Feedback Control of a Modular Proprioceptive Soft Actuator

Abstract

A PneuNets actuator consists of a network of interconnected air chambers embedded in an elastomeric substrate attached to a stretch-limiting layer. It is one of the most common soft robotic bending actuators. Despite the versatility of PneuNets actuators in tasks like locomotion and grasping, precise control of their bending motion requires proprioceptive feedback. In this study, we introduce a modular design for integrating a soft sensor into a PneuNets actuator and couple it with a closed-loop control algorithm to achieve proprioceptive feedback. We embedded an off-the-shelf soft capacitive bend sensor into the strain-limiting layer of the actuator to measure angular displacement. Then, we employed a calibrated proportional-integral-derivative controller and tuned it with image-based linearization to accurately control the actuator to follow an input angle trajectory. The proposed approach enhances control precision and promotes modularity, supporting the development of sustainable soft robots with reusable and interchangeable components.

Introduction

Fluidic actuation is one of the most common actuation modalities used in soft robots and has been developed in different configurations. One key development is PneuNets actuators with an embedded network of channels in elastomers. ¹ The simplicity and versatility of PneuNets offered researchers a platform for developing new materials and fabrication methods for soft robots. While the compliance of soft materials simplifies the control of soft actuators for various tasks, truly autonomous soft robots require a sense of their own physical state to better interact with their surroundings. ² To address this challenge, proprioception and exteroception are realized in PneuNets actuators by integrating stretchable sensors comprised of liquid metals, ³ electroluminescent materials, ⁴ optical waveguides, ⁵ and embedded fluidic channels. ⁶ However, these sensorized actuators require specialized materials and advanced fabrication facilities.

We propose a modular proprioception solution for PneuNets actuators to create affordable soft robotics tools using off-the-shelf materials and components. The working principle of PneuNets bending actuators relies on the asymmetrical elongation of two opposite planes with contrasting stiffness around a central network of interconnected pressurized chambers. ¹ When inflated, the stiffer layer, known as the strain-limiting layer, restricts the elongation of the softer layer, known as the active layer, and the actuator bends inward to accommodate the differential strain. We harnessed the inextensibility of the strain-limiting layer to devise PneuNets actuators with housing for modular integration of a commercial bend sensor (BendLabs, USA) that simultaneously protects the sensor from overstretching. Then, we devised a proportional—integral—derivative (PID) controller that effectively drives the actuator’s angular position by linearizing the bend sensor feedback with visually tracked angles, facilitating precise closed-loop control.

Materials and Methods

Actuator fabrication

We followed the standard procedure for fabricating PneuNets actuators by casting two types of silicone elastomers into 3D-printed molds. ¹ For the active layer, we used a softer silicone elastomer (Ecoflex 00–30, Smooth-On) and for the strain-limiting layer, we used a vinylpolysiloxane with a Shore A32 hardness (Elite Double 32, Zhermack). First, Ecoflex 00–30 was mixed with a 1:1 weight ratio, degassed for 2 minutes using a vacuum chamber (easycomposites DS26-P), poured into the mold, and cured at ${45}^{°}$ C in a convection oven (Binder FD 56) for 20 minutes.

For having a modular design, we constructed a strain-limiting layer with an integrated enclosure that can house a sensor inside. For fabrication, a custom mold with a narrow opening that matches the geometry of the bend sensor was created. The resulting part has a cavity with three transversal bands to hold the bend sensor without applying excessive axial force during bending. The mold parts were produced with a stereolithography 3D printer (Formlabs Form 3) using the standard resin (Formlabs White). We mixed the base and the catalyst components of the Elite Double 32 silicone using a 1:1 weight ratio and subsequently degassed the mixture. The prepolymer mixture was then poured directly into the mold and cured in a convection oven at ${45}^{°}$ C for 10 minutes (Fig. 1a i).

