Bearing Rigidity and Almost Global
Bearing-Only Formation Stabilization
Shiyu Zhao and Daniel Zelazo
S. Zhao and D. Zelazo are with the Faculty of Aerospace Engineering, Israel Institute of Technology, Haifa, Israel.
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Abstract
A fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter-neighbor bearings.
While many previous works focused on the bearing rigidity of two-dimensional frameworks, a first contribution of this paper is to extend these results to arbitrary dimensions.
It is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid.
In this paper, the proposed bearing rigidity theory is further applied to the bearing-only formation stabilization problem where the target formation is defined by inter-neighbor bearings and the feedback control uses only bearing measurements.
Nonlinear distributed bearing-only formation control laws are proposed for the cases with and without a global orientation.
It is proved that the control laws can almost globally stabilize infinitesimally bearing rigid formations.
Numerical simulations are provided to support the analysis.
Bearing rigidity, formation control, attitude synchronization, almost global input-to-state stability
I Introduction
Multi-agent formation control has been studied extensively in recent years with
distance-constrained formation control taking a prominent role [1, 2, 3, 4, 5, 6, 7]. In this setting it is assumed that the target formation is specified by inter-agent distances, and each agent is able to measure relative positions of their neighbors.
Bearing-constrained formation control has also attracted much attention recently [8, 9, 10, 11, 12, 13, 14]. Instead of distances, the formation is specified by inter-agent bearings, and each agent can measure the relative positions or bearings of their neighbors.
Bearing measurements are often cheaper and more accessible than position measurements, spurring interest in cooperative control using bearing-only measurements [15, 9, 10, 11, 16, 17, 12, 13, 18, 14].
This paper studies a bearing-only formation control problem where the target formation is bearing-constrained and each agent has access to the bearing-only measurements of their neighbors.
It is noted that while bearing measurements can be used to estimate relative distances or positions [16, 18, 19], such schemes may significantly increase the complexity of the sensing system in terms of both hardware and software. This then motivates our study focusing on a pure bearing-only control scheme without the need for estimation of additional quantities (e.g., relative position).
Although bearing-only formation control has lately attracted much interest, many problems on this topic remain unsolved.
The studies in [15, 8, 11] considered bearing-constrained formation control in two-dimensional spaces, but required access to position or other measurements in the proposed control laws.
The results reported in [16, 18] only require bearing measurements, but they are used to estimate additional relative-state information such as distance ratios or scale-free coordinates.
The works in [9, 10, 12, 13] studied formation control with bearing measurements directly applied in the control. However, these results were applied to special formations, such as cyclic formations, and may not be extendable to arbitrary formation shapes.
A very recent work reported in [14] solved bearing-only formation control for arbitrary underlying graphs, but only for formations in the plane.
Bearing-only formation control in arbitrary dimensions with general underlying graphs still remains an open problem.
A central tool in the study of bearing-only formation control is bearing rigidity theory^{1}^{1}1Also referred to as parallel rigidity in some literature..
Existing works on bearing rigidity mainly focused on frameworks in two-dimensional ambient spaces [20, 8, 10, 19]. The first contribution of our work, therefore, is an extension of the existing bearing rigidity theory to arbitrary dimensions. We also explore connections between bearing rigidity and distance rigidity, and in particular show that a framework in R2 is infinitesimally bearing rigid if and only if it is also infinitesimally distance rigid.
Based on the proposed bearing rigidity theory, we investigate distributed bearing-only formation control in arbitrary dimensions in the presence of a global reference frame.
We propose a distributed bearing-only formation control law and show by a Lyapunov approach that the control law can almost globally stabilize infinitesimally bearing rigid formations. We also provide a sufficient condition ensuring collision avoidance between any pair of agents under the action of the control.
In the third part of the paper, we investigate bearing-only formation control in the three dimensional space without a global reference frame known to the agents.
Each agent can only measure the bearings and relative orientations of their neighbors in their local reference frames.
