⚡ electrical engineering

Input-Output Extension of Underactuated Nonlinear Systems

This paper proposes a method to enhance the task space capabilities of commercial underactuated systems by integrating auxiliary actuators with a feedback linearizing outer loop, enabling full pose tracking and robust physical interaction without modifying the internal certified low-level controller.

Original authors: Mirko Mizzoni, Amr Afifi, Antonio Franchi

Published 2026-07-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mirko Mizzoni, Amr Afifi, Antonio Franchi

Original paper licensed under CC BY 4.0 (http://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 teach a very talented, but slightly stubborn, robot to dance. This robot is a "quadrotor"—a flying drone with four spinning propellers. It's a superstar at hovering, moving forward, and turning left or right. But it has a secret weakness: it can't just tilt its body sideways without also moving forward, and it can't move sideways without tilting. It's like a bicycle that can only go straight or turn; it can't slide sideways like a crab. This is called being "underactuated," meaning it has fewer controls than the ways it can move in space.

Now, imagine you want this robot to do something fancy, like gently push a wall or hold a tool steady while flying. To do that, it needs full control over all six directions of movement (up/down, left/right, forward/back, and three types of tilting). Usually, to get this super-power, engineers have to build a brand-new, expensive robot from scratch with extra motors. But what if you could just "plug in" a few extra motors to your existing, certified, and safe robot without taking apart its brain? That is the big question this paper tackles. It explores a clever trick to give old, limited robots new superpowers by adding helper motors and using a smart "translator" to make them work together, all while leaving the robot's original, certified safety software completely untouched.

The Paper's Big Idea: The "Translator" Trick

This paper proposes a method to upgrade commercial drones (like the "Flyability Elios 3" used for inspecting tight spaces) by adding two extra propellers. The goal is to let these drones move in ways they were never designed to do—like tilting without moving forward—without ever touching the robot's internal, factory-installed controller.

Think of the drone's internal controller as a highly trained, certified pilot who only knows how to fly a standard four-propeller drone. This pilot is great, but they are also a bit rigid: they only accept high-level commands like "fly forward at 2 meters per second" or "turn yaw at 5 degrees." They don't know how to control the individual speed of each propeller directly, and they certainly don't know what to do with two new propellers you just taped onto the side.

The authors' solution is to build a "translator" (an outer-loop controller) that sits between the human operator and the drone's internal pilot. Here is how the magic happens:

The Setup: The team adds two extra propellers to the drone. These are the "auxiliary actuators."
The Translator's Job: The new controller takes the human's desired path (like "move in a perfect circle while staying perfectly level") and breaks it down. It tells the internal pilot, "Hey, keep doing your standard forward-and-turn thing," but it also secretly uses the two new propellers to handle the tricky parts, like tilting the drone sideways to stay level.
The Result: The internal pilot thinks everything is normal because it's still receiving the exact same type of commands it was certified for. Meanwhile, the new propellers are doing the heavy lifting to give the drone full 6-degree-of-freedom control (moving in any direction and tilting in any way).

What the Paper Found (and What It Didn't)

The researchers didn't just guess this would work; they did the math to prove it's possible under specific conditions. They showed that if you add the extra motors in a certain way, you can "re-interpret" the old commands to control new movements.

The Proof:
They tested this idea using computer simulations. They took a standard quadrotor model and added the two extra propellers.

Scenario 1 (The Dance): They made the drone fly in a complex, wavy path (a Lissajous curve) while dealing with "noise" (like wind or sensor glitches) and slight errors in the drone's weight. The upgraded drone tracked the path perfectly, even when the computer didn't know the exact weight of the drone.
Scenario 2 (The Wall Push): This was the real test. They simulated the drone flying into a wall to push it. A standard drone (without the extra motors and the translator) crashed and lost control immediately because it couldn't handle the physical push. However, the "Quad-IO" platform (the one with the extra motors and the new controller) stayed stable and successfully pushed the wall.

The Limits:
The paper is very clear about what this doesn't do.

It does not require you to rewrite the drone's internal software. The original controller remains "certified" and untouched.
It does not work if you try to control the new motors directly without the translator. The math shows that if you just try to use the new propellers as simple inputs, the system breaks down. You must use the specific feedback-linearization method they described.
The results shown are simulations. The authors tested this on a computer model, not on a real physical drone in a lab yet. They mention that future work will involve testing this on actual hardware.

Why This Matters

The beauty of this approach is that it respects the "safety certificate" of commercial robots. In the real world, you can't just open up a drone used for inspecting nuclear plants or flying in caves and rewrite its code; that would void its safety certification and make it illegal to use.

By keeping the internal controller exactly as the manufacturer left it, this method offers a "plug-and-play" upgrade. It turns a limited, underactuated robot into a fully controllable one, allowing it to perform delicate tasks like physical interaction (pushing, pulling, or holding objects) that were previously impossible for standard commercial drones. The paper suggests that with the right mathematical "translator," we can give old robots new lives without breaking their safety guarantees.

