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Learning Control as Enabling Layer for Embodied Intelligence Research explored with Soft Robotic Swimming in diverse Flow Speeds

This paper demonstrates that augmenting conventional PID control with a Linear Repetitive Learning Estimation Scheme (PID-LRLES) significantly improves the tracking accuracy and repeatability of soft robotic swimming across diverse flow speeds, thereby providing a robust enabling layer for future embodied intelligence research.

Original authors: Fabian Schwab, Federico Allione, Bingcheng Wang, Mohamed El Arayshi, Claudio Mucignat, Ivan Lunati, Cristiano Verrelli, Ardian Jusufi

Published 2026-06-23

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CC BY 4.0

Original authors: Fabian Schwab, Federico Allione, Bingcheng Wang, Mohamed El Arayshi, Claudio Mucignat, Ivan Lunati, Cristiano Verrelli, Ardian Jusufi

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 a robotic fish trying to swim in a river. In a calm, still pond, it's easy for the robot to wiggle its tail in a perfect, rhythmic pattern. But as soon as the water starts flowing, the current pushes against the robot's soft, squishy body. This makes it wobble, drift, and lose its rhythm, much like trying to walk in a straight line while someone keeps pushing you from the side.

This paper, by Schwab and colleagues, introduces a new "brain" for soft robotic fish that helps them stay on track even when the water is moving fast. Here is how they did it, explained simply:

The Problem: The "Stiff" Brain vs. The "Soft" Body

The researchers built a robot fish with a body made of soft silicone, similar to a real fish. To control it, they usually use a standard computer program called PID (think of this as a basic autopilot).

In still water: The PID autopilot works great. It tells the robot to wiggle, and the robot wiggles perfectly.
In flowing water: The PID autopilot gets confused. It tries to correct the robot's movements, but because the water is pushing back in a repeating pattern (every time the tail flicks), the robot starts to lag behind. It's like trying to dance to a song while someone keeps changing the tempo; the dancer (the robot) gets out of sync and starts stumbling. The faster the water flows, the worse the robot gets at following the rhythm.

The Solution: The "Memory" Brain (PID-LRLES)

The team added a special "learning" layer to the robot's brain, called PID-LRLES. You can think of this as giving the robot a short-term memory specifically for the water's rhythm.

How it works: Instead of just reacting to the mistake right now (like the basic PID does), this new system looks at what happened in the last few tail wiggles.
The Analogy: Imagine you are walking on a treadmill that suddenly starts moving faster. A basic walker would just stumble and try to catch up. A "learning" walker would notice, "Oh, every time I take a step, the belt pushes me back a little bit." So, on the next step, they subconsciously push a little harder before the belt even moves them. They learn the pattern of the push and cancel it out in advance.
The Result: The robot learns the specific "push" the water gives it every time it wiggles. After a few seconds (about 10 seconds or 12 wiggles), it adjusts its movements perfectly to cancel out the water's push, keeping its tail moving in a smooth, perfect rhythm regardless of how fast the water is flowing.

The Secret Ingredient: A "Feeling" Sensor

For this learning to work, the robot needs to know exactly how its body is bending. The researchers attached a soft, flexible sensor directly to the robot's spine.

The Metaphor: Think of this sensor like the proprioception in your own body (the sense that tells you where your arm is without looking at it).
Without this sensor, the robot would be guessing how the water is affecting it. With the sensor, the robot "feels" the water pushing on its body in real-time. This feeling allows the "memory" part of the brain to learn the exact pattern of the water's push and correct it.

What They Found

The researchers tested the robot in a water tunnel, increasing the water speed from a calm pool to a fast stream (up to 32.6 cm/s).

One-Time Tuning: They set up the robot's controls in still water and never changed the settings again. They didn't have to re-tune the robot for every new water speed.
Better Accuracy: The robot with the "learning" brain stayed much closer to its perfect swimming rhythm than the robot with the basic brain.
Consistency: The basic robot's performance got worse and more unpredictable as the water got faster. The learning robot stayed steady and reliable, no matter how fast the water was.

Why This Matters

The main goal of this research wasn't just to make a better swimming robot; it was to create a reliable test platform.

Scientists use these robots to study how different tail shapes (like those of ancient extinct fish) affect swimming.
If the robot's swimming is wobbly and inconsistent because of the water, scientists can't tell if a difference in performance is due to the tail shape or just because the robot was having a bad day.
By using this "learning" control, the robot's movements become so consistent that scientists can confidently say, "This tail shape is better because of its shape, not because of random wobbles."

In short, the paper shows that by giving a soft robot a "memory" of the water's rhythm and a "feeling" of its own body, we can make it swim with the same stability and confidence as a real fish, even in a rushing river. This creates a solid foundation for future experiments on how animals swim and how we can build better underwater machines.