To assemble the two parts, we applied a thin layer of Ecoflex 00-30 prepolymer to the surface of the strain-limiting layer and gently pressed the active layer against it and let the silicone cure at room temperature (Fig. 1a ii). Next, a hole was made at the end of the actuator using a wooden stick, silicone tubing was inserted, and fixed in place with silicone adhesive (Silpoxy, Smooth-on). Finally, the bend sensor was inserted into the enclosure (Fig. 1a iii-iv) (see Supplementary Video S1).

System architecture

The PneuNets actuator was controlled with a pneumatic system operated through a microcontroller board (Arduino Uno ATmega328P) and a personal computer (Intel Core i7 vPro 1.9 GHz). The pneumatic system operates in two phases for inflation and deflation. The air and vacuum pressure were generated by a micro diaphragm pump (mini air and vacuum pumps VN2708PM). We used two 3-way pneumatic solenoid valves (Parker X-valve, normally-closed) to control the airflow to and from the actuator. Solenoid valve 1 was used for the control of airflow during inflation phase, while solenoid valve 2 is used for the exhaustion. Both pump and valves were powered by a motor driver shield (DFRobot DRI0039) directly mounted on the microcontroller board. An integrated silicon pressure sensor (NXP Semiconductors MPX5100) was installed at the output of the main pneumatic channel to measure the internal pressure of the actuator. The bend sensor (Bend Labs Digital Flex Sensor, 1-Axis) was connected to the microcontroller through an I2C communication port with a bidirectional logic level converter (SparkFun BOB-12009). Silicone tubing (3 mm and 1.5 mm inner diameter) was used to respectively connect the actuator and the pneumatic components (pressure sensor, pumps, and valves) to the system using barbed connectors (Festo RTU-PK-2/3 and Y-PK-3) (Fig. 1b).

FIG. 1.

Proprioceptive soft actuator. (a) Materials and fabrication method, (i) silicone molding step, (ii) PneuNets actuator assembly, (iii) soft actuator with housing to hold the bend sensor, (iv) proprioceptive soft actuator, (b) Pneumatic and electronic system diagram, (c) Control system schematic, PID controller with linearized (dashed block) feedback. PID, proportional—integral—derivative.

We chose a bidirectional capacitive sensor made of medical-grade silicone (∼Shore A50 hardness) equipped with an integrated low-power analog frontend and I2C communication (1-Axis Bend Labs Digital Flex Sensor). The sensor outputs the angular displacement Δθ computed from the angle between two vectors tangential to the ends of the sensor. The slender geometry of this sensor allows for easy integration. The sensor has a lifetime of >1M cycles and a linear response with zero drift and produces repeatable ( ${0.18}^{°}$ repeatability) and precise angular output regardless of path, bending radius, and strain.

The firmware uploaded to the microcontroller was compiled in Arduino IDE and is responsible for controlling the pump speed and the state of the valves, acquiring data from the bend sensor and the pressure sensor, and communicating with the central processing unit through a serial port (115200 bps). The frontend software running on the computer is a Python script compiled in Spyder environment. This script generates the input trajectories for the soft actuator and logs sensor output data.

We used a tracking software ⁷ to trace the tip motion of the actuators from video recordings made with a single lens digital reflex camera (D3400, Nikon) to estimate the bend angle over time.

PID controller

In this work, we applied a PID controller to regulate the input voltage of the pump and activate solenoid valves to adjust the supplied airflow to the actuator in response to the measured bending angle by the sensor. The output of a PID controller is described by the formula: $$u\left[n\right]=K_{p}e\left[n\right]+K_{i}T{\displaystyle {\sum }_{k=0}^{n}e\left[k\right]}+\frac{K_d}{T}(e\left[n\right]-e\left[n-1\right])$$(1)

