⚡ electrical engineering

Cooperative transportation of a cable–suspended underactuated planar load through model-based motion planning

This paper presents a computationally efficient, model-based motion planning approach that utilizes input-output normal forms and explicit numerical integration to enable two independent carts to cooperatively transport a cable-suspended underactuated load with exact trajectory tracking and verified tension feasibility, achieving significantly reduced tracking errors and oscillations compared to classical model-free methods in experimental industrial settings.

Original authors: Paolo Boscariol, Dario Richiedei, Iacopo Tamellin, Alberto Trevisani

Published 2026-07-02

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Paolo Boscariol, Dario Richiedei, Iacopo Tamellin, Alberto Trevisani

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to carry a heavy, wobbly chandelier across a room using two people walking on a straight track. The chandelier hangs from the people by two ropes. If you just walk at a steady pace, the chandelier will swing like a pendulum, making it hard to place it exactly where you want.

This paper describes a clever mathematical "recipe" for two robots (called "carts") to carry a suspended load without it swinging, ensuring it lands perfectly still at the exact right spot.

Here is the breakdown of how they did it, using simple analogies:

The Problem: The "Swinging Chandelier"

In many factories, robots need to move heavy things. Sometimes, they use cables to hang the load. The problem is that these loads are underactuated.

The Analogy: Think of a child on a swing. You can push the swing (the cart), but you can't directly control the child's angle (the load). If you push the swing too hard or too fast, the child swings wildly.
The Challenge: The robots need to move the load along a specific path (like a straight line or a lift) at a specific speed. If they just move the robots smoothly, the hanging load will still wobble because of physics. Usually, to stop the wobble, you need sensors on the load itself to tell the robots when to stop. But in many industrial machines, you can't add sensors to the hanging part; you can only control the carts.

The Solution: "Reverse Engineering" the Motion

The authors didn't try to guess how to move the robots. Instead, they worked backward.

The Goal: They decided exactly where the load must be at every single moment in time. No compromises.
The Math Trick: They used a special mathematical tool (called "input-output normal form") to ask a question: "If the load needs to be here, moving this fast, with no wobble, what exactly must the ropes be pulling with?"
The Calculation: Once they knew how hard the ropes needed to pull, they calculated exactly where the two robot carts needed to be to create that specific pull.

Think of it like a conductor leading an orchestra. Instead of telling the musicians to just "play loud," the conductor calculates the exact note every instrument must play to create a perfect harmony. The authors calculated the exact "notes" (positions and speeds) the robots needed to play to keep the load steady.

The "Safety Check"

A unique part of their method is that it checks for "rope slack."

The Metaphor: Imagine trying to pull a sled with a rope. If you push the sled forward too fast, the rope goes slack, and you lose control. The authors' math checks every step of the plan to ensure the ropes are always tight (pulling, not pushing). If the plan requires the rope to go slack, the computer knows the plan is impossible and flags it before the robots even move.

The Real-World Test

They tested this on a high-tech industrial conveyor system (B&R ACOPOStrak) with two carts and a steel bar hanging between them. They compared their "smart math" method against a standard "dumb" method (which just tries to move the carts smoothly without thinking about the swinging load).

The Results:

The "Dumb" Method: The load swung wildly. When it stopped, it was still wobbling. The error (how far off the target it was) was huge—up to 40 millimeters (about 1.5 inches).
The "Smart" Method: The load followed the path perfectly. When it stopped, it was almost instantly still. The error was tiny—only 0.7 millimeters (thinner than a credit card).
The Improvement: Their method reduced the error by more than 90%.

Why This Matters

The best part is that this "smart math" doesn't need to change the robots' internal software.

The Analogy: Imagine you have a car with a locked dashboard. You can't change the engine or the brakes. But you can write a perfect driving script (a GPS route) that tells the car exactly how to turn the wheel and press the gas so it drives perfectly without needing to touch the engine.
This method works exactly like that. It calculates the perfect route for the robots and feeds it to the existing machine. It's fast, simple, and works on machines that are already built and locked down.

In short: The paper shows how to use math to predict exactly how two robots should move to carry a swinging load without it ever wobbling, making industrial transport much more precise and reliable.

Technical Summary: Cooperative Transportation of a Cable–Suspended Underactuated Planar Load

Problem Statement
The paper addresses the challenge of cooperative transportation and manipulation of a cable-suspended, underactuated planar load using two independent carts moving along a common linear track. The system consists of a rigid beam (the load) with three degrees of freedom (horizontal position, vertical position, and orientation) suspended by two rigid cables of fixed length from two carts. Because the system is controlled by only two independent forces (the cart actuators) but possesses three degrees of freedom, it is inherently underactuated.

The core difficulty lies in the fact that the load is "kinematically undetermined"; its position is not uniquely defined by the cart positions alone due to the pendulum-like dynamics and nonlinear coupling. Traditional kinematics-based planning fails here because the load can oscillate or displace even when cart positions are fixed. Furthermore, in industrial scenarios requiring precise synchronization (e.g., manufacturing or packaging lines), the load must track a prescribed trajectory exactly in both time and space without modifying the desired path. A significant constraint is that the control architecture is often "collocated," meaning sensors and actuators are located only on the carts, with no direct feedback from the load. This makes disturbance rejection and oscillation suppression difficult, as the cart controllers are largely insensitive to the load's dynamic state due to the small load-to-motor inertia ratio.

