⚡ electrical engineering

Robust Optimal Tracking Control of a Nonlinear Underactuated 3-DOF Mechanical Crane under Payload-Mass Uncertainty Using an Effective-Input Transformation

This paper proposes a robust optimal control framework for a nonlinear 3-DOF underactuated mechanical crane that utilizes an effective-input transformation to handle payload-mass uncertainty, ensuring stable trajectory tracking and anti-sway performance across varying load conditions without linearizing the system dynamics.

Original authors: Rehan Ali Rahimon, Irfan Ali, Abdul Qadir

Published 2026-09-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Rehan Ali Rahimon, Irfan Ali, Abdul Qadir

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

In the bustling world of heavy industry, from the docks of a shipping port to the steel skeleton of a new skyscraper, the mechanical crane is a workhorse of precision. Its job is deceptively simple: lift a heavy load, move it across a distance, and set it down exactly where it belongs. Yet, the physics of this task is fraught with a stubborn complication. When a crane hoists a weight, that weight does not simply hang still; it swings like a pendulum. If the operator moves the crane too quickly, the load swings wildly, making it impossible to place with accuracy and creating a dangerous hazard. The challenge is compounded by the fact that the crane has fewer controls than it has moving parts. The machine can push the cart forward and pull the cable up or down, but it has no direct lever to stop the swinging motion of the load itself. The swing must be stopped indirectly, by carefully timing the movements of the cart and the cable. This makes the crane an "underactuated" system, a term engineers use to describe machines where the number of controls is smaller than the number of ways the machine can move.

Adding to this difficulty is the reality of the real world: the weight of the load is rarely known with perfect certainty. A crane might be calibrated for a specific weight, but if the actual load is heavier or lighter, the physics of the swing change, and a control strategy that worked perfectly yesterday might fail today. Researchers have long sought a way to build a control system that can handle these unknowns without needing to simplify the complex, twisting math of the crane's motion. They needed a method that could keep the load steady and on course, even when the weight hanging from the cable was a mystery.

A team of researchers has now developed a new approach to this problem, creating a robust control framework designed specifically for a three-dimensional crane system. Their work focuses on a crane with three main ways of moving: the horizontal position of the cart, the length of the cable, and the angle of the swinging load. While the cart and the cable are directly controlled by motors, the swing angle is left to the laws of physics. The researchers tackled the issue of unknown payload mass by first creating a mathematical "translation" of the crane's behavior. They transformed the difficult, underactuated equations into a simpler, fully actuated form. Imagine taking a complex puzzle where some pieces are missing and rearranging the view so that every piece seems to be there, allowing for a standard solution to be found. In this new view, the system behaves as if it had a third control input for the swing, even though the physical machine still only has two motors.

Once this virtual, fully actuated version of the crane was established, the team designed an optimal controller. This is a mathematical rule that calculates the best possible action to take at every moment to minimize errors and keep the system stable. They used a rigorous method based on a continuous equation that balances the desire for quick movement against the need for smooth, stable control. Crucially, this design did not rely on simplifying the crane's motion into a straight line or a simple curve; it worked directly with the full, complex, non-linear reality of the machine. After calculating the ideal actions for this virtual system, the researchers used a specific mathematical tool to translate those actions back into the two real commands the crane's motors could actually execute: how hard to push the cart and how hard to pull the cable. This translation process ensured that the physical motors could follow the plan, even though the plan was originally drawn for a system with more controls than the crane actually possessed.

To test if this method truly worked, the researchers ran extensive computer simulations using a powerful mathematical software. They set the crane to move a cart to a specific spot, shorten the cable to a precise length, and eliminate any swinging. They then tested the system under four different conditions: with no extra weight, and with additional loads of 1.5 kilograms, 3.0 kilograms, and 4.0 kilograms. The baseline weight of the suspended load was 0.5 kilograms, meaning the total weight varied from 0.5 kg up to 4.5 kg. The goal was to see if the controller could guide the crane to its target and stop the swing, regardless of how heavy the load became.

The results showed that the system was remarkably resilient. For the lightest load, the crane moved smoothly to its destination, and the swing vanished quickly. As the weight increased, the behavior of the system changed in predictable ways. The heavier loads caused more noticeable shaking and oscillation in the cable length and the swing angle during the initial movement. The cart's position, however, remained steady and accurate across all tests, showing that the horizontal movement was largely unaffected by the changing weight. Even with the heaviest load of 4.5 kilograms, the system did not fail. The oscillations were larger and lasted a bit longer, but they remained within safe limits and eventually died out, leaving the load perfectly still at the target location.

The study also examined the forces the motors had to exert to achieve this stability. When the load was light, the motors applied a brief burst of force and then settled down. As the weight increased, the motors had to work harder, applying larger and more complex bursts of force to counteract the increased inertia and the stronger tendency of the heavy load to swing. Despite these increased demands, the forces never grew out of control. The motors applied just enough power to correct the path and dampen the swing, and once the crane reached its target, the forces returned to zero. This confirmed that the controller did not require constant, frantic adjustments to keep the system stable; it simply guided the machine to a calm stop.

