Learning-Based Geometric Leader-Follower Control for Cooperative Rigid-Payload Transport with Aerial Manipulators

1. Problem Statement

The paper addresses the challenge of cooperative transport of a rigid payload by multiple aerial manipulators (UAVs with robotic arms) under rigid grasp constraints.

• Core Difficulty: Unlike cable-suspended loads, rigid grasps create a tightly coupled system where the dynamics of the agents and the payload are interdependent via holonomic constraints. This introduces internal force redundancy (multiple force distributions can produce the same payload motion) and complex differential-algebraic equation (DAE) dynamics.
• Uncertainty: Real-world cooperative manipulation suffers from model mismatches (payload mass/inertia variations), unmodeled aerodynamic disturbances, and actuator dynamics. Traditional geometric controllers often assume perfect models, leading to degraded performance or grasp failure under uncertainty.
• Goal: Develop a control framework that ensures uniform ultimate boundedness (UUB) of tracking errors with high probability, explicitly accounting for learning uncertainty and the hierarchical nature of the leader-follower architecture.

2. Methodology

The proposed solution integrates Geometric Control on SE(3) with Gaussian Process (GP) regression within a Leader-Follower architecture.

A. Unified Geometric Modeling

• System Dynamics: The authors derive a unified multibody Lie-group formulation for $N$ aerial manipulators and a rigid payload.
• DAE Formulation: The system is modeled as a coupled DAE system. It explicitly captures:
  • Agent dynamics (translational, rotational, and joint).
  • Payload dynamics (linear and angular).
  • Holonomic Constraints: Rigid grasp conditions ($\phi(z)=0$) ensuring the contact points on the UAVs match the attachment points on the payload.
  • Contact Wrenches: The mapping of individual contact forces ($\lambda$) to the net payload wrench ($W_L$) via the grasp matrix $G(R_L)$.

B. Learning-Augmented Control Architecture

The control strategy is hierarchical:

1. Leader (Payload Level):
   • Generates a desired payload wrench ($W_{L,d}$) based on geometric tracking errors (position and attitude on SE(3)).
   • GP Compensation: Augments the nominal wrench command with a feedforward term derived from a GP model ($\hat{f}_{uk,L}$) that learns unknown payload-level disturbances (e.g., mass mismatch, external forces).
2. Follower (Agent Level):
   • Wrench Allocation: Solves the underdetermined wrench allocation problem to distribute the desired wrench among agents while regulating internal forces (forces that do not affect payload motion but affect grasp stability).
   • Realization: Each agent tracks its assigned contact forces.
   • GP Compensation: Each agent uses its own GP model ($\hat{f}_{uk,j}$) to compensate for local unmodeled dynamics (e.g., aerodynamics, joint friction), improving the accuracy of contact force realization.

C. Gaussian Process (GP) Integration

• Uncertainty Quantification: GPs are used to model unknown disturbance functions. The GP provides both a mean (for feedforward compensation) and a variance (predictive uncertainty).
• High-Probability Bounds: Theoretical bounds on the learning error are derived using Reproducing Kernel Hilbert Space (RKHS) theory. These bounds depend on the GP predictive covariance and are explicitly incorporated into the stability analysis.
• Data Handling: The system uses streaming datasets that are "frozen" at update intervals to compute GP posteriors, ensuring computational feasibility during flight.

D. Stability Analysis

• Lyapunov Analysis: A Lyapunov function is constructed for the closed-loop system.
• Theoretical Guarantee: The authors prove that the payload tracking errors are Uniformly Ultimately Bounded (UUB) with high probability.
• Scaling Law: The ultimate bound on the error scales linearly with the GP predictive uncertainty (both at the payload and agent levels) and the realization mismatch. As more data is collected and uncertainty decreases, the tracking performance asymptotically approaches the nominal ideal.

3. Key Contributions

1. Unified Geometric DAE Model: Development of a rigorous geometric model for cooperative rigid-payload transport that explicitly handles contact wrenches, internal force redundancy, and coupled agent-payload dynamics.
2. Hierarchical Learning Framework: A novel leader-follower control architecture where learning is applied at two distinct levels:
   • Payload Level: Compensates for global disturbances affecting the net wrench.
   • Agent Level: Compensates for local disturbances to improve force realization, reducing the "wrench mismatch" passed to the payload.
3. Rigorous Stability with Learning: A stability proof establishing high-probability UUB where the error bound is explicitly a function of the GP predictive uncertainty. This bridges the gap between data-driven learning and rigorous control theory.
4. Handling Redundancy: A systematic approach to handling the non-injective grasp map, separating the generation of the desired wrench from the allocation of internal forces.

4. Simulation Results

The authors validated the framework using a team of 2 aerial manipulators transporting a rigid payload with 4 contact points.

• Scenarios: Three cases were compared:
  • (C1) Nominal (No learning).
  • (C2) Payload learning only.
  • (C3) Payload + Agent learning.
• Findings:
  • Error Reduction: Case (C3) achieved the tightest tracking bounds, significantly outperforming the nominal case (C1).
  • Role of Agent Learning: Comparing (C2) and (C3) demonstrated that agent-level learning reduces the wrench mismatch ($\Delta W$) caused by imperfect force realization. This confirms the modularity of the design: better agent performance directly improves payload stability.
  • Uncertainty Correlation: The simulation showed that regions of high GP predictive uncertainty correlated with larger transient tracking errors, validating the theoretical link between GP variance and control performance.
  • Force Behavior: The contact forces remained bounded and smooth, with no high-frequency chattering, and internal forces were successfully regulated without affecting the net payload motion.

5. Significance

This paper represents a significant advancement in cooperative aerial manipulation by:

• Bridging Theory and Practice: It moves beyond simplified load models (cables) to handle the complex, coupled dynamics of rigid grasps, which are essential for construction and heavy-lifting applications.
• Robustness to Uncertainty: By integrating GP learning with geometric control, it provides a mathematically guaranteed method to handle the inevitable model mismatches and disturbances in real-world UAV operations.
• Scalable Architecture: The leader-follower approach allows for scalable coordination where the "leader" defines the task and "followers" handle the execution, with learning mechanisms ensuring robustness at every level.
• Theoretical Rigor: Unlike many learning-based control papers that rely on empirical results, this work provides probabilistic stability guarantees, making it suitable for safety-critical applications where performance bounds must be known a priori.

In conclusion, the proposed framework offers a robust, theoretically grounded solution for cooperative transport of heavy payloads, explicitly quantifying how learning uncertainty impacts control performance and demonstrating that reducing this uncertainty through data collection directly improves tracking accuracy.