Technical Summary: Input-Output Extension of Underactuated Nonlinear Systems

Problem Statement
Commercial underactuated (UA) robotic systems, such as standard quadrotors, are characterized by having fewer actuators than degrees of freedom (DoF). While these platforms are cost-effective and certified for specific tasks (e.g., aerial surveillance), they lack the full six-degree-of-freedom (6D) control required for advanced applications like aerial physical interaction (APhI). Existing solutions often require designing fully actuated platforms (e.g., hexarotors with tilted rotors), which necessitates significant hardware redesign and forfeits the safety certifications and reliability of commercial off-the-shelf units. Furthermore, many advanced control strategies assume full access to low-level actuation or system models, which is not feasible for closed-source commercial platforms where internal controllers cannot be modified.

The core problem addressed is how to enhance the task-space capabilities of a commercial UA system by adding auxiliary actuators to achieve full 6D pose tracking, while strictly preserving the integrity, certification, and structure of the system's existing internal low-level controller.

Methodology
The authors propose an "Input-Output Extension" framework that integrates auxiliary actuators via a feedback-linearizing outer-loop controller. The system is modeled as an interconnected set of four subsystems:

The Plant: The physical dynamics of the vehicle.
The Primary Actuation System: The original actuators controlled by the internal, pre-certified controller.
The Internal Controller: A "black box" component that accepts high-level commands (e.g., velocity, yaw rate) and outputs control signals. It is assumed to be unmodifiable and operates under the assumption that no auxiliary inputs exist ( $c_2 = 0$ ).
The Auxiliary Actuation System: New actuators added to the platform.

The methodology relies on feedback linearization to reinterpret the original high-level commands. By defining a specific reference assignment, the outer-loop controller cancels the influence of the internal controller's error dynamics, effectively exposing a "virtual input" ( $u_1$ ) corresponding to the highest-order derivatives of the original outputs. This virtual input is then combined with the new auxiliary inputs ( $u_2$ ) to control previously inaccessible degrees of freedom.

The paper establishes two necessary and sufficient conditions (Theorem 1) for the existence of a compensator that renders the composite system input-state-output linear and decoupled:

Invariance of Relative Degree: The auxiliary actuation system must not alter the vector relative degree of the primary output ( $y_1$ ). This ensures the original input-output mapping remains valid for the internal controller.
Nonsingularity of the Full Interaction Matrix: A combined interaction matrix, accounting for both primary and auxiliary dynamics, must be nonsingular in the operating neighborhood.

Key Contributions

Input-Output Extension Framework: A novel control architecture that allows commercial UA systems to control additional DoFs (e.g., roll and pitch angles in a quadrotor) by adding hardware without modifying the certified internal software.
Preservation of Certification: The approach interacts with the internal controller solely through certified input-output interfaces, leaving its structure and parameters unchanged.
Re-targeting Mechanism: The method reinterprets legacy high-level commands to control unactuated variables while using new inputs to compensate for dynamic coupling effects introduced by the coexistence of legacy and auxiliary actuation.
Theoretical Conditions: Rigorous derivation of conditions under which the extended system achieves full decoupled control.

Results
The proposed method was validated through numerical simulations using a commercial quadrotor (Flyability Elios 3) retrofitted with two additional rotors.

Robustness: The system demonstrated robustness against parametric uncertainties (mass and inertia mismatches), actuation noise, and internal reference perturbations.
Physical Interaction: In a scenario involving physical contact with a wall (simulating APhI), the modified platform (Quad-IO) maintained stable tracking of a straight-line trajectory. In contrast, a standard quadrotor without the extension diverged after 11 seconds, highlighting the instability of the baseline system under contact forces.
Trajectory Tracking: The system successfully tracked complex Lissajous-type paths in the horizontal plane while maintaining constant altitude, even under uncertainty.

Significance and Claims
The paper claims to address a critical gap in the state of the art: enabling general commercial underactuated systems to perform complex 6D tasks without sacrificing the safety and usability guarantees of their original certification. The significance lies in offering a practical, cost-effective alternative to designing new fully actuated platforms. The authors emphasize that their approach allows for the "re-targeting" of existing platforms, making them capable of physical interaction tasks that were previously impossible for standard quadrotors.

The paper modestly notes a current limitation: the method requires knowledge of the internal controller's structure (specifically the interaction matrix and drift terms), which must be either provided by the manufacturer or identified. Future work is proposed to investigate data-driven methods, such as reinforcement learning, to relax this assumption and to validate the approach on physical hardware.

The Paper's Big Idea: The "Translator" Trick

What the Paper Found (and What It Didn't)

Why This Matters

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