💻 computer science

Modeling and Control of a Pneumatic Soft Robotic Catheter Using Neural Koopman Operators

This paper proposes a neural network-enhanced Koopman operator framework that jointly learns system dynamics and control strategies to achieve high-precision, radiation-minimized open-loop control of pneumatic soft robotic catheters, outperforming existing model-based and Koopman variants in experimental positioning and orientation tasks.

Original authors: Yiyao Yue, Noah Barnes, Lingyun Di, Olivia Young, Ryan D. Sochol, Jeremy D. Brown, Axel Krieger

Published 2026-03-05

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CC BY 4.0

Original authors: Yiyao Yue, Noah Barnes, Lingyun Di, Olivia Young, Ryan D. Sochol, Jeremy D. Brown, Axel Krieger

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 navigate a tiny, flexible garden hose through a maze of delicate, winding tunnels to reach a specific spot on the wall. This is essentially what a doctor does during a heart procedure, using a catheter to treat irregular heartbeats. But doing this manually is like trying to thread a needle while wearing thick boxing gloves—it's hard, shaky, and relies heavily on the doctor's skill.

To make this easier, scientists have built robotic soft catheters. These are like smart, squishy hoses that can bend and twist on command. However, controlling them is a nightmare for computers because they are nonlinear. In simple terms, if you push the hose a little bit, it might bend a tiny bit; but if you push it a little more, it might suddenly snap into a completely different shape. It's unpredictable, like trying to predict exactly how a wet noodle will flop when you flick it.

The Problem: The "Guessing Game"

To control this robot, you need a map (a model) that tells the computer exactly how the hose will move.

Old maps were too simple. They assumed the hose always bends in a perfect circle, which isn't true.
Super-detailed maps existed but were so heavy and complex that the computer couldn't calculate them fast enough to be useful in real-time.
The Radiation Issue: Usually, doctors use X-rays (fluoroscopy) to see where the catheter is. But X-rays are like a bright, harmful spotlight. You don't want to leave it on too long. The goal is to navigate the robot to the target without constantly looking at the X-ray screen (open-loop control). This means the robot's brain needs to be so good at predicting its own movement that it doesn't need to peek.

The Solution: The "Neural Koopman" Translator

The authors of this paper invented a new way to teach the robot's brain. They used something called a Neural Koopman Operator.

Here is the best way to understand it:
Imagine the robot's movement is a chaotic dance in a dark room. It's hard to predict where the dancer will step next.

The "Lift" (Koopman Theory): The Koopman operator is like a magical pair of glasses that lifts the dancer into a different dimension—a "lifted space." In this new dimension, the chaotic dance suddenly looks like a simple, straight line. It turns a messy, unpredictable curve into a clean, straight path that a computer can easily calculate.
The "Neural" Part: Traditionally, scientists had to manually design the rules for these magical glasses, which often failed because they were too rigid. This paper uses Neural Networks (AI) to learn the perfect rules automatically. The AI figures out exactly how to translate the messy real-world movement into the clean, predictable "lifted" world, and then translates it back.

Think of it like a universal translator. The robot speaks "Chaos," and the AI translates it into "Simple Math" for the computer to solve, then translates the solution back into "Chaos" for the robot to execute.

How They Tested It

The team built a prototype catheter and tested it in two ways:

The Target Game: They asked the robot to move to specific spots on a table without looking at a camera. The "Neural Koopman" robot was incredibly accurate, hitting the target within 2 millimeters (about the width of a pencil lead). Other methods missed by much larger margins.
The Heart Simulation: They built a 3D-printed model of a human heart chamber (an atrium) with a fake heart wall. The robot had to navigate through a narrow entry and touch specific spots on the wall.
- The Result: The new method was the winner. It was more accurate, more consistent, and even faster than the other methods. It reached the target in less time, which is crucial in surgery to save time and reduce stress.

Why This Matters

This research is a big step toward safer, more precise heart surgeries.

Less Radiation: Because the robot can predict its own path so well, doctors might not need to keep the X-ray machine on as long, protecting both the patient and the doctor from radiation.
Better Precision: The robot can reach tiny, hard-to-get spots that human hands might struggle with.
The Future: This isn't just about heart catheters; it's a blueprint for controlling any soft, squishy robot, from surgical tools to search-and-rescue bots that can squeeze through rubble.

In short, the authors taught a squishy robot how to "think" in straight lines so it can navigate a messy world with the precision of a surgeon's scalpel, all without needing to constantly peek at an X-ray.

1. Problem Statement

The paper addresses the challenges associated with pneumatic soft robotic catheters (PSRCs) used in cardiac interventions, specifically for treating atrial fibrillation.

Nonlinearity and Complexity: PSRCs are inherently underactuated and exhibit highly nonlinear, complex behaviors that are difficult to model using traditional analytical methods (e.g., Piecewise Constant Curvature or Cosserat rod models), which often rely on simplifying geometric assumptions or require precise material parameters.
Sensing Limitations: In clinical settings, continuous imaging (fluoroscopy) is limited due to radiation exposure risks. This necessitates open-loop control strategies where the catheter must reach target poses reliably without real-time feedback.
Limitations of Existing Data-Driven Methods: While the Koopman operator offers a data-driven framework to lift nonlinear dynamics into a linear space for predictive control, traditional approaches rely on manually chosen basis functions (e.g., monomials). These often fail to capture the complex behaviors of soft robots, leading to poor generalization and control performance. Purely neural dynamic models, conversely, lack the structured linear representation needed for efficient optimal control.

