A Novel Adaptive Formation Control Strategy for Teams of Unmanned Vehicles Under Complete Dynamic Uncertainty

1. Problem Statement

The paper addresses the formation control problem for a team of unmanned systems (Multi-Agent Systems or MAS) operating in completely uncertain dynamic environments. Key challenges include:

• Complete Uncertainty: The dynamics of the agents are modeled as general nonlinear mechanical systems where the inertia matrix ($M$), Coriolis/centripetal forces ($C$), damping/friction ($D$), and gravitational forces ($g$) are entirely unknown and time-varying.
• Complex Leader Dynamics: Unlike traditional studies that assume linear, time-invariant leaders with no inputs, this work considers a virtual leader driven by bounded, time-varying external inputs. This allows for more complex and realistic reference trajectories.
• Distributed Constraints: The system operates under a distributed framework where agents only have access to local information and communication with neighbors, without a central controller.
• Goal: Design a control strategy that enables a group of agents to track a virtual leader while maintaining a specific geometric formation, despite the lack of knowledge regarding system dynamics and the presence of external disturbances.

2. Methodology

The authors propose a two-layer hierarchical control architecture based on Radial Basis Function Neural Networks (RBF NNs) and Cooperative Deterministic Learning.

A. System Modeling

• Agent Dynamics: Modeled as nonlinear mechanical systems with unknown parameters.
• Virtual Leader: Defined by a state-space model with bounded, time-varying inputs ($r$), generating complex reference trajectories.
• Communication Topology: A fixed directed graph where the leader has a path to at least one follower, and the follower subgraph is undirected.

B. Control Architecture

The proposed scheme consists of two distinct layers:

1. Upper Layer: Cooperative Nonlinear Estimator

   • Since not all agents can directly access the leader's state, a cooperative discontinuous nonlinear estimation protocol is designed.
   • This layer estimates the leader's position and velocity ($\hat{p}_i, \hat{v}_i$) using local neighbor information.
   • Theoretical analysis proves that the estimation error converges exponentially to zero, ensuring all agents have a reliable reference trajectory ($\hat{p}_{r,i} = \hat{p}_i + p^*_i$) to follow.

2. Lower Layer: Cooperative Deterministic Learning Control

   • Backstepping Design: A backstepping technique is used to define tracking errors and virtual control inputs.
   • RBF NN Approximation: The unknown nonlinear dynamics (including the inertia matrix and external forces) are approximated using RBF Neural Networks. The unknown function $G(x_i)$ is represented as $W^{*T}S(x_i) + \epsilon$.
   • Cooperative Weight Update: A novel cooperative learning law is introduced. Agents update their neural network weights not only based on their own tracking errors but also by sharing estimated weights with neighbors. This facilitates knowledge consensus and accelerates learning across the network.
   • Control Law: The final control input $\tau_i$ combines the NN approximation of uncertainties, a feedback term for tracking errors, and the cooperative weight update term.

3. Key Contributions

The paper introduces three primary innovations compared to existing literature:

1. Generalized System Uncertainty: The approach handles systems with completely unknown inertia matrices and nonlinear dynamics, removing the restrictive assumption of known or partially known inertial properties found in previous works.
2. Cooperative Online Knowledge Sharing: The control protocol enables cooperative online learning. By utilizing estimated NN weights from neighboring agents, the system enhances knowledge consensus, allowing the team to learn the uncertain dynamics more efficiently than isolated agents.
3. Complex Reference Trajectories: The framework supports a generalized leader model with time-varying external inputs, enabling the tracking of diverse and expressive trajectories rather than simple linear paths.

4. Results and Validation

The effectiveness of the proposed strategy was validated through rigorous theoretical analysis and numerical simulations in MATLAB.

• Theoretical Guarantees:
  • Stability: Using Lyapunov stability theory, the authors proved that all closed-loop signals are Uniformly Ultimately Bounded (UUB).
  • Convergence: The tracking errors ($z_1, z_2$) and weight estimation errors are proven to converge exponentially to a small neighborhood of zero.
• Numerical Simulation:
  • Setup: A simulation of four agents with completely uncertain dynamics (unknown $M, C, D, g$) tracking a virtual leader with a sinusoidal time-varying input.
  • Estimation Performance: The cooperative estimator successfully tracked the leader's state with exponential convergence of errors to zero.
  • Formation Tracking: The agents successfully maintained a predefined geometric formation (a specific 3D configuration) while tracking the leader's complex trajectory.
  • Error Analysis: Position tracking errors in all three dimensions converged rapidly to a negligible neighborhood around zero, demonstrating high accuracy despite total system uncertainty.

5. Significance and Applications

This work bridges a critical gap in multi-agent control by solving the integrated challenge of cooperative adaptive formation control under complete uncertainty.

• Robustness: The method is highly robust to environmental disturbances and internal model uncertainties, making it suitable for unstructured environments.
• Scalability: The distributed nature of the algorithm avoids the single point of failure and communication bottlenecks associated with centralized control.
• Practical Applications: The proposed strategy is particularly relevant for:
  • Persistent Aerial Surveillance: Coordinating unmanned airships or drones for long-duration missions.
  • Search and Rescue: Navigating cluttered terrains (e.g., forests) where GPS is unreliable and dynamics are unpredictable.
  • Atmospheric Data Collection: Maintaining formation while collecting data in varying wind conditions.
  • Cooperative Payload Transport: Moving heavy loads where individual agent dynamics are difficult to model precisely.

In conclusion, the paper presents a mathematically rigorous and practically viable solution for deploying intelligent, adaptive teams of unmanned vehicles in highly uncertain and dynamic real-world scenarios.