Algebraic-Connectivity-Based Multi-USV Distributed Formation Method via Adding a Reverse Edge

Question

1. What is the Laplacian matrix of a directed graph?

2. What is a Directed Acyclic Graph (DAG) and its topological ordering?

3. What is the length of a stem in a DAG?

4. What are the roles in formation containment tracking tasks?

8. What is the topology structure of Graph G in the Main Results section?

Accepted Answer

The Laplacian matrix of a directed graph is defined as L = D - T, where D represents the in-degree of the graph and T is the adjacency matrix. It is a square matrix with diagonal elements representing the in-degree of each node and off-diagonal elements as 0. The Laplacian matrix is used to analyze the connectivity and properties of the graph. It has an eigenvector [1, 1, ..., 1]^T with a non-negative eigenvalue, and the smallest non-zero eigenvalue is called the algebraic connectivity. This matrix plays a crucial role in understanding the network's behavior and achieving consensus among agents in the network.

Accepted Answer

A Directed Acyclic Graph (DAG) is a directed graph without directed cycles. It consists of nodes and edges, where each edge connects one node to another, ensuring no node can loop back to itself. DAGs always have a topological ordering, a sequence of nodes where the starting node of each edge comes before its ending node. This ordering is unique for DAGs but not for general graphs. Examples of DAGs include singly linked chains, grids, trees, and mushrooms. A local DAG structure called stem, with a singly linked network structure, is used for local quantitative analysis and reduces the workload of analyzing large networks. The topological ordering of a DAG ensures that for each edge, the start node always precedes the end node. For example, one topological ordering of a graph could be (A, C, E, B, D).

Accepted Answer

The length of a stem in a Directed Acyclic Graph (DAG) is defined as the number of nodes between the start node and the end node, excluding the start and end nodes themselves. It is calculated by subtracting the start node from the end node and adding 1. For example, in a stem structure with start node h = 5 and end node = 8, the length of the stem would be 8 - 5 + 1 = 4. This length represents the number of nodes that are part of the stem structure, connecting the start and end nodes in the topological ordering of the DAG. The length of a stem is an important characteristic in analyzing the structure and properties of a DAG, as it provides insights into the connectivity and relationships between nodes within the graph.

Accepted Answer

In formation containment tracking tasks, there are three roles: tracking leader, formation leader, and followers. The tracking leader determines the macroscopic motion trajectory of the entire system. The formation leader tracks the tracking leader according to the formation set by the motion control protocol. Followers converge into the convex hull formed by the formation leader based on the motion control protocol. These roles are essential for coordinating the multi-agent system's movement and achieving the desired formation and tracking objectives.

Accepted Answer

The topology relationship between the formation leader and followers in multi-USV systems satisfies the condition that for any follower, there is at least one formation leader that has a directed path to it. This relationship is represented by the Laplacian matrix, which can be divided into different components based on the order of tracking leader, formation leaders, and followers. The Laplacian matrix helps in understanding the connectivity and dynamics of the multi-USV system. Additionally, the assumption that the motion angle deviation of each USV can be ignored due to the spatial distance between them allows for the use of an approximate first-order linear point-mass motion model to describe each USV. The dynamic descriptions of the formation leaders and followers are given by equations that represent their position and control input. The tracking leader's position control input is bounded, and the followers converge to the convex hull formed by the formation leader. The formation tracking position error and relative error between the formation leader and its neighbors are defined to monitor the tracking performance. The formation containment tracking control protocol is used to ensure the desired formation tracking and containment of the USVs in the system.

Accepted Answer

In general, no specific topology is required between the tracking leader and followers. The tracking leader is not connected to the followers. The topology between the tracking leader and the formation leader is ignored, focusing only on the topology between the formation leader and the followers. Assuming a graph with leaders and followers, the Laplacian matrix is used to determine the convergence of followers to the convex hull formed by the formation leaders. The specific convergence position is given by two lemmas.

Accepted Answer

The goal is to propose a tree-like topology structure based on a single-chain DAG, where followers are interconnected with formation leaders. This topology includes M formation leaders and (N - M) followers, aiming to describe communication connections while satisfying specific requirements for algebraic connectivity and convergence rate. The design considers adding reverse edges to decrease algebraic connectivity and maximize system convergence. The topology structure is crucial in achieving efficient communication and control within the formation.

