Preliminaries
This section introduces the notation, an overview of port-Hamiltonian systems and graph theory, the model of the agents to be used, and the problem statement. These concepts will be used throughout the remainder of the paper.
Port-Hamiltonian Systems
The port-Hamiltonian dynamics is given by the port-Hamiltonian dynamical model defined in the paper.
Problem Statement and Solution Approach
Two objectives are addressed: distance control and collision avoidance between agents. A physically inspired strategy emulates the interconnection among the agents using sets of virtual springs and dampers. A collection of single sets is defined as an Elementary Coupling Block (ECB), resembling an interconnected physical structure.
The mesh of ECBs enables the achievement of both objectives. Since the forces of the springs vanish when they are de-energized, selecting their lengths allows the desired distances between agents to be established. As the spring forces tend to infinity when the distances approach zero, they also provide a collision-avoidance strategy. When the distance between agents exceeds the desired value, the springs apply an attractive force. The dampers attenuate oscillations in the mesh.
A virtual coupling should emulate what a real physical coupling would produce if it were present. To implement these virtual couplings, the efforts that physical couplings would generate are translated into force references for the agent actuators—the wheel torques generated by the motors—which are responsible for producing the actual efforts.
Interconnected Physical Structure
This section presents the port-Hamiltonian and bond-graph energy-based models for both a single ECB and the multi-agent system. These models underscore the energy storage, the interconnection structure, and the dynamic behavior of the ECB spring.
State-Feedback Formation Control Law
The CbI approach provides a static state-feedback solution to the original formation-control problem, with the control law the state-feedback formation-control expression defined in the paper.
The corresponding control block diagram is presented in the paper.
Discussions
The advantages of the control strategies include the following:
- In Leader Agent Control, both formation control and collision prevention are carried out through a single controller, which also has a physical interpretation.
- Only Position Control ensures that the orientation of the topology is always as desired, with no risks of collisions between any of the agents.
- Position Control lacks a physical interpretation, and the control is more focused on each individual agent rather than the formation itself.
Formation controls can be classified based on the controlled variable: position, displacement, or distance. Position Control is a position-based formation control method, while Leader Agent Control is a hybrid between distance and displacement control. Although this strategy actively controls distance, calculating the control signals involves measuring global displacements between agents for computation of the Formation Matrix. Furthermore, since it is necessary to know the positions of the leader agent and its neighbor, it could also fall into the classification of position-based control.
Both formation schemes can also be differentiated based on their interpretation. Leader Agent Control is based on basic mechanical elements—springs and dampers—and treats the multi-agent system as a semi-rigid body composed of interconnected subsystems through the ECBs. Therefore, its results have a physical interpretation.
Position Control introduces a virtual reference structure, where the control-law expressions emerge from the geometry of the structure. Hence, its interpretation is purely mathematical.
In a physical scenario, the proposed strategy must be implemented in a centralized scheme because computing the Formation Matrix requires the global positions of the agents. These positions can be obtained through the fusion of camera measurements, IMU data, and odometry.
Frequently Asked Questions
What are the two main objectives of the proposed strategy? The objectives are distance control and collision avoidance between agents.
What is an Elementary Coupling Block? An ECB is a set of virtual springs and dampers used to emulate physical couplings between agents.
How do the springs contribute to collision avoidance? Their forces tend to infinity as the distance between agents approaches zero.
Why must the proposed strategy use centralized implementation in a physical scenario? Computing the Formation Matrix requires the global positions of the agents.