💻 computer science

Inductance-Based Force Self-Sensing in Fiber-Reinforced Pneumatic Twisted-and-Coiled Actuators

This paper presents a self-sensing fiber-reinforced pneumatic twisted-and-coiled actuator that utilizes a conductive nickel wire and inductance feedback to achieve accurate, low-hysteresis force estimation through a hybrid observer combining an Extended Kalman Filter with constrained optimization, thereby overcoming the intrinsic hysteresis and proprioception limitations of traditional designs.

Original authors: Yunsong Zhang, Tianlin Li, Mingyang Yang, Feitian Zhang

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yunsong Zhang, Tianlin Li, Mingyang Yang, Feitian Zhang

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

The Big Idea: Giving Soft Robots a "Sixth Sense"

Imagine you have a very strong, stretchy rubber band that acts like a muscle. This is a Twisted-and-Coiled Actuator (TCA). It's great for soft robots because it's light, flexible, and powerful. But there's a problem: it's "blind."

When you squeeze it with air pressure, it contracts. But if you don't have a camera or a heavy sensor attached to it, the robot has no idea how hard it's pulling or how far it has stretched. It's like trying to lift a heavy box in the dark; you don't know if you're lifting 5 pounds or 50 pounds until you drop it.

The Solution: The researchers at Peking University figured out how to turn the muscle itself into a sensor. They made the "muscle" smart enough to feel its own strength without needing any extra gadgets attached to it.

1. The "Magic Wire" (The Hardware)

Usually, these rubber muscles are reinforced with a strong nylon thread to keep them from bursting. The researchers swapped this nylon thread for a conductive nickel wire.

The Analogy: Think of the muscle as a spring. If you wrap a copper wire around that spring, the whole thing becomes a giant coil (like an inductor in a radio).
What happens: When the muscle stretches or squeezes, the shape of that wire changes slightly. Just like stretching a spring changes how it bounces, changing the wire's shape changes its electrical inductance (a measure of how it stores magnetic energy).
The Trick: By measuring this tiny electrical change, the robot can "feel" what the muscle is doing.

2. The "Hysteresis" Problem (The Confusing Memory)

Here is where it gets tricky. In the world of soft robots, things have "memory." This is called hysteresis.

The Analogy: Imagine a rubber band. If you stretch it to 10 inches, the force you feel is different than if you pull it back from 11 inches to 10 inches. The rubber band "remembers" if you were stretching it out or letting it snap back.
The Problem: If you try to guess the robot's length just by looking at the electrical signal, you get confused. The signal might say "10 inches," but is the robot stretching out to 10 inches, or shrinking back to 10 inches? It's a dead end.

3. The "Aha!" Moment (Force vs. Length)

The researchers discovered a secret shortcut. They found that while the electrical signal is confused about length, it is crystal clear about force.

The Metaphor: Imagine a person holding a heavy backpack.
- If you ask, "How far did you walk?" (Length), the answer is fuzzy because they might be walking forward or backward.
- But if you ask, "How heavy is the backpack?" (Force), the answer is always the same, no matter which way they are walking.
The Discovery: The electrical signal (inductance) is a perfect mirror of the force the muscle is pulling, even when the muscle is stretching and shrinking. It ignores the confusing "memory" of the length and focuses purely on the tension.

4. The "Smart Brain" (The Hybrid Observer)

Now that they know the signal tells them the force, they need to figure out the length and make sure the robot moves smoothly. They built a "Smart Brain" (a mathematical algorithm) to do this.

The Analogy: Imagine you are driving a car in thick fog. You can't see the road (the length), but your engine sound (the inductance) tells you exactly how hard the engine is working (the force).
The Brain's Job:
1. Listen: It hears the engine sound (measures inductance).
2. Guess: It knows the engine sound means "500 lbs of force."
3. Calculate: Using a map of how the car usually drives (a physics model), it calculates: "If the engine is working this hard, we must be going up a hill at 30 mph."
4. Correct: It uses a "filter" (like a noise-canceling headphone) to ignore static and guess errors, ensuring the robot doesn't get dizzy or confused.

5. The Results: Blindfolded Success

They tested this new "blind" robot against robots with expensive, heavy sensors attached.

The Test: They made the robot pull on a weight, stretch, and shrink, sometimes changing the weight suddenly.
The Outcome: The "blind" robot (using only the wire) performed almost exactly as well as the robot with the heavy external sensors. It could hold a specific weight perfectly and follow a moving target accurately.
Why it matters: This means soft robots can be lighter, cheaper, and more flexible because they don't need to carry heavy, rigid sensors. They can "feel" themselves.

Summary

The paper is about giving soft robots a superpower: the ability to feel their own strength using a special wire woven into their skin. By realizing that this wire is great at sensing "how hard" the robot is pulling (even if it's bad at sensing "how far" it has moved), the researchers built a smart computer program that uses that "hardness" to guess the position. This allows soft robots to move precisely without needing bulky, expensive equipment attached to them.