The controller was calibrated following the Ziegler–Nichols method ⁸ and then fine-tuned empirically by using a trial-and-error approach. The final values of the tuned PID gains were found to be: $K_p=5(1/\mathrm{deg})$ , $K_i=1$ (1/deg s), and $K_d=0.15$ (s/deg). The sample time of the system is $T=20$ milliseconds. The control sequence starts when the input trajectory block generates the desired angular profile ${\theta }_d\left[n\right]$ in a sequence of points. Then, the PID controller calculates the control action $u\left[n\right]$ that corresponds to the angular position error $e\left[n\right]$ . With this control action, the logical comparator defines the actions for the pneumatic system (valves and pump) to inflate and deflate the actuator. The control variables for the system regulate the voltage of the pump $V\left[n\right]$ from 0 to 5 volts, realized by adjusting the duty cycle of the pulse width modulation signal of the 8 bits microcontroller’s output pin in the range [0, 255], and the on/off states of the solenoid valves ${\phi }_1\left[n\right]$ and ${\phi }_2\left[n\right]$ . The measurement of the bend sensor ${\theta }_s\left[n\right]$ was postprocessed by an alpha filter ( $\alpha =0.9$ ), a digital low pass filter ${\theta }_f\left[n\right]$ that attenuates high-frequency noise, before estimating the position error defined as: $${\theta }_f\left[n\right]=\alpha {\theta }_s\left[n\right]+(1-\alpha ){\theta }_f\left[n-1\right]$$(2)

Finally, the control system output is the bend angle of the actuator $\theta \left[n\right]$ (Fig. 1c). We limit the maximum pressure at $p_{\mathit{max}}=27$ kPa and the maximum bending angle at ${{\theta }_{\mathit{max}}=150}^{°}$ to prevent the actuators from overinflating. After the calibration procedure, the controller was able to maintain a stable bending angle in the range of ${\theta =0}^{°}-{110}^{°}$ and respond to step changes in the desired angular positions.

Results and Discussion

To ensure the precision of our system, we followed a systematic performance evaluation. Initially, we compared the compliance of the actuator with and without the bend sensor by monitoring the pressure response of the actuator during inflation from ${\theta =0}^{°}$ to ${\theta =110}^{°}$ to prove that the bend sensor does not significantly alter the pressure response of the system, and the flexibility of the actuator is preserved (Fig. 2a). This result is remarkable because commercial resistive bend sensors commonly used in soft actuators are stiff, which restricts the deformation range of the actuators and can cause damage to the actuator and impede durability. ⁹ Next, we evaluated the PID controller when the proprioceptive actuator was set to follow predefined input trajectories. We considered an input trajectory, where the soft actuator followed a stepped ramp curve, rising from ${\theta =0}^{°}$ to ${\theta =110}^{°}$ and returning to ${\theta =0}^{°}$ in increments of ${\Delta \theta =10}^{°}$ . The duration of each step was 40 seconds.

FIG. 2.

Experimental results. (a) Pressure response of the actuator with and without the embedded bend sensor, (b) Actuator response to a ramp input using a PID controller, (c) Linear regression between tracker and sensor data, (d) Actuator ramp response to a PID controller with linearized feedback. (e) Actuator response for an arbitrary generated activation sequence, and sequential snapshots of the bending response. PID, proportional—integral—derivative.

We logged the filtered angular displacement ${\theta }_f\left[n\right]$ measured by the bend sensor over time and in parallel recorded a video of the displacement of the markers mounted on the tip of the actuator. From the video recordings, we tracked the angular displacement of the actuator ${\theta }_t\left[n\right]$ at the same time. We found that the controller successfully followed the desired trajectory with a mean absolute error $e_{\mathit{MAE}}^d={\sum }_{n=1}^{n=N}\left|{\theta }_d-{\theta }_f\right|/N={1.21}^{°}$ between desired and filtered sensed angle. However, comparing the filtered angular displacement measured by the sensor ${\theta }_f\left[n\right]$ , to the video recorded displacement ${\theta }_t\left[n\right]$ , shows a deviation with a larger absolute mean error of ${e_{\mathit{MAE}}^t={\sum }_{n=1}^{n=N}\left|{\theta }_t-{\theta }_f\right|/N=4.27}^{°}$ , which is more pronounced for ${\theta >70}^{°}$ (Fig. 2b). This deviation, particularly at larger angles, is caused by the inherent nonlinearity of actuator that may press the sensor inside the enclosure and affect the measurements (see Supplementary Video S2). In the next section, we will overcome this limitation through a linearization process.

