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...ij ] ∈ Rp×p, the directed graph of S, denoted by Γ(S), is the directed graph on p nodes i, i ∈ {1, 2, . . . , p}, such that there is an edge in Γ(S) from node j to node i if and only if sij = 0 (cf. =-=[20]-=-). Again, we assume that there is no self edge (i, i). B. Existing Consensus Algorithms for Double-integrator Dynamics Consider vehicles with double-integrator dynamics given by ṙi = vi, v̇i = ui, i ...

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...r indirectly to all of the vehicles in the team. I. INTRODUCTION Consensus means that multiple vehicles reach an agreement on a common value. Consensus algorithms have a historical perspective in [1]–=-=[4]-=-, to name a few, and have recently been studied extensively in the context of cooperative control of multiple autonomous vehicles (see [5] and references therein). Some results in consensus algorithms...

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...–[17], to name a few. In particular, formation keeping strategies are addressed in [7], [9] for multi-vehicle formation maneuvers under the assumption of a bidirectional ring interaction graph. Refs. =-=[12]-=-, [13] study flocking algorithms that guarantee velocity matching, flock centering, and collision avoidance for a group of vehicles under undirected information exchange. In [11] the problem of decent...

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...ly or indirectly to all of the vehicles in the team. I. INTRODUCTION Consensus means that multiple vehicles reach an agreement on a common value. Consensus algorithms have a historical perspective in =-=[1]-=-–[4], to name a few, and have recently been studied extensively in the context of cooperative control of multiple autonomous vehicles (see [5] and references therein). Some results in consensus algori...

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...ue. Consensus algorithms have a historical perspective in [1]–[4], to name a few, and have recently been studied extensively in the context of cooperative control of multiple autonomous vehicles (see =-=[5]-=- and references therein). Some results in consensus algorithms can be understood in the context of connective stability [6]. Consensus algorithms are primarily studied for vehicles with single-integra...

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... to name a few. In particular, formation keeping strategies are addressed in [7], [9] for multi-vehicle formation maneuvers under the assumption of a bidirectional ring interaction graph. Refs. [12], =-=[13]-=- study flocking algorithms that guarantee velocity matching, flock centering, and collision avoidance for a group of vehicles under undirected information exchange. In [11] the problem of decentralize...

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...y in the context of cooperative control of multiple autonomous vehicles (see [5] and references therein). Some results in consensus algorithms can be understood in the context of connective stability =-=[6]-=-. Consensus algorithms are primarily studied for vehicles with single-integrator kinematics in the literature. For vehicles with double-integrator dynamics, consensus related problems have been studie...

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...1p is a p × 1 column vector of all ones. In the case of undirected graphs, 0 is a simple eigenvalue of L and all of the other eigenvalues are positive if and only if the undirected graph is connected =-=[18]-=-. In the case of directed graphs, 0 is a simple eigenvalue of L and all of the other eigenvalues have positive real parts if and only if the directed graph contains a directed spanning tree [19]. Give...

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...sensus algorithms are primarily studied for vehicles with single-integrator kinematics in the literature. For vehicles with double-integrator dynamics, consensus related problems have been studied in =-=[7]-=-–[17], to name a few. In particular, formation keeping strategies are addressed in [7], [9] for multi-vehicle formation maneuvers under the assumption of a bidirectional ring interaction graph. Refs. ...

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...he literature. For vehicles with double-integrator dynamics, consensus related problems have been studied in [7]–[17], to name a few. In particular, formation keeping strategies are addressed in [7], =-=[9]-=- for multi-vehicle formation maneuvers under the assumption of a bidirectional ring interaction graph. Refs. [12], [13] study flocking algorithms that guarantee velocity matching, flock centering, and...

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...ected [18]. In the case of directed graphs, 0 is a simple eigenvalue of L and all of the other eigenvalues have positive real parts if and only if the directed graph contains a directed spanning tree =-=[19]-=-. Given a matrix S = [sij ] ∈ Rp×p, the directed graph of S, denoted by Γ(S), is the directed graph on p nodes i, i ∈ {1, 2, . . . , p}, such that there is an edge in Γ(S) from node j to node i if and...

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...on avoidance for a group of vehicles under undirected information exchange. In [11] the problem of decentralized stabilization of vehicle formations is studied under directed information exchange. In =-=[14]-=-, second-order consensus algorithms are proposed and analyzed under directed information exchange, where it is shown that both the interaction graph and the coupling strength of relative velocities be...

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...eraction graph. Refs. [12], [13] study flocking algorithms that guarantee velocity matching, flock centering, and collision avoidance for a group of vehicles under undirected information exchange. In =-=[11]-=- the problem of decentralized stabilization of vehicle formations is studied under directed information exchange. In [14], second-order consensus algorithms are proposed and analyzed under directed in...

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...y undirected connected interaction graph. IV. CONSENSUS WITHOUT RELATIVE VELOCITY MEASUREMENT Note that (4) requires measurement of relative velocities between neighboring vehicles. Motivated by [7], =-=[21]-=-, we propose a consensus algorithm without relative velocity measurement based on a passivity approach as ˙̂xi = Γx̂i + n∑ j=1 aij(ri − rj) yi = PΓx̂i + P n∑ j=1 aij(ri − rj) ui = − n∑ j=1 aij(ri − rj...

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...e result in the general case of directed information exchange. Refs. [16], [17] further extend [14] to incorporate a group reference velocity. A second-order consensus algorithm is also considered in =-=[15]-=- under undirected information exchange. The main purpose of the current paper is to extend some existing results in consensus algorithms for double-integrator dynamics in four aspects. First, we propo...

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...that both the interaction graph and the coupling strength of relative velocities between neighboring vehicles affect the convergence result in the general case of directed information exchange. Refs. =-=[16]-=-, [17] further extend [14] to incorporate a group reference velocity. A second-order consensus algorithm is also considered in [15] under undirected information exchange. The main purpose of the curre...

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...any α > 0. It is left to show that the inequality (21) guarantees that all of the eigenvalues of Γ corresponding to µi that satisfies Re(µi) < 0 and Im(µi) = 0 have negative real parts. Motivated by =-=[10]-=-, [14], we use Fig. 1 to show the notations used in the proof. We only need to consider µi that satisfies Re(µi) < 0 and Im(µi) > 0 since any µi that satisfies Re(µi) < 0 and Im(µi) < 0 is a complex c...