Technical Summary: Learning Control as an Enabling Layer for Embodied Intelligence in Soft Robotic Swimming

Problem Statement
Soft robotic platforms are valuable for studying body-caudal undulatory locomotion and testing biomechanical hypotheses (robophysics). However, their compliant bodies are difficult to control precisely under changing hydrodynamic loading. Conventional proportional–integral–derivative (PID) feedback can stabilize periodic undulation in static water but fails in dynamic flow. Specifically, PID accumulates flow-dependent tracking delays and increasing inter-trial variability due to system nonlinearities. This variability obscures the effects of morphology and environmental flow, making it difficult to attribute differences in thrust or stability to specific physical parameters rather than control-induced noise. While learning-based control has emerged as a complement to PID for soft robots with uncertain parameters, previous implementations of Linear Repetitive Learning Estimation Schemes (LRLES) were validated only in static water. It remained unclear whether such schemes could maintain tracking accuracy and repeatability under dynamic, hydrodynamic loading.

Methodology
The authors evaluated a control architecture augmenting a standard PID controller with a Linear Repetitive Learning Estimation Scheme (PID-LRLES) on a soft robotic fish platform.

Platform: The robot features a compliant body-caudal structure actuated by paired soft pneumatic muscles and instrumented with an embedded soft capacitive bending sensor. The sensor provides real-time proprioceptive feedback at 1 kHz, measuring the actual body deformation rather than relying on open-loop commands.
Control Architecture:
- Baseline: A conventional fixed-gain PID controller tuned in static water.
- Proposed: PID-LRLES, which adds a learning term in parallel to the PID block. This scheme generalizes classical integral action to periodic (non-constant) references. It utilizes a finite-dimensional state-space realization (using Padé approximations for delays) to estimate and compensate for periodic disturbances cycle-to-cycle. The learning transfer function is designed with poles having negative real parts to avoid the long-term instability typical of classical repetitive controls.
Experimental Setup: Experiments were conducted in a recirculating flow tank at five bulk flow speeds ranging from 0 to 32.6 cm s⁻¹ (Reynolds numbers up to ~7.5 × 10⁴).
Protocol: Controller gains were tuned once in static water and held fixed across all flow conditions to isolate the contribution of the learning term. Each trial involved a 30-second closed-loop stage where the robot tracked a periodic bending-envelope reference. Five independent repetitions were recorded per condition for both controllers.
Metrics: Tracking accuracy was quantified by the Root-Mean-Square Error (RMSE) of the bending envelope. Repeatability was assessed via the inter-trial spread of RMSE, analyzed using a paired Wilcoxon signed-rank test.

Key Contributions

Validation under Dynamic Flow: The study provides a full evaluation of PID-LRLES under controlled dynamic flow, demonstrating that the learning control improves not only tracking accuracy but also the robustness and repeatability of swimming experiments.
Integration of Soft Proprioception: The work integrates embedded soft proprioceptive feedback (capacitive bending sensor) into a closed-loop control system, allowing the controller to observe true body deformation and the periodic hydrodynamic bias induced by the flow.
Reduction of Variability: The authors demonstrate a significant reduction in tracking-error variability compared to conventional PID control, even with fixed gains.
Enabling Layer for Robophysics: The approach is positioned as an enabling layer for future robophysical studies, allowing researchers to isolate the effects of morphology, sensing, and environmental flow by minimizing control-induced variability.

Results

Tracking Accuracy: While the PID baseline maintained stable oscillation, it exhibited a growing tracking delay and amplitude deficit as flow speed increased (peak deviation rising from ~0.02 rad at 0 cm/s to ~0.04 rad at 32.6 cm/s). In contrast, PID-LRLES tracked the reference more closely across all speeds, maintaining the envelope within ±0.015 rad of the reference even at the highest flow speed after a transient period.
Error Variability: The mean RMSE for PID-LRLES was lower than PID at every flow speed. Crucially, the PID baseline showed a monotonic increase in mean RMSE with flow speed, whereas PID-LRLES showed a much flatter dependence (rising only from 0.015 to 0.018 rad).
Statistical Significance: A paired Wilcoxon signed-rank test on the per-trial RMSE distributions (pooled across five flow speeds, $n_{pairs}=25$ ) confirmed that PID-LRLES significantly reduced tracking-error variability compared to the PID baseline ( $p = 1.8 \times 10^{-4}$ ).
Robustness: The learning term effectively absorbed the recurring component of the hydrodynamic disturbance, removing the trial-to-trial drift that the fixed-gain PID could not compensate for.

Significance and Claims
The paper claims that by reducing flow-dependent control-induced variability, the PID-LRLES approach provides a robust enabling layer for robophysical studies. Specifically:

It allows observed differences in thrust, pitch torque, or swimming stability to be confidently attributed to morphology or environmental flow rather than control artifacts.
It demonstrates that learning-enhanced closed-loop control can function effectively with gains tuned only in static water, removing the burden of per-flow recalibration.
It highlights the complementary nature of soft sensing and repetitive learning: the sensor exposes the periodic hydrodynamic bias, while the learning term absorbs it over past oscillation cycles.
The architecture is computationally efficient (22 states), allowing deployment on low-power, embedded real-time hardware without GPU support, making it suitable for untethered autonomous soft robots.

The authors note limitations, including testing at a single flapping frequency, the use of a rigid mast (precluding free-swimming tests), and the restriction to envelope tracking rather than direct thrust or efficiency metrics. However, they assert that the demonstrated kinematic repeatability is a necessary prerequisite for reliable hydrodynamic comparisons across different morphologies.