Accepted Answer

The topology structure of Graph G is constructed by n 'minimal units'; the ellipsis in Figure 3 and Figure 4 means there are edges connecting other followers, and d can be any constant less than n and larger than or equal to 2. For example, when n = 4, d = 3, the topology structure is as in Figure 5. The entire graph G is a DAG, and the numbering of the nodes connected to each other follows the topological order. The Laplacian matrix of graph G is a lower triangular matrix. Due to the fact that the addition of a reverse edge is limited to followers only (the formation leader cannot receive signals from followers), according to Lemma 1, in order to study the effect of adding reverse edges on the algebraic connectivity of stem structures, it is necessary to pay attention to the in-degree of the nodes in the stem structure, i.e., nodes numbered from n + 1 to 2n must satisfy the requirements of Lemma 1 for in-degree. By observation, the in-degree of follower node n + 1 is d, while the in-degree of all other follower nodes is d + 1. If we choose follower node n + 1 as the head node and another follower node as the tail node, the in-degree between the follower nodes does not satisfy the condition of Equation (3), i.e., the in-degree of the tail node must not be greater than the in-degree of the head node and all intermediate nodes. Therefore, a virtual node with an in-degree of 1, denoted as 2n + 1, was added to the original DAG G. This node can be directly connected to any follower node except for follower node n + 1 (since node n + 1 is the first among all follower nodes in the topological order, it cannot serve as the tail node). When a follower node is needed as the tail node, a virtual node with an in-degree of 1 can be added after it, and it can be used as the tail node for the reverse edge analysis. The specific characteristics are given by Lemma 4. The topology between followers is a stem, as shown in Figure 6. Then, the algebraic connectivity l min = 1 for the new graph G.

Accepted Answer

The impact of adding a reverse edge in a Directed Acyclic Graph (DAG) is given by Proposition 1. It considers a graph G with 2n+1 nodes that has a topological order and whose Laplacian matrix is a lower triangular matrix. The graph contains a stem consisting of n+1 nodes, each numbered in topological order as n+1, n+2, ..., 2n+1. The node 2n+1 is a virtual node, and the in-degrees of these nodes have specific characteristics. The impact of adding a reverse edge on the algebraic connectivity of the graph is analyzed using Lemma 1, and the eigenvalues corresponding to the nodes within the scope of the reverse edge will change. The formation leader can be ignored in the subsequent analysis of algebraic connectivity. The number of possible reverse edges that can be added in a stem structure with t nodes is EQUATION. When the tail node of a reverse edge is located at the highest indexed follower node, the number of possible reverse edges that can be added is t-1. The analysis focuses on the case where the tail node of the reverse edge is the follower node with the largest topological order, connecting the virtual node 2n+1 to the follower node with the largest topological order 2n. This analysis helps in identifying the reverse edge that can reduce the algebraic connectivity and minimize the loss of algebraic connectivity in a stem structure.

Accepted Answer

The designed topology for USV formation consists of 5 formation leaders (USV 1 to USV 5) and 5 followers (USV 6 to USV 10), with USV 11 being a fictitious follower. The tracking leader is USV 0, and the connections between the tracking leader and formation leaders are represented by dashed lines. The formation is centered around the tracking leader, forming a regular pentagon shape. The formation vector h L (t) is defined for the formation leaders, with h i (t) = cos(t + 2(i-1)p 5 ) - sin(t + 2(i-1)p.

Accepted Answer

Adding reverse edges to a USV formation helps in achieving more evenly distributed convergence positions of followers and maintains a fixed angular difference with the formation leader. This improves obstacle avoidance ability in complex sea areas. Without reverse edges, followers may be too close to the formation leader, compromising safety and stability. The addition of reverse edges results in a rotational symmetry in the relationship between the convergence positions of followers and the position of the formation leader. The algebraic connectivity of the follower network decreases with the addition of reverse edges, affecting the convergence rate. The impact of reverse edges on algebraic connectivity is related to the range covered by the reverse edge and the in-degree of the nodes within that range, and is independent of the location of the reverse edge.

Algebraic-Connectivity-Based Multi-USV Distributed Formation Method via Adding a Reverse Edge

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Most frequently asked questions

1. What is the Laplacian matrix of a directed graph?

2. What is a Directed Acyclic Graph (DAG) and its topological ordering?

3. What is the length of a stem in a DAG?

4. What are the roles in formation containment tracking tasks?

5. What is the relationship between formation leader and followers in multi-USV systems?

6. What topology is required between tracking leader and followers?

7. What is the goal of the topology structure in formation control?

8. What is the topology structure of Graph G in the Main Results section?

9. What is the impact of adding a reverse edge in a DAG?

10. What is the designed topology for USV formation?

11. How does adding reverse edges affect the convergence positions of followers in a USV formation?

Citations

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References

On Certain Integrals of Lipschitz-Hankel Type Involving Products of Bessel Functions

Consensus strategies for cooperative control of vehicle formations

Containment Control in Mobile Networks

Theory and Experiment on Formation-Containment Control of Multiple Multirotor Unmanned Aerial Vehicle Systems

An Error Transformation Approach for Connectivity-Preserving and Collision-Avoiding Formation Tracking of Networked Uncertain Underactuated Surface Vessels

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