1. Problem Statement

Fiber-Reinforced Pneumatic Twisted-and-Coiled Actuators (FR-PTCAs) are promising soft robotic actuators due to their high power density, compliance, and energy efficiency. However, their practical deployment in closed-loop control systems is hindered by two main issues:

Lack of Intrinsic Proprioception: They lack built-in sensing capabilities, requiring external sensors (e.g., encoders, load cells) which add mass, stiffness, and complexity, compromising the soft robot's compliance.
Modeling and Sensing Challenges: Unlike thermal TCAs, pneumatic variants suffer from strong hysteresis in their force-length relationships. Furthermore, traditional self-sensing methods (e.g., resistance-based) often struggle with thermal drift or strong hysteresis, making accurate real-time state estimation difficult. Specifically, the relationship between inductance and actuator length is highly hysteretic and ambiguous, rendering direct length-based self-sensing unreliable.

2. Methodology

A. Design and Fabrication

The authors modified the standard FR-PTCA fabrication process by replacing the conventional non-conductive nylon monofilament with a conductive nickel wire.

Structure: The actuator consists of a TPU tube reinforced by a helically wrapped nickel wire, which is then twisted and coiled.
Mechanism: The nickel wire serves a dual purpose: it provides structural reinforcement (maintaining anisotropy) and acts as an inductive sensor coil.
Key Insight: While the inductance-length relationship is hysteretic, experimental characterization revealed that the inductance-force relationship is deterministic and exhibits low hysteresis at constant internal pressures. This is because the inductance couples more strongly to the internal mechanical stress (tension) of the reinforcement fiber than to the global axial strain.

B. Modeling

Dynamic Model: A control-oriented linear model was established relating Force ( $F$ ), Pressure ( $P$ ), and Length ( $x$ ): $F = k(x - x_0) + cP$ .
Self-Sensing Model: A parametric, data-driven model was developed to map Inductance ( $L$ ) to Force ( $F$ ) and Pressure ( $P$ ). The model accounts for non-linearities using pressure-dependent coefficients:
$L(F, P) = \lambda_1(P)F^{\lambda_2(P)}e^{\lambda_3(P)F^{\lambda_4(P)}} + \lambda_5(P)$
where coefficients are linear functions of pressure.

C. Hybrid Observer Design

To address the non-monotonic nature of the inductance-force curve (where one inductance value could correspond to multiple force states) and measurement noise, the authors developed a Nonlinear Hybrid Observer:

Extended Kalman Filter (EKF): Predicts the state based on a constant velocity model to ensure temporal continuity.
Constrained Optimization: An intermediate layer solves a non-linear inverse problem to estimate the force ( $F^*$ $F^{*}$ ) by minimizing a cost function comprising:
- Model Fidelity: Matching the predicted inductance to the measured value.
- Continuity Constraint: Penalizing deviations from the EKF's predicted trajectory to resolve ambiguity between rising/falling branches.
- Regularization: Stabilizing the solution near zero-gradient regions (peaks) to prevent noise-induced jumps.

D. Control Strategy

A closed-loop control framework was implemented using a PID controller with model-based feedforward compensation. The feedback signal is derived entirely from the inductance-based state estimates, eliminating the need for external sensors.

3. Key Contributions

Novel Actuator Design: Fabrication of a self-sensing FR-PTCA using conductive nickel wire, preserving mechanical compliance while enabling intrinsic inductive sensing.
Discovery of Low-Hysteresis Coupling: Identification and characterization of a deterministic, low-hysteresis relationship between actuator inductance and output force (at constant pressure), contrasting with the highly hysteretic inductance-length relationship.
Hybrid EKF-Optimization Observer: Development of a robust state estimation algorithm that resolves the non-monotonicity of the inductance mapping, enabling accurate real-time force and displacement estimation without external instrumentation.
Integrated Control Framework: Demonstration of a self-sensing closed-loop control system that achieves performance comparable to external sensor-based systems.

4. Experimental Results

State Estimation Accuracy:
- Force Estimation: The hybrid observer achieved a Root Mean Square Error (RMSE) of 0.1745 N (3.79% NRMSE) and a Mean Absolute Error (MAE) of 0.1358 N.
- Displacement Estimation: Indirectly inferred via the dynamic model, achieving an RMSE of 10.89 mm. While less accurate than force estimation due to unmodeled hysteresis in the stiffness model, it remains sufficient for control tasks.
Closed-Loop Control Performance:
- Compared to open-loop control, the self-sensing feedback significantly reduced tracking errors.
- Force Tracking: The self-sensing method reduced RMSE by 42.3% to 68.1% across various trajectories, performing nearly identically to systems using external load cells (difference in RMSE < 0.01 N).
- Displacement Tracking: Reduced RMSE by 51.5% to 73.3% compared to open-loop control.
Robustness: Under dynamic load perturbations (randomly added/removed weights), the self-sensing system rapidly tracked force changes with an error bounded within 0.09 N, demonstrating stability and lack of drift.

5. Significance

This work provides a robust solution for sensor-minimal soft robotics. By leveraging the unique electromechanical coupling of FR-PTCAs, the authors demonstrated that high-fidelity force estimation and effective closed-loop control are achievable without external rigid sensors. This preserves the fundamental compliance and lightweight nature of soft robots, addressing a critical bottleneck in the deployment of soft robotic systems for applications requiring precise interaction with unstructured environments. The proposed hybrid observer architecture also offers a generalizable approach for handling non-monotonic sensor mappings in other soft actuation systems.