We propose a distributed control law to control both the position and the orientation of each agent.
It is shown that the orientation will synchronize and the target formation is almost globally stable.
This paper is organized as follows.
Section II presents the bearing rigidity theory that is applicable to arbitrary dimensions.
Section III studies bearing-only formation control in arbitrary dimensions in the presence of a global reference frame, and Section IV studies the case without a global reference frame.
Simulation results are presented in Section V.
Conclusions and future works are given in Section VI.
Notations
Given Ai∈Rp×q for i=1,…,n, denote diag(Ai)≜blkdiag{A1,…,An}∈Rnp×nq.
Let Null(⋅), Range(⋅), and rank(⋅) be the null space, range space, and rank of a matrix, respectively.
Denote Id∈Rd×d as the identity matrix, and 1≜[1,…,1]T.
Let ∥⋅∥ be the Euclidian norm of a vector or the spectral norm of a matrix, and ⊗ the Kronecker product.
For any x=[x1,x2,x3]T∈R3, the associated skew-symmetric matrix is denoted as
[x]×≜⎡⎢⎣0−x3x2x30−x1−x2x10⎤⎥⎦.
(1)
An undirected graph, denoted as G=(V,E), consists of a vertex set V={1,…,n} and an edge set E⊆V×V with m=|E|.
The set of neighbors of vertex i is denoted as Ni≜{j∈V:(i,j)∈E}.
An orientation of an undirected graph is the assignment of a direction to each edge.
An oriented graph is an undirected graph together with an orientation.
The incidence matrix H∈Rm×n of an oriented graph is the {0,±1}-matrix with rows indexed by
edges and columns by vertices: [H]ki=1 if vertex i is the head of edge k, [H]ki=−1 if vertex i is the tail of edge k, and [H]ki=0 otherwise. For a connected graph, one always has H1=0 and rank(H)=n−1 [21].
Ii Bearing Rigidity in Arbitrary Dimensions
In this section, we propose a bearing rigidity theory that is applicable to arbitrary dimensions.
The basic problem that the bearing rigidity theory studies is whether a framework can be uniquely determined up to a translation and a scaling factor given the bearings between each pair of neighbors in the framework.
This problem can be equivalently stated as whether two frameworks with the same inter-neighbor bearings have the same shape.
We first define some necessary notations.
Given a finite collection of n points {pi}ni=1 in Rd (n≥2, d≥2), a configuration is denoted as p=[pT1,…,pTn]T∈Rdn.
A framework in Rd, denoted as G(p), is a combination of an undirected graph G=(V,E) and a configuration p, where vertex i∈V in the graph is mapped to the point pi in the configuration.
For a framework G(p), define
eij≜pj−pi,gij≜eij/∥eij∥,∀(i,j)∈E.
(2)
Note the unit vector gij represents the relative bearing of pj to pi.
This unit-vector representation is different from the conventional ways where a bearing is described as one angle (azimuth) in R2, or two angles (azimuth and altitude) in R3.
Note also that eij=−eji and gij=−gji.
We now introduce an important orthogonal projection operator that will be widely used in this paper.
For any nonzero vector x∈Rd (d≥2), define the operator P:Rd→Rd×d as
P(x)≜Id−x∥x∥xT∥x∥.
For notational simplicity, denote Px=P(x).
Note Px is an orthogonal projection matrix which geometrically projects any vector onto the orthogonal compliment of x.
It can be verified that PTx=Px, P2x=Px, and Px is positive semi-definite.
Moreover, Null(Px)=span{x} and the eigenvalues of Px are {0,1(d−1)}.
In the bearing rigidity theory, it is often required to evaluate whether two given bearings are parallel to each other.
The orthogonal projection operator provides a convenient way to describe parallel vectors in arbitrary dimensions.
Lemma 1.
Two nonzero vectors x,y∈Rd are parallel if and only if Pxy=0 (or equivalently Pyx=0).
Proof.
The result follows from Null(Px)=span{x}.
∎
Remark 1.