Methodology
The authors propose a model-based motion planning approach that treats the problem as a dynamics-model inversion. Rather than relying on optimization or heuristic smoothing (like jerk-bounded profiles), the method computes the exact reference trajectories for the actuated carts required to force the load to follow a specific, pre-defined trajectory.

The methodology proceeds as follows:

System Modeling: The system is modeled using a set of index-3 nonlinear Differential-Algebraic Equations (DAEs) based on redundant coordinates. However, for the planning algorithm, the authors utilize the input-output normal form of the load dynamics. The load is treated as an unconstrained body acted upon by cable tensions, with the kinematic constraints of the cables explicitly included.
Inverse Dynamics Formulation: The controlled outputs are defined as the load's center of mass coordinates ( $x_G, y_G$ ). By expressing the load dynamics in terms of cable tensions ( $\tau$ ) and utilizing the input-output normal form, the authors derive an algebraic relationship between the desired load acceleration and the required cable tensions. This involves a "decoupling matrix" that maps control inputs to output accelerations.
Numerical Solution: Since the system is non-flat (it does not admit a purely algebraic inverse dynamics solution), the method employs explicit numerical integration. Starting from known initial conditions, the algorithm uses a fourth-order Runge-Kutta scheme to forward-integrate the equations of motion. At each time step, it:
- Computes the load state ( $\chi, \dot{\chi}$ ).
- Solves the nonlinear kinematic constraint equations to determine the required cart positions ( $x_C$ ).
- Calculates the necessary cable tensions algebraically.
- Derives the cart velocities and accelerations via differentiation of the kinematic constraints.
Feasibility Verification: A critical feature of the approach is the explicit calculation of cable tensions at every step. This allows for an immediate check of trajectory feasibility by ensuring tensions remain positive (taut cables) and within physical bounds.
Implementation: The resulting reference trajectories (position, velocity, and acceleration) for the carts are computed offline and uploaded to the industrial controller. The native cascade control of the carts then tracks these references.

Key Contributions

Exact Trajectory Tracking: The method ensures the load tracks a prescribed trajectory exactly in time and space without altering the desired path, a requirement often unmet by model-free approaches.
Collocated Control Compatibility: The approach is designed specifically for systems where load feedback is unavailable. It relies entirely on the model to shape the cart motion, making it compatible with proprietary industrial controllers that cannot be modified to accept external load feedback.
Computational Efficiency: By using explicit integration schemes (Runge-Kutta) rather than complex symbolic flatness calculations or heavy optimization, the method maintains a low computational burden suitable for offline planning and implementation on resource-constrained industrial hardware.
Built-in Feasibility Check: The formulation naturally provides cable tension values, enabling designers to verify if a trajectory is physically realizable (i.e., cables do not go slack) before execution.

Experimental Results
The method was validated on an industrial-grade setup using a B&R ACOPOStrak permanent magnet linear synchronous motor system with two carts (0.7 kg each) and a steel bar load (0.244 kg). Two test cases were performed: a horizontal translation (0.3 m) and a combined horizontal-vertical lifting motion (200 mm horizontal, 50 mm vertical), both executed in 1.5 seconds using ninth-degree rest-to-rest polynomial profiles.

The proposed "Model inversion" method was compared against a "Rigid motion planning" approach (which neglects underactuation and treats the load as a rigid body attached to the carts).

Horizontal Translation: The model-based approach reduced the maximum tracking error in the horizontal direction from 40.9 mm to 0.7 mm (a 98.3% reduction) and in the vertical direction from 5.2 mm to 0.5 mm (90.7% reduction). RMS errors were similarly reduced by over 85%.
Lifting Motion: Similar improvements were observed, with maximum horizontal tracking error reduced by 98% and vertical error by 85.6%. The contour error (tracking capability in Cartesian space) was reduced by approximately 94%.
Residual Oscillations: The proposed method significantly suppressed residual oscillations after motion completion compared to the rigid planning approach, which exhibited persistent vibrations.

Significance and Claims
The paper claims that the proposed approach offers a practically viable solution for precision cooperative transportation in industrial automation. Its primary significance lies in its ability to achieve high-precision tracking and oscillation suppression without requiring modifications to the native control architecture or the addition of load-side sensors. This makes it particularly attractive for industries using closed proprietary controllers where internal feedback loops cannot be altered.

The authors emphasize that the method's reliance on explicit numerical integration allows for easy implementation and low computational cost, while the explicit tension calculation provides a valuable tool for system sizing and trajectory feasibility analysis during the design phase. The results confirm that model-based planning is superior to heuristic, model-free methods for underactuated cable-suspended systems, achieving tracking errors in the sub-millimeter range (peak 0.7 mm) with negligible residual oscillations.