2. Methodology

The authors propose a Neural Koopman Operator framework integrated with Model Predictive Control (MPC) to model and control the PSRC.

A. Neural Koopman Operator Framework

Instead of manually selecting basis functions, the framework uses a feed-forward neural network (FNN) to learn the "lifting" transformation end-to-end.

Architecture:
- Encoders: A state encoder ( $\Phi_e^x$ ) maps the original state $x_k$ to a high-dimensional lifted space. Crucially, a separate input encoder ( $\Phi_e^u$ ) maps the control input $u_k$ (and potentially the state) into the lifted space. This allows the system to learn nonlinear input effects (NINK variant), unlike standard Koopman methods that assume linear input influence.
- Linear Evolution: In the lifted space, the system dynamics are approximated as linear: $\gamma(x_{k+1}) = A\Phi_e^x(x_k) + B\Phi_e^u(x_k, u_k)$ .
- Decoders: A decoder network ( $\Phi_d^x$ ) maps the lifted state back to the original physical state.
Training: The network is trained in two stages:
1. Encoder Training: Minimizes the prediction error of the lifted state evolution over a horizon $R$ .
2. Decoder Training: Minimizes the reconstruction error between the predicted lifted state and the actual physical state.
3. An R-step loss is used to encourage long-horizon accuracy rather than just one-step prediction.

B. Control Strategy: Open-Loop MPC

The learned linear model in the lifted space is used within an MPC framework to generate control inputs without continuous feedback.

Optimization: The controller solves a quadratic optimization problem in the lifted space to find the optimal sequence of encoded inputs ( $\tilde{u}^*$ ) that minimizes the distance to the target pose.
Decoding: Since the optimal input $\tilde{u}^*$ exists in the encoded space, a decoder network ( $\Phi_d^u$ ) translates it back into the physical pressure commands ( $u^*$ ) for the syringe pumps.
Baseline Comparisons: The proposed method (NNKM) is compared against:
- LNKM: Neural Koopman with linear input assumption.
- MBKM: Traditional Koopman with manually selected monomial basis functions.
- SSM: Linear state-space model in original space.
- PCC: Piecewise Constant Curvature model with Jacobian-based control (standard model-based approach).

3. Key Contributions

Neural Koopman Framework for Soft Robotics: A novel architecture that jointly learns state and input encoders, the Koopman operator, and decoders. It preserves the linear structure required for control while capturing complex nonlinear dynamics through learned basis functions.
Nonlinear Input Handling: The introduction of a dedicated input encoder (NINK) allows the model to capture the nonlinear relationship between pneumatic pressure inputs and system dynamics, addressing a key limitation of previous Koopman applications.
Open-Loop Control for Clinical Scenarios: The integration of this framework with MPC enables reliable open-loop control, aiming to reduce radiation exposure by minimizing the need for continuous fluoroscopy during cardiac ablation.
Comprehensive Validation: The method is validated through both interactive position control and a clinically inspired atrial ablation simulation.

4. Experimental Results

Experiments were conducted on a dual-segment PSRC fabricated from hyperelastic material, using a 3D-printed atrium-like cavity for validation.

Modeling Performance:
- The NINK model achieved the highest accuracy, with an average RMSE of 0.6 mm (x), 1.1 mm (y), and 2.2° (orientation).
- It significantly outperformed the Linear Input Neural Koopman (LINK), Monomial Basis (MBKM), and Linear State-Space (SSM) models, particularly in predicting the $x$ -position, which has a narrow workspace range.
Interactive Position Control (Experiment 1):
- NNKM reached 100% of targets within a 1.5 mm error threshold and 100% within a 3.5 mm threshold.
- Baseline methods (MBKM, SSM) failed to reach targets accurately, often causing oscillation due to poor model prediction.
Pose Control in Atrium Model (Experiment 2):
- NNKM achieved an average position error of 2.1 ± 0.4 mm and orientation error of 4.9 ± 0.6°.
- Efficiency: NNKM was the fastest, reaching the target 8% faster than the next best method (LNKM) and 29% faster than the linear state-space model.
- Comparison: While the PCC baseline showed lower variance than some linear models, it suffered from higher mean errors due to geometric assumptions (perfect arcs) that do not hold for real soft robots experiencing stretching and twisting.

5. Significance

This work demonstrates that Neural Koopman Operators can effectively bridge the gap between data-driven modeling and control theory for soft robots.

Clinical Impact: By enabling accurate open-loop control, the proposed framework has the potential to significantly reduce radiation exposure for patients and operators during cardiac ablation procedures.
Robustness: The method outperforms both traditional physics-based models (which struggle with manufacturing inconsistencies) and other data-driven baselines, proving its suitability for the highly nonlinear and uncertain environment of soft robotic catheters.
Future Directions: The authors plan to extend this to full-shape modeling, constrained motion planning, and force estimation for tissue interaction, paving the way for safer autonomous surgical interventions.