Most existing works use the notion of normal vectors to describe parallel vectors in R2 [20, 8, 10].
Specifically, given a nonzero vector x∈R2, denote x⊥∈R2 as a nonzero normal vector satisfying xTx⊥=0. Then any vector y∈R2 is parallel to x if and only if (x⊥)Ty=0.
This approach is applicable to two dimensional cases but difficult to extend to arbitrary dimensions.
Moreover, it is straightforward to prove that in R2 the use of the orthogonal projection operator is equivalent to the use of normal vectors based on the fact that Px=x⊥(x⊥)T/∥x⊥∥2 for x∈R2.
We are now ready to define the fundamental concepts in bearing rigidity.
These concepts are defined analogously to those in the distance rigidity theory.
Definition 1 (Bearing Equivalency).
Frameworks G(p) and G(p′) are bearing equivalent if P(pi−pj)(p′i−p′j)=0 for all (i,j)∈E.
Definition 2 (Bearing Congruency).
Frameworks G(p) and G(p′) are bearing congruent if P(pi−pj)(p′i−p′j)=0 for all i,j∈V.
Fig. 1: The two frameworks are bearing equivalent but not bearing congruent.
The bearings between (p1,p3) or (p2,p4) of the frameworks are different.
By definition, bearing congruency implies bearing equivalency.
The converse, however, is not true, as illustrated in Figure 1.
Definition 3 (Bearing Rigidity).
A framework G(p) is bearing rigid if there exists a constant ϵ>0 such that any framework G(p′) that is bearing equivalent to G(p) and satisfies ∥p′−p∥<ϵ is also bearing congruent to G(p).
Definition 4 (Global Bearing Rigidity).
A framework G(p) is globally bearing rigid if an arbitrary framework that is bearing equivalent to G(p) is also bearing congruent to G(p).
By definition, global bearing rigidity implies bearing rigidity.
As will be shown later, the converse is also true.
We next define infinitesimal bearing rigidity, which is one of the most important concepts in the bearing rigidity theory.
Consider an arbitrary orientation of the graph G and denote
ek≜pj−pi,gk≜ek/∥ek∥,∀k∈{1,…,m}
(3)
as the edge vector and the bearing for the kth directed edge. Denote e=[eT1,…,eTm]T and g=[gT1,…,gTm]T.
Note e satisfies e=¯Hp where ¯H=H⊗Id. Define the bearing functionFB:Rdn→Rdm as
FB(p)≜[gT1⋯gTm]T∈Rdm.
The bearing function describes all the bearings in the framework.
The bearing rigidity matrix is defined as the Jacobian of the bearing function,
R(p)≜∂FB(p)∂p∈Rdm×dn.
(4)
Let δp be a variation of the configuration p.
If R(p)δp=0, then δp is called an infinitesimal bearing motion of G(p). This is analogous to infinitesimal motions in distance-based rigidity.
Distance preserving motions of a framework include rigid-body translations and rotations, whereas bearing preserving motions of a framework include translations and scalings.
An infinitesimal bearing motion is called trivial if it corresponds to a translation and a scaling of the entire framework.
Definition 5 (Infinitesimal Bearing Rigidity).
A framework is infinitesimally bearing rigid if all the infinitesimal bearing motions are trivial.
Up to this point, we have introduced all the fundamental concepts in the bearing rigidity theory.
We next explore the properties of these concepts.
We first derive a useful expression for the bearing rigidity matrix.
Lemma 2.
The bearing rigidity matrix in (4) can be expressed as
R(p)=diag(Pgk∥ek∥)¯H.
(5)
Proof.
It follows from gk=ek/∥ek∥,∀k∈{1,…,m} that
∂gk∂ek=1∥ek∥(Id−ek∥ek∥eTk∥ek∥)=1∥ek∥Pgk.
As a result, ∂FB(p)/∂e=diag(Pgk/∥ek∥) and consequently
R(p)=∂FB(p)∂p=∂FB(p)∂e∂e∂p=diag(Pgk∥ek∥)¯H.
