Behavioral Data-Driven Optimal Trajectory Generation for Rotary Cranes

This paper proposes a behavioral data-driven framework based on Willems' fundamental lemma to generate optimal, open-loop slewing trajectories for rotary cranes that effectively suppress load sway and reduce operation time and energy consumption without requiring explicit system modeling or extensive datasets.

The Gist

Imagine a construction crane as a giant, human-sized pendulum. When a worker spins the crane's arm (the boom) to move a heavy load, the weight hanging from the cable doesn't just follow the arm; it swings back and forth like a child on a swing. If the worker spins too fast or stops too abruptly, the load swings wildly, which is dangerous and wastes time.

For decades, engineers have tried to solve this using two main methods:

1. The "Math Model" Approach: They build a complex mathematical equation of how the crane should behave. But real cranes are messy (wind, worn-out gears, flexible cables), so the math is never 100% right, leading to imperfect movements.
2. The "Trial and Error" Approach: They rely on highly skilled human operators who use their intuition to spin the crane slowly and smoothly to avoid swinging. But skilled operators are rare, and humans get tired.

The New Solution: The "Memory Bank" Method

This paper introduces a third way: Behavioral Data-Driven Control. Instead of trying to write a perfect math book about the crane, the authors let the crane "teach" them how it moves by recording its actual movements.

Here is how they did it, using simple analogies:

1. The "Motion Dictionary" (Willems' Fundamental Lemma)

Imagine you want to learn how to speak a new language. Instead of studying grammar rules (the math model), you listen to thousands of hours of native speakers (the data). Eventually, you build a mental "dictionary" of phrases.

The authors did this with the crane. They ran the crane through many different smooth movements, recording every input (how fast they spun the motor) and every output (how the load swung). They organized this data into a giant "Motion Dictionary" (called a Hankel matrix). This dictionary contains thousands of tiny snippets of the crane's past behavior.

2. The "Puzzle Solver" (Convex Optimization)

Now, imagine you need to move the load from Point A to Point B as fast as possible without it swinging.

• Old Way: You try to calculate the perfect path using a formula that might be slightly wrong.
• New Way: You take your "Motion Dictionary" and ask a computer: "Can you stitch together a few snippets from this dictionary to create a new path that gets us from A to B, stops exactly there, and keeps the load still?"

The computer solves this as a puzzle. It looks for the perfect combination of past movements to create a new, smooth, optimal path. Because it's stitching together real, proven movements, the result is naturally smooth and safe, even if the crane has weird quirks or noise.

3. The "Smooth Jazz" vs. "Robot Dance"

The paper compares their new method against a traditional "Model-Based" method (the old math approach).

• The Old Method was like a robot dancing: it tried to follow a strict, pre-calculated line. Because the math wasn't perfect, it missed the target and took a long time (about 35 seconds) to stop the swinging.
• The New Method was like a jazz musician improvising: it found a path that was 50% faster (about 15 seconds), stopped much more precisely, and reduced the swinging (sway) by up to 35%.

Why This Matters

The authors tested this on a real, scaled-down crane in a lab. They found that by using the "Motion Dictionary" approach:

• No Expert Needed: You don't need a PhD in physics to tune the system; you just need good data.
• Handles Real Life: It works even with noisy sensors and imperfect hardware because it learns from the actual behavior, not a theoretical ideal.
• Saves Time and Energy: The crane moves faster and uses less energy because the path is perfectly optimized for that specific machine.

The Catch

The paper notes that this method isn't magic. It requires:

1. Good Data: You have to collect a lot of diverse movement data first (like recording many hours of native speakers).
2. Tuning: You have to adjust a few "knobs" (hyperparameters) to tell the computer how much to prioritize speed vs. smoothness.

In Summary:
Instead of trying to write a perfect textbook on how a crane moves, the authors built a library of how the crane actually moves. They then used a computer to mix and match those real movements to create a perfect, fast, and safe path for the crane to follow, outperforming traditional math-based methods in speed and stability.

Technical Summary

Technical Summary: Behavioral Data-Driven Optimal Trajectory Generation for Rotary Cranes

Problem Statement
The automation of rotary cranes in the construction industry faces significant challenges due to the underactuated nature of crane dynamics and the difficulty of obtaining precise load position measurements in real-world environments. Traditional model-based control strategies rely on accurate parametric models and expert tuning, often failing to capture system complexities or handle modeling inaccuracies, which leads to inefficient energy use, missed targets, and excessive load sway. Conversely, many learning-based approaches require extensive datasets and high computational resources. Furthermore, full feedback automation is often impractical due to the difficulty of measuring suspended loads on long, flexible cables and the risk of feedback loops amplifying oscillations in the presence of disturbances. Consequently, crane operations often rely on open-loop trajectory planning, which is limited by the fidelity of the underlying physical models.

