for S ⊆ V \ Va is modular.

Proposition 21. Let Qa(S) be defined as in (29). The setfunction

f` (S) = logdet[Qa(SR ∪ S) +[

02n×2n 02n×2ns02ns×2n I2na

]]

(31)



12

where S ⊆ V \(Va∪SR), and rank [f` (∅)] = rank [Qa(SR)] =2n is submodular.

As per §4.2, the greedy Alg. 1 can be applied to the setfunction (30) to select leaders that improve the convergenceperformance even when then the framework is not anchoredinfinitesimally bearing rigid. The logdet measure is applica-ble as a selection criteria once SR leaders are selected whichare sufficient to establish anchored infinitesimal bearingrigidity with rank [Qa(SR)] = 2n. Additionally, positivelinear combinations of the set functions (28), (30) and (31)are also submodular functions by Prop. 4. The suboptimal-ity guarantees of Thm 5 would then apply to the greedyselection process. From the initial graph in Fig. 4d with fouranchors present, Fig. 6a shows the selection of leaders underthe measure fr (S) + αft (S).

5.3 Edge Weight Selection

The edge weights w composed of the m initial edges andma anchored edges can be optimized similarly to §4.3.In the anchored case, the worst case convergence rate isdictated by the smallest nonzero eigenvalue of Qa, namelyλ2na+1(Qa). Examining a reduced form of the anchored

rigidity matrix defined as Rra =[ R

∂ĝa(S)∂f

], with the

associated reduced unitless anchored rigidity matrix R̃rathen for Qra := R̃TraDk̄

[m2

kI2]R̃ra ∈ R2n×2n,

λ2na+1(Qa) = λ1(Qra).

This follows from the removal of the nullspace of Qathrough the reduced anchored rigidity matrix. The de-pendence of Qra on w can be explicitly represented asQra =

∑s∈E ws̄Qrau({s}), where the matrix Qau({s})

represents the unweighted contribution of edge s ∈ E toQra. Subsequently, the maximization of λ2na+1(Qa) can berepresented as a linear matrix inequality (LMI) as shown in(32a).The maximum control effort umax of each agent in V \ Vacan be distributed across the positive weights w throughthe inequalities (32b). It can often be desirable to explicitlyencode the control effort limit devoted to the anchor edgeswa by the leader set V`. This flexibility can help avoid for-mation trajectories that are dominated by the anchor edges.In this direction, one approach is to upper bound the anchoredges weights contribution as

∑j∈Na(i) wij ≤ βumax for

i ∈ V` and β ∈ (0, 1]. To this end, denoting the submatrix ofthe incidence matrix corresponding to anchor edges as Hathen the anchor edge weight bound is equivalent to (32c).For β = 1, the anchor weight bound is inactive and anchoredges are constrained in the same fashion as the other edgesin the network through inequalities (32b).An attraction of the unforced dynamics (10b) is if smallexternal signals are applied through ũi in (10a) then theformation transiently maintains its shape as it convergesto a new translated and scaled formation [6]. A challengewith the external controls signal adopted in (27) is that theeffect of the anchor edges can represent a large externalsignal that dramatically warps the formation shape. Thiscan be mitigated though an edge weight design that limits

Algorithm 3 Convex optimization problem to maximizeλ2na+1(Qa) with respect to the weight vector w given afixed edge set E.

maximizew

δ

s.t.∑s∈E

ws̄Qrau({s}) � δI, (32a)

w ≥ 0, I(V \ Va)T∣∣∣HT ∣∣∣w ≤ umax1, (32b)

I(V`)T∣∣∣HTa ∣∣∣wa ≤ βumax1 (32c)

the maximum control effort due to anchor edges. Inequality(32c) when β is close to 0 has this desired effect. Thedrawback from this approach is that it can induce a slowerconvergence to the desired formation. A demonstration ofthis is shown in Fig. 6b with the convergence of β = 1 muchfaster than that of β = 0.1. The trade-off is that for β = 1then formation shape is largely distorted under scaling asindicated in the snapshots of the formation trajectory in Fig.7a-c.

An alternative, which is a marriage of fast convergence andsmall external signals in the face of a warped formationshape, is to replace the static weight wij on the anchor edgeswith a state-dependent edge weight wij . The weight wijwould ideally be small when the formation is “warped”and large otherwise. A local measure of the alignment ofthe formation is the error ‖ui‖. A proposed state dependentanchor weight wij(‖ui‖) for j ∈ Na(i) where i ∈ V` is

w̄ij(‖ui‖) =I({i})T

∣∣HT ∣∣w − ‖ui‖I({i})T |HTa |wa

wij .

Here, I({i})T∣∣HT ∣∣w and I({i})T ∣∣HTa ∣∣wa corresponds to

the maximum available control effort available to agent iand to the anchor edges, respectively. As the formation be-comes well aligned, ‖ui‖ decreases and wij increases. Simi-larly, as the formation locally starts to warp the external sig-nal supplied through the anchor edges is reduced. It shouldbe noted that as ‖ui‖ ≤ I({i})T

∣∣HT ∣∣w − I({i})T ∣∣HTa ∣∣wathen wij ≥ wij . This positive weight is required for theconvergence properties of Thm 18 to be applicable. Further,the maximum control effort of umax is preserved in this caseas

‖ui + ũi‖ ≤ ‖ui‖+ ‖ũi‖ (from the triangle inequality)

≤∑

j∈N(i)

wij +∑

j∈Na(i)

wij(‖ui‖) (as∥∥∥(r̂Tij f̂⊥ij )r̂⊥ij∥∥∥ ≤ 1)

≤∑

j∈N(i)

wij +∑

j∈Na(i)

I({i})T∣∣HT ∣∣w − ‖ui‖

I({i})T |HTa |wawij

= ‖ui‖+I({i})T

∣∣HT ∣∣w − ‖ui‖I({i})T |HTa |wa

∑j∈Na(i)

wij

≤ ‖ui‖+ I({i})T∣∣∣HT ∣∣∣w − ‖ui‖ ≤ umax.