∎
The expression (5) can be used to characterize the null space and the rank of the bearing rigidity matrix.
Lemma 3.
A framework G(p) in Rd always satisfies span{1⊗Id,p}⊆Null(R(p)) and rank(R(p))≤dn−d−1.
Proof.
First, it is clear that span{1⊗Id}⊆Null(¯H)⊆Null(R(p)).
Second, since Pekek=0, we have R(p)p=diag(Pek/∥ek∥)¯Hp=diag(Pek/∥ek∥)e=0 and hence p⊆Null(R(p)).
The inequality rank(R(p))≤dn−d−1 follows immediately from span{1⊗Id,p}⊆Null(R(p)).
∎
For any undirected graph G=(V,E), denote Gκ as the complete graph over the same vertex set V, and Rκ(p) as the bearing rigidity matrix of the framework Gκ(p).
The next result gives the necessary and sufficient conditions for bearing equivalency and bearing congruency.
Theorem 1.
Two frameworks G(p) and G(p′) are bearing equivalent if and only if R(p)p′=0, and bearing congruent if and only if Rκ(p)p′=0.
Proof.
Since R(p)p′=diag(Id/∥ek∥)diag(Pgk)¯Hp′=diag(Id/∥ek∥)diag(Pgk)e′, we have
R(p)p′=0⇔Pgke′k=0,∀k∈{1,…,m}.
Therefore, by Definition 1, the two frameworks are bearing equivalent if and only if R(p)p′=0.
By Definition 2, it can be analogously shown that frameworks are bearing equivalent if and only if Rκ(p)p′=0.
∎
Since any infinitesimal motion δp is in Null(R(p)), it is implied from Theorem 1 that R(p)(p+δp)=0 and hence G(p+δp) is bearing equivalent to G(p).
We next give a useful lemma and then prove the necessary and sufficient condition for global bearing rigidity.
Lemma 4.
A framework G(p) in Rd always satisfies span{1⊗Id,p}⊆Null(Rκ(p))⊆Null(R(p)) and dn−d−1≥rank(Rκ(p))≥rank(R(p)).
Proof.
The results that span{1⊗Id,p}⊆Null(Rκ(p)) and dn−d−1≥rank(Rκ(p)) can be proved similarly as Lemma 3.
For any δp∈Null(Rκ(p)), we have Rκ(p)δp=0⇒Rκ(p)(p+δp)=0.
As a result, G(p+δp) is bearing congruent to G(p) by Theorem 1.
Since bearing congruency implies bearing equivalency, we further know R(p)(p+δp)=0 and hence R(p)δp=0.
Therefore, any δp in Null(Rκ(p)) is also in Null(R(p)) and thus Null(Rκ(p))⊆Null(R(p)).
Since R(p) and Rκ(p) have the same column number, it follows immediately that rank(Rκ(p))≥rank(R(p)).
∎
Theorem 2 (Condition for Global Bearing Rigidity).
A framework G(p) in Rd is globally bearing rigid if and only if Null(Rκ(p))=Null(R(p)) or equivalently rank(Rκ(p))=rank(R(p)).
Proof.
(Necessity)
Suppose the framework G(p) is globally bearing rigid.
We next show that Null(R(p))⊆Null(Rκ(p)).
For any δp∈Null(R(p)), we have R(p)δp=0⇒R(p)(p+δp)=0.
As a result, G(p+δp) is bearing equivalent to G(p) according to Theorem 1.
Since G(p) is globally bearing rigid, it follows that G(p+δp) is also bearing congruent to G(p), which means Rκ(p)(p+δp)=0⇒Rκ(p)δp=0.
Therefore, any δp in Null(R(p)) is in Null(Rκ(p)) and thus Null(R(p))⊆Null(Rκ(p)).
Since Null(Rκ(p))⊆Null(R(p)) as shown in Lemma 4, we have Null(R(p))=Null(Rκ(p)).