The researchers found that while the crane's ability to move horizontally was robust against changes in weight, the vertical and swinging motions were more sensitive. The heavier the load, the more the system had to fight against the natural physics of the swing. However, the proposed control method successfully managed this sensitivity. It kept the errors bounded, meaning the deviations from the target never grew too large, and it ensured that all movements eventually converged to the desired state. The work demonstrates that it is possible to design a control system for a complex, underactuated machine that can handle significant uncertainty in the load's mass without needing to simplify the underlying physics. By using a clever transformation to view the problem differently and then translating the solution back to the real world, the team created a method that keeps the load steady, the path accurate, and the operation safe, even when the weight hanging from the cable is unknown.

Technical Summary: Robust Optimal Tracking Control of a Nonlinear Underactuated 3-DOF Mechanical Crane under Payload-Mass Uncertainty

Problem Statement
Mechanical cranes are inherently underactuated nonlinear systems where the number of independent actuator inputs (trolley force and hoisting force) is fewer than the number of generalized coordinates (trolley displacement, cable length, and payload swing angle). This configuration creates strong nonlinear coupling between actuated and unactuated states, complicating controller design. Furthermore, practical operation involves significant uncertainty in the suspended payload mass. Variations in payload mass alter the system's inertia, Coriolis, centrifugal, and gravitational dynamics, which can degrade trajectory-tracking accuracy and anti-sway performance. Existing approaches often rely on linearization or small-angle approximations, which may limit performance when operating directly on the original nonlinear equations.

Methodology
The study proposes a robust optimal control framework that operates directly on the nonlinear 3-DOF crane model without linearization. The methodology is structured in two primary stages:

Effective-Input Transformation: To address the underactuation mismatch, the authors introduce an effective-control transformation. This technique converts the original underactuated dynamics into an equivalent fully actuated representation within the generalized-coordinate acceleration space. By defining a three-dimensional virtual effective input ( $u_{eff}$ ), the controller synthesis problem is simplified. The transformation accounts for nominal system dynamics and isolates uncertainties (specifically payload mass variations) into a bounded nonlinear term.
Robust Optimal Control Synthesis: An optimal state-feedback law is designed for the transformed fully actuated system using an infinite-horizon quadratic performance index. The control gain is derived by solving the continuous-time algebraic Riccati equation (CARE). The design incorporates a Lyapunov-based stability analysis to ensure robustness against bounded nonlinear and inertia-related uncertainties. The stability proof demonstrates that if the optimal control stabilizes the nominal transformed system, it also stabilizes the uncertain nonlinear system, provided specific bounding conditions on the uncertainty are met.
Physical Input Recovery: Since the physical crane possesses only two actuators but the controller generates a three-dimensional effective input, the physical trolley ( $F_x$ ) and hoisting ( $F_l$ ) forces are recovered using the left Moore–Penrose pseudoinverse. This step maps the virtual control action back to the actual physical actuator commands, providing the minimum-norm least-squares solution when exact reconstruction is not possible.

Key Contributions
The paper outlines four principal contributions:

Nonlinear Formulation: The controller is derived directly from the original nonlinear 3-DOF crane equations, avoiding small-angle assumptions or linearization.
Transformation Strategy: The introduction of an effective-input transformation to obtain an equivalent fully actuated representation in generalized-coordinate acceleration space, facilitating standard optimal control design for underactuated systems.
Robust Synthesis: The development of a robust optimal feedback controller via the algebraic Riccati equation, with physical actuator forces reconstructed via the Moore–Penrose pseudoinverse.
Numerical Validation: A comprehensive numerical investigation of robustness against payload variations (0, 1.5, 3.0, and 4.0 kg additional mass) using the nonlinear closed-loop equations solved directly via Mathematica's NDSolve.

Results
Numerical simulations were conducted for a 3-DOF crane with a nominal suspended mass of 0.5 kg and additional payload masses up to 4.0 kg (total mass 4.5 kg). The desired trajectory involved moving the trolley to 4.0 m, setting the cable length to 1.5 m, and eliminating payload swing.

Tracking Performance: For all investigated payload conditions, the tracking errors for trolley position, cable length, and payload swing angle converged to zero.
Payload Sensitivity: The trolley-position channel exhibited high insensitivity to payload variations, maintaining smooth convergence. In contrast, the cable-length and payload-swing channels showed increased transient oscillations as the payload mass increased.
Stability and Boundedness: Despite the pronounced transient oscillations in the cable-length and swing-angle responses for heavier payloads (particularly at 3.0 kg and 4.0 kg), all oscillations remained bounded and decayed over time. The system successfully stabilized at the desired equilibrium without requiring sustained control effort once the target was reached.
Control Effort: The physical actuator forces ( $F_x$ and $F_l$ ) remained bounded. As payload mass increased, the amplitude and duration of the transient control forces increased, particularly in the hoisting channel, reflecting the controller's compensation for higher inertia and nonlinear coupling.

Significance
The study demonstrates that the proposed robust optimal framework effectively preserves trajectory tracking and stabilization for a nonlinear underactuated crane across a significant range of payload uncertainties. The primary significance lies in the ability to handle the underactuated nature of the crane and payload mass variations without linearizing the system dynamics. The results confirm that while heavier payloads induce more pronounced transient oscillations in the unactuated coordinates (swing and cable length), the control strategy ensures these oscillations are damped and the system converges to the desired state. The work validates the utility of effective-input transformations combined with optimal control and pseudoinverse recovery for managing complex, uncertain underactuated mechanical systems.

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