Methodology
This paper proposes a behavioral data-driven framework for generating open-loop slewing trajectories that suppress load sway while minimizing operation time and energy consumption. The approach is grounded in Willems' fundamental lemma and behavioral systems theory, which allows for the characterization of system behavior directly from measured input-output data, bypassing the need for explicit parametric modeling.

The core methodology involves the following steps:

1. Data Collection and Representation: The system is excited using smooth, piecewise-constant input signals (specifically, sums of sines with randomized frequencies) to capture nonlinear dynamics within a specific operational region. The collected input-output data (slewing angular velocity and load/boom angles) are organized into a Hankel matrix.
2. Nonparametric Modeling: The paper treats the Hankel matrix as a dictionary of motion primitives. By leveraging the superposition principle in linear subspaces, the method represents the system's behavior as a linear combination of past trajectories. To handle the nonlinear nature of the crane and noise in the data, the framework employs $\ell_1$-norm regularization to promote sparsity in the combination coefficients, effectively selecting relevant motion primitives. Additionally, singular value truncation is used to attenuate noise.
3. Optimization Formulation: The trajectory generation problem is formulated as a convex optimization problem. The objective is to find a trajectory that interpolates between given initial and target resting conditions (zero velocity and sway) while minimizing a cost function that balances:
   • Load Sway: Keeping the load close to its downward equilibrium.
   • Time: Minimizing the distance to the target boom position to ensure swift arrival.
   • Energy: Minimizing the squared control effort.
   • Smoothness: Penalizing the total variation of the boom angle and velocity to prevent abrupt changes that induce oscillations.
   • Constraints: Strict bounds are imposed on actuator limits and load sway angles to ensure safety.

Key Contributions

• First Application of Behavioral Theory to Cranes: To the authors' knowledge, this is the first work to employ behavioral systems theory for crane control.
• Nonparametric Trajectory Generation: The method generates optimal trajectories directly from noisy data without requiring explicit parameter identification, nonlinear ODE solvers, or complex modeling assumptions.
• Practical Workflow: The paper presents a systematic workflow for data collection, hyperparameter tuning (via grid search on nonparametric simulation performance), and trajectory generation, making the framework applicable to other crane systems.
• Handling Nonlinearity and Noise: By interpreting the Hankel matrix as a dictionary of local linear approximations and using sparse regularization, the method effectively captures nonlinear dynamics and remains robust to measurement noise.

Results
The proposed method was validated on a 1:9 scaled laboratory rotary crane and compared against an established model-based optimal trajectory generation approach (using a multi-waypoint method with sequential quadratic programming).

• Performance Metrics: The data-driven approach achieved up to a 35% reduction in load sway, a 43% reduction in tracking error, and a 50% reduction in travel time compared to the model-based benchmark.
• Experimental Validation: In experimental trials, the generated trajectories successfully moved the load between specified angles with high precision (tracking error of 0.002 rad) and minimal residual oscillation, closely matching the predicted behavior despite open-loop execution.
• Simulation: Simulative studies confirmed that the method could capture nonlinear dynamics (specifically radial load sway) that linearized models failed to represent, provided sufficient data diversity and appropriate hyperparameter tuning.

Significance and Claims
The paper claims that this work reduces the complexity of trajectory optimization for nonlinear, underactuated systems to linear algebraic operations and convex optimization problems. It demonstrates that crane automation can be achieved without relying on expert on-site operators, detailed system modeling, high-fidelity simulators, or complex nonlinear solvers.

The authors maintain a modest stance regarding the scope of their claims:

• They acknowledge that the method requires careful tuning of hyperparameters and that the quality of the results is heavily dependent on the design of the input signals used for data collection.
• They note that while the method is data-driven, it is not "data-greedy" compared to other learning-based methods, as relatively short data sequences were sufficient.
• They explicitly state that a rigorous theoretical analysis of the behavioral framework for general nonlinear dynamics remains an open problem.
• Future work is identified as extending the approach to full-scale cranes, accounting for unforeseen disturbances (e.g., wind, torsional effects), and potentially applying Data-enabled Predictive Control (DeePC) for closed-loop operation.

The study concludes that by operating directly on data, the proposed framework offers a practical, reliable, and efficient alternative to traditional model-based approaches for suppressing load sway in rotary cranes.