Fig. 7 compares snapshots of the trajectories for the state-dependent solution with β = 0.1 to the non state-dependentdynamics with β = 1. Reduction to the distortion of the



13

(a) (b)

Figure 6: a) Weighted anchor selection using the functionfr (S) + αft (S) and b) convergence rates for aggressiveanchor weights (β = 1), conservative anchor weights(β = 0.1) and state-dependent conservative anchor weights(β = 0.1, wk̄). Anchor and leader nodes are represented bya filled square and circles, respectively.

(a) (b) (c)

(d) (e) (f)

Figure 7: a-c) Trajectories for β = 1 and d-f) trajectoriesfor β = 0.1 with state-dependent weighting. Anchor andleader nodes are represented by a filled square and circles,respectively

formation shape is noted and in addition the convergencerate in compared as shown in Fig. 6b.

6 CONCLUSION

In this paper, we built on our previous work on bearing-based dynamics for formation shape acquisition. The workexamined connections between the graph topology and theperformance of the dynamics. A bearing rigidity matrixweighted by the edge lengths on the underlying graph wasshown to be critical in dictating stability and convergence ofthe resulting protocol. These connections allowed submod-ular optimization to be leveraged to select favorable edgesbased on guaranteeing bearing rigidity and promoting fastconvergence. Further, the worst case convergence of thedynamics was examined in relation to the edge weightson the graph and a convex optimization framework waspresented to select optimal edge weights.An anchored version of the dynamics was also presentedwhereby a subset of the agents, called leaders, had accessto bearing measurements to fixed landmarks. Anchoredinfinitesimally bearing rigidity was proven to be a necessary

and sufficient condition for the convergence of the dynamicsto a unique formation. The stability and convergence of theanchored dynamics was shown to be dependent on sub-modular set function on the selected leader agents, provingthe greedy leader selection process exhibited close to opti-mal performance. A similar convex optimization problemto the unanchored case was presented to select optimaledge weights to improve the worst-case convergence of theanchored dynamics. With the motivation of preserving theformation shape in transients, a state-dependent weightedversion of the anchored dynamics was proposed. In supportof the dynamic weight selection, anchored bearing simu-lations demonstrated that the formation shape was morereadily preserved compared to fixed edge weights.

The state-dependent weights on the anchor edges intro-duced in the final section of this work can be considered as alocal memory-less feedback approach to preserve the forma-tion scale. A direction of future work is to consider dynamicedge weights based on performance history. The objectivewould be to have a learned formation shape-preservationmeasure which would be used to inform the weight selec-tion. On a more general front, the underlying machinery ofthe bearing dynamics is rigidity theory. A fruitful subjectof future work would be developing a generalization ofthis theory with the goal of showing that efficient topology,leader, and weight selection can be attained based on sub-modular optimization and convex optimization for a largeclass of formation control problems.

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Eric Schoof (S’12) received the B.S. degree inApplied and Computational Mathematical Sci-ences from the University of Washington, andthe Ph.D. degree in Aeronautics and Astronau-tics at the University of Washington in 2017. Heis currently a Research Fellow in the departmentof Electrical and Electronic Engineering at theUniversity of Melbourne.His research interests are networked dynamicsystems and cooperative robotics with applica-tions to multi-agent robotics and human-swarm

interactions.

Airlie Chapman (M’14) received the B.S. de-gree in aeronautical engineering and the M.S.degree in engineering research from the Uni-versity of Sydney, Australia, in 2006 and 2008,respectively, and the M.S. degree in mathemat-ics and the Ph.D. degree in aeronautics andastronautics from the University of Washington,Seattle, in 2013. She is currently a Lecturer inthe department of Mechanical Engineering at theUniversity of Melbourne.Her research interests are networked dynamic

systems and graph theory with applications to multi-agent systems.

Mehran Mesbahi (F’15) received his Ph.D. de-gree from the University of Southern California,Los Angeles, in 1996. He was a member of theGuidance, Navigation, and Analysis Group at theJet Propulsion Laboratory, California Institute ofTechnology, from 1996 to 2000 and an Assis-tant Professor of Aerospace Engineering andMechanics, University of Minnesota from 2000to 2002. He is currently a Professor of Aero-nautics and Astronautics, Adjunct Professor ofMathematics, and Executive Director of the Joint

Center for Aerospace Technology Innovation, at the University of Wash-ington. His research interests are distributed and networked aerospacesystems, systems and control theory, and engineering applications ofoptimization and combinatorics. Mehran Mesbahi is a Fellow of IEEE; hereceived the NSF CAREER Award in 2001, the NASA Space Act Awardin 2004, the UW Distinguished Teaching Award in 2005, the UW Collegeof Engineering Innovator Award in 2008, and the Graduate Instructor ofthe Year Award in 2015.

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