To rectify the disparity between sensor ${\theta }_f\left[n\right]$ and tracker ${\theta }_t\left[n\right]$ measurements, we conducted a linear regression analysis using the curve fitting toolbox in MATLAB. The corrected angle is obtained by a linear function ${\theta }_l\left[n\right]=a{\theta }_f\left[n\right]+b$ that relates the filtered sensed angle ${\theta }_f\left[n\right]$ and the tracked angle ${\theta }_t\left[n\right]$ with determined coefficients $a=1.15$ , and $b=-5.18$ (Fig. 2c). Following this evaluation, we repeated the same activation sequence (desired input shown in Fig. 2b) with linearization. The results were compelling, the actuator followed the desired trajectory, exhibiting a mean absolute error $e_{\mathit{MAE}}^d={\sum }_{n=1}^{n=N}\left|{\theta }_d-{\theta }_l\right|/N={1.21}^{°}$ between the setpoint and the linearized values. Moreover, this linearization showcases a notable reduction in the absolute mean error between the linearized sensor measurements ${\theta }_l\left[n\right]$ and the tracked positions ${\theta }_t\left[n\right]$ , now reduced to a mere ${e_{\mathit{MAE}}^t={\sum }_{n=1}^{n=N}\left|{\theta }_t-{\theta }_l\right|/N=1.23}^{°}$ (Fig. 2d and Supplementary Video S2). Next, using the same linearization parameters tuned for the range of angles from ${\theta =0}^{°}$ to ${\theta =110}^{°}$ , we carried out another activation sequence with arbitrary step changes of the desired angle ${\theta }_d\left[n\right]$ within $0^{°}$ to ${80}^{°}$ range at 30 seconds intervals. Our analysis revealed a mean absolute error $e_{\mathit{MAE}}^d={\sum }_{n=1}^{n=N}\left|{\theta }_d-{\theta }_l\right|/N={2.9}^{°}$ between the desired and the linearized angle, and with a disparity mean absolute error $e_{\mathit{MAE}}^t={\sum }_{n=1}^{n=N}\left|{\theta }_t-{\theta }_l\right|/N={1.63}^{°}$ between linearized and tracked data (Fig. 2e) (see Supplementary Video S3). These results confirm the algorithm’s robustness across diverse conditions and the applicability of the proposed compensation method to arbitrary input trajectories within the calibration range $\theta \in \left[0°,110°\right]$ (Fig. 2e).

Conclusion

We developed a sensorized soft actuator with integrated modular proprioception as a platform for closed-loop control of soft robots. A capacitive bend sensor was embedded in the strain-limiting layer of a PneuNets soft actuator, while an integrated pressure sensor measured the internal pressure during actuation. A PID controller with a linearization compensation strategy was employed to control the angular displacement of the soft actuator with high repeatability under ramped and arbitrary activation sequences. The work presented here demonstrates an easy-to-implement approach using off-the-shelf components to control soft robots. The control method relies on a purely data-driven approach that utilizes sensor outputs without the analytical description of the system dynamics, which is highly complex due to the nonlinear nature of soft materials and large deformations. The proposed platform is an accessible testbed for developing more advanced control algorithms. ¹⁰ Moreover, through a modular design approach, our work contributes towards developing sustainable soft robots with reusable and replaceable components. However, the scalability of our approach may be influenced by the form factors, size, and sensor technology, which may require adjusting the design of the actuator and identifying calibration parameters.