(Sufficiency) Suppose Null(R(p))=Null(Rκ(p)).
Any framework G(p′) that is bearing equivalent to G(p) satisfies R(p)p′=0.
It then follows from Null(R(p))=Null(Rκ(p)) that Rκ(p)p′=0, which means G(p′) is also bearing congruent to G(p).
As a result, G(p) is globally bearing rigid.
Because R(p) and Rκ(p) have the same column number, it follows immediately that Null(Rκ(p))=Null(R(p)) if and only if rank(Rκ(p))=rank(R(p)).
∎
The following result shows that bearing rigidity and global bearing rigidity are equivalent notions.
Theorem 3 (Condition for Bearing Rigidity).
A framework G(p) in Rd is bearing rigid if and only if it is globally bearing rigid.
Proof.
By definition, global bearing rigidity implies bearing rigidity.
We next prove the converse is also true.
Suppose the framework G(p) is bearing rigid.
By the definition of bearing rigidity and Theorem 1, any framework satisfying R(p)p′=0 and ∥p′−p∥≤ϵ also satisfies Rκ(p)p′=0, i.e.,
R(p)(p+δp)=0⇒Rκ(p)(p+δp)=0,∀δp,∥δp∥≤ϵ,
where δp=p′−p.
It then follows from R(p)p=0 and Rκ(p)p=0 that
R(p)δp=0⇒Rκ(p)δp=0 for all ∥δp∥≤ϵ.
This means Null(R(p))⊆Null(Rκ(p)) in spite of the constraint of ∥δp∥.
Since Null(Rκ(p))⊆Null(R(p)) as shown in Lemma 4, we further have Null(R(p))=Null(Rκ(p)) and consequently G(p) is globally bearing rigid.
∎
We next give the necessary and sufficient condition for infinitesimal bearing rigidity.
Theorem 4 (Condition for Infinitesimal Bearing Rigidity).
For a framework G(p) in Rd, the following statements are equivalent:
G(p) is infinitesimally bearing rigid;
rank(R(p))=dn−d−1;
Null(R(p))=span{1⊗Id,p}=span{1⊗Id,p−1⊗¯p}, where ¯p=(1⊗Id)Tp/n is the centroid of {pi}i∈V.
Proof.
Lemma 3 shows span{1⊗Id,p}⊆Null(R(p)).
Observe 1⊗Id and p correspond to a rigid-body translation and a scaling of the framework, respectively.
The stated condition directly follows from Definition 5. Note also that {1⊗Id,p−1⊗¯p} is an orthogonal basis for span{1⊗Id,p}.
∎
The special cases of R2 and R3 are of particular interest.
A framework G(p) is infinitesimally bearing rigid in R2 if and only if rank(R(p))=2n−3, and in R3 if and only if rank(R(p))=3n−4.
Note Theorem 4 does not require n≥d.
The following result characterizes the relationship between infinitesimal bearing rigidity and global bearing rigidity.
Theorem 5.
Infinitesimal bearing rigidity implies global bearing rigidity.
Proof.
Infinitesimal bearing rigidity implies Null(R(p))=span{1⊗Id,p}.
Since span{1⊗Id,p}⊆Null(Rκ(p))⊆Null(R(p)) as shown in Lemma 4, it immediately follows from Null(R(p))=span{1⊗Id,p} that Null(Rκ(p))=Null(R(p)), which means G(p) is globally bearing rigid according to Theorem 2.
∎
The converse of Theorem 5 is not true, i.e., global bearing rigidity does not imply infinitesimal bearing rigidity.
For example, the collinear framework as shown in Figure 2(a) is globally bearing rigid but not infinitesimally bearing rigid.
We have at this point discussed three notions of bearing rigidity: (i) bearing rigidity, (ii) global bearing rigidity, and (iii) infinitesimal bearing rigidity.
According to Theorem 3 and Theorem 5, the relationship between the three kinds of bearing rigidity can be summarized as below: