Safe Payload Transfer with Ship-Mounted Cranes: A Robust Model Predictive Control Approach

1. Problem Statement

The paper addresses the challenge of safely transferring payloads using ship-mounted cranes in dynamic, unstructured marine environments.

• Core Challenge: Ship-mounted cranes operate on moving platforms subject to significant external disturbances (waves, wind) causing complex base motions (roll, pitch, yaw). This creates underactuated dynamics where the payload swings uncontrollably, and the system is highly sensitive to modeling uncertainties.
• Safety Requirements: The system must perform precise payload insertion (e.g., inserting a cylindrical object into a receptacle) while strictly avoiding collisions with:
  • Obstacles on the ship deck.
  • The walls of the target receptacle.
  • The deck surface outside the target lip.
• Limitations of Existing Methods: Traditional Model Predictive Control (MPC) often struggles to balance robustness, safety, and performance. Conventional robust MPC can be overly conservative (reducing performance), while standard Control Barrier Functions (CBFs) applied in discrete time may fail to guarantee safety between sampling instants (inter-sample violations), especially under unmodeled dynamics.

2. Methodology

The authors propose a Robust Model Predictive Control (MPC) framework integrated with Robust Zero-Order Control Barrier Functions (R-ZOCBFs).

A. System Modeling

• Dynamics: A 5-DOF electromechanical crane model is derived using Euler-Lagrange equations. It accounts for the ship's time-varying orientation ($ds(t)$) as an external disturbance.
• Uncertainty: The model includes an uncertainty vector $\delta_e(t)$ representing unmodeled dynamics (e.g., wind, friction) and measurement errors. The system is treated as a control-affine, time-varying nonlinear system with sample-and-hold effects.

B. Safety Formulation

The safety constraints are defined via time-varying safe sets:

1. Collision Avoidance (Bounding Boxes): Time-varying rectangular bounding boxes are generated around the payload to avoid static obstacles on the deck. These are represented by six linear safety functions.
2. Target Insertion: A smooth, non-linear safety function ($h_t$) is designed for the cylindrical target. It ensures the payload remains aligned with the target and above the target surface until insertion, preventing contact with the target rim or the deck.
3. Unified Safe Set: The overall safe set is the intersection of the bounding box constraints and the target constraint.

C. Control Framework: Robust MPC with R-ZOCBF

• R-ZOCBF (Robust Zero-Order Control Barrier Function): To address inter-sample safety violations common in discrete-time systems, the authors utilize ZOCBFs. They extend this to a Robust version (R-ZOCBF) that accounts for bounded uncertainties ($\delta_e$).
  • The constraint ensures that the safety function remains non-negative not just at the next sample time, but throughout the entire sampling interval, even under worst-case disturbances.
• Online Parameter Adaptation: A key innovation is a sampling-based optimization scheme to adaptively tune the robustness parameter ($\delta_t$) in real-time.
  • Instead of using a fixed, conservative bound, the controller samples the state space around the current state to estimate the worst-case effect of uncertainty on the safety function.
  • This allows the controller to reduce conservativeness when uncertainties are low and tighten constraints only when necessary, balancing safety with trajectory tracking performance.
• MPC Implementation: The controller solves a nonlinear MPC problem using the acados software package (SQP Real-Time Iteration). The cost function minimizes control effort, tracking error, and payload swing rates while satisfying the R-ZOCBF constraints.

3. Key Contributions

1. R-ZOCBF for Sampled-Data Systems: The paper introduces and experimentally validates a robust zero-order CBF framework specifically designed for sampled-data systems with unmodeled dynamics, guaranteeing safety between time steps.
2. Online Robustness Adaptation: A novel optimization-based method is proposed to tune the robustness margin online. This reduces the conservatism typically associated with robust control, allowing for more aggressive and accurate payload transfer.
3. Integrated Perception and Control: The system combines real-time visual perception (using SAM2 for segmentation and RANSAC for pose estimation) with the MPC controller to generate dynamic bounding boxes and target constraints.
4. Experimental Validation: The first hardware demonstration of this framework on a 5-DOF crane mounted on a Stewart platform, which simulates realistic ocean wave motions.

4. Results

The approach was evaluated through both simulations and hardware experiments.

• Simulation:
  • Compared against a nominal MPC (without robustness margins), the proposed Robust MPC successfully prevented collisions with the target and environment under sinusoidal base motion and bounded state uncertainties.
  • The nominal controller violated safety constraints (collided with the target), while the R-ZOCBF controller maintained safety.
• Hardware Experiments:
  • Setup: A custom 8ft carbon fiber crane on a Stewart platform simulated sea states (lateral sinusoidal motion, $\pm 0.05$m).
  • Performance: The system successfully inserted a PVC pipe (payload) into a 5cm aperture target while avoiding obstacles.
  • Comparison:
    • Nominal Safety: The system failed to maintain safety under disturbance, resulting in collisions.
    • Robust Safety (R-ZOCBF): The system maintained the safety function $h(t) \geq 0$ throughout the maneuver, successfully inserting the payload despite significant base excitation and model uncertainties.
  • Adaptation: The online tuning of the robustness parameter allowed the system to remain close to the reference trajectory without being overly conservative, demonstrating effective trade-off management.

5. Significance

• Safety-Critical Robotics: This work provides a rigorous framework for safe robot operation in highly uncertain, dynamic environments, moving beyond theoretical guarantees to experimental validation.
• Marine Logistics: The methodology directly addresses critical challenges in offshore logistics, ship-to-ship transfer, and subsea operations where precision is compromised by environmental factors.
• General Applicability: While focused on cranes, the authors note that the R-ZOCBF and online adaptation strategies are applicable to other robotic tasks requiring safe parts mating, assembly, and insertion under uncertainty (e.g., spacecraft docking, peg-in-hole assembly).
• Bridging Theory and Practice: By integrating advanced perception (VOS models) with robust control theory, the paper demonstrates a viable path toward fully autonomous, safe robotic systems in unstructured real-world settings.