A retrofit sensing strategy for soft fluidic robots

Soft robots are intrinsically capable of adapting to different environments by changing their shape in response to interaction forces. However, sensory feedback is still required for higher level decisions. Most sensing technologies integrate separate sensing elements in soft actuators, which presents a considerable challenge for both the fabrication and robustness of soft robots. Here we present a versatile sensing strategy that can be retrofitted to existing soft fluidic devices without the need for design changes. We achieve this by measuring the fluidic input that is required to activate a soft actuator during interaction with the environment, and relating this input to its deformed state. We demonstrate the versatility of our strategy by tactile sensing of the size, shape, surface roughness and stiffness of objects. We furthermore retrofit sensing to a range of existing pneumatic soft actuators and grippers. Finally, we show the robustness of our fluidic sensing strategy in closed-loop control of a soft gripper for sorting, fruit picking and ripeness detection. We conclude that as long as the interaction of the actuator with the environment results in a shape change of the interval volume, soft fluidic actuators require no embedded sensors and design modifications to implement useful sensing.

objects. Moreover, we demonstrate our approach by retrofitting it to a range of existing pneumatic soft actuators and grippers powered by positive and negative pressure. Finally, we show the robustness of our fluidic sensing strategy in closed-loop control of a soft gripper for practical applications such as sorting, fruit picking and ripeness detection. Based on these results, we conclude that as long as the interaction of the actuator with the environment results in a shape change of the interval volume, soft fluidic actuators require no embedded sensors and design modifications to implement useful sensing. We believe that the relative simplicity, versatility, broad applicability and robustness of our sensing strategy will catalyze new functionalities in soft interactive devices and systems, thereby accelerating the use of soft robotics in real world applications.
The intrinsic compliance of soft robots provides adaptability to unknown environments (1)(2)(3). For example, a soft robotic gripper passively adapts its body shape, making it possible to grasp various objects without the need for active sensing (4,5). However, when it comes to more advanced tasks such as identifying and sorting objects, sensory feedback from the gripper becomes essential to achieve closed-loop control in gripping and manipulation (6). Benefiting from advances in soft materials, soft robotic sensing has been enabled by embedding flexible or stretchable sensors made from piezoresistive and piezocapacitive polymer composites (7), liquid metals (8), electrically and ionically conductive hydrogels (9), and polymeric optical waveguides (10). Both proprioception (sensing of self-deformation) and exteroception (sensing of external stimuli) of soft robots have been successfully demonstrated with embedded sensors.
Moreover, multimodal sensing, i.e., the simultaneous perception of multiple physical parameters, has been achieved by machine learning (11,12) and embedding various sensors into the soft actuator (13)(14)(15). A common feature in all these sensing strategies for soft robotic applications is the separation of actuation and sensing elements (16,17). This is a result of the compliance of the soft systems, which complicates integration and reduces reliability of the sensors that need to be embedded in the soft actuator, therefore placing considerable constraints on the design of both the sensors and actuators.
As fluidic actuation represents a plurality in soft robotics (18), sensing strategies based on fluidic media, either gas or liquid, have been investigated to reduce the integration difficulties of actuation and sensing elements, such as fluidic resistance sensing (19,20), fluidic pressure sensing (21-32) and electrical resistance sensing (33). Most fluidic sensing strategies incorporate an additional cavity to the soft actuator (21, 24-26, 28, 29). Since the enclosed cavity contains a fixed amount of fluid, deformation of the actuator or contact with the environment changes the volume of the cavity and thus increases or decreases the internal pressure. Interestingly, this pressure response can be measured remotely by connecting the cavity and electronic pressure sensor via a tube, such that no electronic components need to be embedded in the soft actuator.
A particularly interesting yet simple method uses the cavity of the soft actuator itself to sense external force by measuring and analyzing the fluidic pressure of the soft actuator (34,35).
The benefit of such a self-sensing approach has also been demonstrated in dielectric elastomer actuators (36-38) and electrohydraulic actuators (39-41). In these systems the electrical characteristics of the actuator can be measured to infer the mechanical deformation while it is being actuated, hence no additional sensors and associated electronics are needed (37). While fluidic self-sensing has originally been demonstrated in a potential medical application (34), a natural question to ask is how widely applicable, versatile and robust such an approach is. To answer this question, we need to gain a better understanding of the underlying principles that allow for fluidic self-sensing, and determine if we can infer the interaction of a wide variety of soft actuator with their environment by measuring and analyzing the fluidic response of the enclosed cavity. And if so, we want to determine how easy it is to integrate and retrofit such a sensing approach, and if interactions with the environment can be robustly measured.
To achieve this, in this work we will first experimentally show how the response of a typical soft fluidic bending actuator changes when interacting with the environment. We next introduce several strategies to sense these interactions without the need to embed additional sensing elements in the soft actuator. We demonstrate how to apply our fluidic sensing strategy to a soft gripper, and to enable a versatile range of sensing applications such as size, shape, surface roughness and stiffness sensing of objects. To demonstrate that the sensing approach can be retrofitted, we apply the sensing strategy to a filament actuator, a McKibben actuator, a thermoplastic polyurethane (TPU) actuator, a soft suction gripper specifically designed for medical applications and two commercially available soft grippers. We furthermore developed a basic model based on a linear extension actuator to study the underlying factors that determine the sensing resolution. Finally, we show that our fluidic sensing strategy is robust enough to implement closed-loop control in gripping and sorting applications.

Fluidic sensing of the soft robot-environment interaction
We start by looking into the characteristic behavior of a typical soft PneuNet bending actuator (42) when interacting with the environment. We inflate a soft actuator onto a rigid plate from different heights h, and characterize the pressure-volume response for each height, i.e., the pressure P as a function of supplied air volume V , at different heights (fitting curves in Importantly, by definition any physical interaction with the environment leads to a change in the internal geometric volume of a soft actuator because of the compliance of the soft body. In order to effectively sense these differences in the pressure-volume response, and with that the interaction of the soft actuator with the environment, we connect the soft actuator to a pressurized air tank via a solenoid valve and measure the equilibrium pressure using an external pressure sensor after opening of the valve (orange line in Fig. 1a). Depending on the initial distance h of the actuator to the surface, the equilibrium pressure P eq will be slightly different. Interestingly, for this specific actuator design and interaction, the relationship between the equilibrium pressure P eq and height h can be approximated by a linear relationship. Therefore, the initial distance between the actuator and plate can be inferred from the fluidic signal based on the calibration curve ( Fig. 1b and Fig. S2a) with an accuracy of ±1.7 mm (Fig. S3a). The force applied by the actuator on the plate can also be inferred from the equilibrium pressure P eq (Fig. S4). Note that since the steel air tank has a linear pressure-volume relationship, varying the tank size effectively changes the slope of the tank's pressure-volume curve. This changes the intersection points with the actuator's pressure-volume curves in Fig. 1a, making it possible to tune the sensing resolution (Fig. S5).
While here we connect the actuator to a steel tank and solenoid valve to implement sensing, depending on the available equipment and precision requirement, the interaction of a soft actuator with the environment can also be characterized by controlling the pressure and measuring the volume flow input (blue line in Fig. 1a, Fig. 1c and Fig. S2b) with a sensing accuracy of ±4.7 mm (Fig. S3b), or controlling the volume flow input and measuring pressure (red line in Fig. 1a, Fig. 1d and Fig. S2c) with a sensing accuracy of ±3.4 mm (Fig. S3c).
Our sensing strategy can also be directly applied to a soft gripper to sense the size of objects. We demonstrate size sensing of cylindrical objects using a soft gripper consisting of four 6 PneuNet bending actuators (Fig. 1e, Fig. S6 and Supplementary Video 1). To enable sensing, the actuators are jointly connected to the external system that contains an air tank, a solenoid valve and a pressure sensor. Fig. 1f shows that when gripping larger objects, a higher equilibrium pressure P eq is reached. Interestingly, the relationship between the equilibrium pressure P eq and cylinder diameter D can also be fitted with a linear function within the grasping range of the gripper (d ⪆ 20 mm). Repeated tests on a similar gripper showed the same results, where we found that the pressure measurements vary within ± 0.08 kPa over 100 cycles (Fig. S7).

Time-enabled sensing versatility
Having demonstrated the basic principles of our sensing approach, we next show that versatile sensing applications can be achieved by measuring the pressure response of the soft actuator over time. To show how we can extract more information from the pressure-time response, we first revisit height sensing where so far we only considered the equilibrium pressure at a specific moment in time (Fig. 1). Instead, if we correlate the pressure-time response of the actuator to a reference response, i.e., free actuation without interacting with the environment, we can determine the moment of contact ( Fig. 2a-b). In Fig. 2b we evaluate ∆P = P − P ref over time from the onset of actuation and determine the time of first contact t c1 when ∆P > 0.
We can then use t c1 to infer the initial distance h between the actuator and plate (Fig. 2c) based on the tip displacement-time curve of the actuator in the reference response, with an accuracy of -2.9 to 3.8 mm (Fig. S3d). Note that the sensing speed of this strategy is dominated by the actuation speed, which is determined by the flow resistance between the air tank and actuator ( Fig. S8). Interestingly, a higher sensing response speed can be achieved by measuring the time of contact t c1 (Fig. 2c) compared to the equilibrium P eq (Fig. 1b), because the measurement of t c1 does not require the system to reach equilibrium.
Based on this approach, we show how we can sense (i) the shape of objects, (ii) the stiffness Figure 2: Time-enabled sensing versatility. a-c, Inferring the initial distance h between the actuator and plate from the time of first contact t c1 . h = 20 mm in a and b. The dashed line in c represents the vertical displacement-time curve of the tip of actuator in the case of free actuation. d, e, Shape sensing of rectangular objects by measuring both the time of first and second contact of the soft gripper. f, g, Stiffness sensing by measuring both the time of first contact and the equilibrium pressure, which indicate the vertical displacement of the actuator at the first contact (h c = h) and equilibrium (h e ≥ h depending on the object stiffness), respectively. h = 5 mm in g. h, i, Surface scanning and profile reconstruction of two different surface profiles. of a soft substrate and (iii) the profile of a surface. As a first demonstration of the sensing versatility, we use our previously introduced gripper to sense the aspect ratio of rectangular objects. response when the soft gripper grips a rectangular object, indicating the length and width of the object, respectively. While this could also be achieved by individually addressing each actuator, which would likely make it easier to extract shape information from the soft gripper, it would also require additional hardware that might not be needed or available in specific applications.
As a second demonstration of the sensing versatility, we achieve multimodal sensing of distance and stiffness when the actuator interacts with a soft plate (Fig. 2f). Here, the time of first contact t c1 extracted from the pressure-time response (Fig. 2g, Supplementary Video 2) indicates the initial distance between the actuator and plate, while the equilibrium pressure P eq indicates the final vertical displacement of the actuator. By comparing these two displacements, we can extract the indentation depth of the soft actuator, which can be correlated to the stiffness of the plate when considering the stiffness of the soft actuator. Therefore, measuring t c1 and P eq together makes it possible to compare the stiffness values of objects (Fig. 2g).
As a final demonstration of the sensing versatility, we use the actuator as a profilometer by considering the variations in the equilibrium pressure when moving the actuator along a surface ( Fig. 2h-i and Supplementary Video 2). To show this, we move the actuator horizontally along a surface with a robotic arm and measure the pressure response continuously. The profile of the object can be reconstructed using a calibration curve (Fig. 1b) and a reference pressure response, which rules out the influence of system leakage or other variations over time. Note that in this sensing application the sharpness of the tip of the soft actuator will determine the resolution of the sensing signal. Figure 3: Retrofitting the fluidic sensing approach to a filament actuator (a-c), a McKibben actuator (d-f) and a 3D-printed bending actuator (g-i). The filament (a) and McKibben (d) actuator are used as a muscle to rotate an arm towards a stopper. TPU bending actuator (g) wrapping around a cylinder with a diameter D. Corresponding pressure-volume relation for the soft actuator (solid) and tank (dashed) (b, e, h) and equilibrium pressure in the system (c, f, i) for different positions of the stopper or cylinder diameter D. Experimental results from five tests are shown for each θ (b, e) and each D (h). Scale bars, 30 mm. Figure 4: Retrofitting the fluidic sensing approach to a suction cup (a-e) and two commercial soft grippers (f-k). a, Front and bottom views of the suction gripper. Scale bar, 10 mm. b, Schematic of the suction gripper attaching to a soft object. c, Smoothened pressure-volume responses of the suction gripper (pink) and air tank (blue) when attaching to silicone samples with different shore moduli. d, Experimental sensing results obtained using two different initial pressures P 0 in the air tank. e, Force-pressure responses from three pulling tests on the suction gripper when attached to a silicone sample with a shore hardness of OO-30. Commercial vacuum (f) and pressurized (i) grippers gripping a cylindrical object with a diameter D. Scale bar, 50 mm. Corresponding pressure-volume relation for the soft gripper (solid) and tank (dashed) (g, j) and equilibrium pressure in the system (h, k) for objects. Experimental results from five tests are shown for each D (g, j). 11

Retroffiting the fluidic sensing approach
In all demonstrations so far, we used one or more identical soft bending actuator. However, our sensing approach can also be retrofitted to a broad range of fluidic actuators without the need for any design changes. To demonstrate the wide applicability, using our approach we sensorize a filament actuator, a McKibben actuator, a 3D-printed bending actuator (43), a suction cup, and two commercial soft grippers (Fig. 3, Fig. 4  To determine if our sensing approach can also be used for higher actuation pressures, we next retrofit our sensing strategy to a 3D-printed TPU bending actuator that requires an actuation pressure around 200 kPa (43). In previous tests with the bending actuator, we only consider a single contact between the soft actuator and the environment. Since the TPU bending actuator forms a circular shape at higher pressures (43), we tested our sensing strategy with conformal grasping (49), where the soft actuator interacts with the cylindrical object at multiple contact points (Fig. 3g). We find that the conformal grasping of cylindrical objects with various diameters results in different pressure-volume responses of the soft actuator (Fig. 3h) and that we can also correlate the equilibrium pressure with the diameter of the grasping object, even for these higher pressure ranges (Fig. 3i).
To test our retrofitting approach with soft grippers, we first apply it to a suction cup (Fig. 4a) that was specifically designed for tissue gripping in Minimal Invasive Surgery (MIS) (50). The requirements for the foldability, adaptability and biocompatibilty of the tissue gripper make it challenging to embed sensors in the gripper itself to obtain sensory feedback during operation.
With our fluidic sensing strategy, the pressure sensor can be connected remotely to the gripper outside of the human body and no additional design change of the gripper is needed. Once vacuum is applied to the soft gripper, the connected surface gets pulled into the gripper, reducing its internal geometric volume (Fig. 4b). The surface stiffness influences the pressure-volume response of the gripper through the amount of reduced internal geometric volume of the gripper (fitting curves in Fig. 4c, and test results in Fig. S9). The final equilibrium pressure P eq can be used to infer the stiffness of the surface that is attached to the gripper, where the sensing resolution can be tuned by the initial pressure P 0 in the air tank ( Fig. 4d and Fig. S10). Furthermore, we found that the equilibrium pressure in the gripper changes almost linearly with the pulling force applied on the gripper ( Similarly, we demonstrate that we can retrofit sensing to commercial soft grippers ( Fig. 4fk), both to grippers powered by vacuum pressure (51) and by positive pressure (52). Even though the pressure-volume relation for both grippers is relatively different, in both cases we find a linear correlation between the size of cylindrical objects and the measured equilibrium pressure. Note that because the internal volumes of the actuators and grippers are different in these examples, we had to replace some of the external hardware. For example, the internal volume of the air tank, which determines the slope of the tank's pressure-volume response, is chosen based on a compromise between initial tank pressure and sensing resolution (Fig. S5).
Characterizing the sensing resolution  Table 1). Even though the relative pressure difference (pressure difference due to interaction with the environment in comparison to maximum pressure obtained in the actuator) might be smaller for actuators that require higher inflation pressure (e.g., TPU and filament actuators), the absolute pressure change for all actuators we tested is in the same order of magnitude ( Table 1).
Note that the average sensing resolution can be determined by dividing the absolute pressure change by the tested range of sensing target and is therefore not affected by a lower relative pressure difference.
In order to find out the underlying factors that determine the sensing resolutions of the soft actuators and grippers, we consider both the initial and final states of the system. At the initial state, the air tank with an internal geometric volume v tank is pressurized at p tank = p 0 , and the actuator with an internal geometric volume v act = v 0 is at atmosphere pressure p act = p atm . At the final state, the air tank and the actuator reach the same pressure p tank = p act = p 1 , and the internal geometric volume of the actuator becomes v act = v 1 . Assuming constant temperature, since the total amount of air mass inside the system (air tank and the actuator) stays constant, according to the ideal gas law, we have Note that the absolute pressure here is indicated in lowercase letter to distinguish it from the relative pressure (with respect to atmospheric pressure) that is used elsewhere in the manuscript.
When the total amount of air inside the system remains constant, v 1 only depends on the inter-action of the soft actuator with the environment, i.e., the sensing target ξ. Therefore, the sensing resolution dp 1 /dξ can be written as where, for example for the gripping test of the cylinders ξ = D, i.e., the diameter of the cylindrical objects in Fig. 4f and i. Moreover, dv 1 /dξ represents the sensitivity of the internal geometric volume of the gripper to gripping cylindrical objects with different diameters. Equation 2 shows that essentially, the variation of internal geometric volume when the soft actuator interacts with the environment in different ways causes the pressure change, which can be used to infer the interaction. However, it is not trivial to compare the sensing resolutions of the soft actuators and grippers in Fig. 3 and Fig. 4, because these actuators vary in actuation pressure, internal volume and sensing targets. To give an example, since the initial internal geometric volumes v 0 of both commercial grippers in Fig. 4f and i are known, we can determine v 1 at equilibrium from equation 1 based on experimental measurements of p 1 (Table S1 and Fig. S11), from which we can then obtain dp 1 /dD according to equation 2. We find that the sensitivity of the internal geometric volume of the gripper to gripping different cylindrical objects equals |dv 1 /dD| = 0.16 ml/mm for the vacuum gripper and |dv 1 /dD| = 0.13 ml/mm for the PneuNet gripper, while the magnitude of the sensing resolution |dp 1 /dD| = 0.08 kPa/mm of the vacuum gripper is twice that of the PneuNet gripper (|dp 1 /dD| = 0.04 kPa/mm). According to equation 2, the smaller term Finally, it should be noted that the overall sensing accuracy is determined by both the sensing resolution of the actuator or gripper and the sensing accuracy of the pressure sensor used, and luckily, there are ample (relatively cheap) pressure sensors on the market that span various pressure ranges with high enough accuracy for our purpose (Table S2). For example, in the tests with the PneuNet actuator in Fig. 1b, we used a ±34.5kPa pressure sensor with an accuracy of ±0.25% (of Full Scale Span). By dividing the pressure sensor error (±0.1725kPa) by the average sensing resolution (0.04kPa/mm), we can obtain an overall sensing accuracy of ±4.31mm for the PneuNet actuator. This sensing accuracy is valid for one-time measurement with the pressure sensor. In this work, however, we always do the pressure measurement over a period of time (which we were able to reduce to 0.05 seconds in Fig. S20) and calculate the average value, which gives a higher overall sensing accuracy, e.g., ±1.7mm with a 95% confidence interval ( Fig. S3a) for the PneuNet actuator.

Closed-loop control with fluidic sensing
With the insights gained on sensing performance, we finally demonstrate that the fluidic sensing strategy is robust for closed-loop control in three applications: size sorting, tomato picking and ripeness detection. In the first closed-loop control experiment ( Every time a new object is gripped, its size is measured via the feedback pressure in the gripper that is averaged over a fixed time window (red band in Fig. 5b) after actuation. When the feedback pressure is smaller than the pressures in the sorted array, the robotic arm directly moves the new object to the end of the queue, e.g., the first and last actuation cycles in Fig. 5b. Otherwise, the robotic arm moves objects in the sorted array to make the correct position available for the new object, e.g., the third and sixth actuation cycles in Fig. 5b. The closed-loop control makes it possible to successfully sort the four cylindrical objects with random input order (Fig. S13, Fig. S15 and Supplementary Video 4). The fluidic sensing strategy also works for sorting random objects with irregular shape (Fig. S16, Fig. S17 and Supplementary Video 4).
Note that no calibration curve is needed here since the comparison between the equilibrium pressure values is sufficient for sorting.
In the second closed-loop control experiment (Fig. 5c-d and Supplementary Video 5), we perform a tomato picking experiment. We artificially increase the pressure in every step to demonstrate that the proposed sensing approach can provide feedback for picking automation. Note that in a more realistic setting, one would have likely used the highest pressure immediately. During each picking cycle, the algorithm compares the equilibrium pressure P eq in the gripper to a corresponding reference measurement P ref in free space, and evaluates ∆P = P eq − P ref to determine a successful or unsuccessful picking. When ∆P is smaller than a threshold, the algorithm regards it as an unsuccessful picking as the actuators are not deformed by the tomato. It then starts the next picking cycle with a higher actuation pressure.
Once ∆P is larger than a threshold (0.2 kPa), the algorithm regards it as a successful picking.
The gripper places the tomato on the table and moves to the next tomato in line. Note that the slip of the tomato out of the gripper in an unsuccessful picking can also be detected from the abrupt decrease in the pressure-time curve, which could potentially provide additional sensing feedback. We tested a total of nine tomatoes in three runs, out of which six tomatoes were successfully picked and placed, one tomato was not picked by the gripper, the other two were successfully picked but not recognized because the tomato slipped into the palm of the gripper after being picked from the stem, resulting in a ∆P smaller than the threshold ( Fig. S18 and Supplementary Video 5). As our soft gripper was not specifically designed for picking tomatoes, the design of the gripper should be optimized for this task. Importantly, this would not affect the retrofit implementation of our sensing approach. The red band in the plot represents the pressure feedback measurement, and the yellow band represents the object movement. The four pressure feedback measurements (average ± standard deviation) are 34.80 ± 0.07 kPa, 35.36 ± 0.07 kPa, 35.05 ± 0.08 kPa, 34.13 ± 0.07 kPa, respectively. c, Snapshots of the tomato picking experiment representing the pressure reference measurement and picking of the three tomatoes, respectively. d, gripper actuation pressure and TCP coordinates over time during the tomato picking. The red band represents the pressure feedback measurement, and the yellow band represents the tomato placement on the table. The solid line in f,h,j,l represents that the object diameter inferred by t c equals that inferred by ∆P eq , the dashed line represents the object diameter inferred by ∆P eq is 3 mm smaller than that inferred by t c . For the ripe tomatoes and rigid dummy, a measurement above the dashed line is considered as a successful sensing event. Conversely, for the overripe tomato, a measurement below the dashed line indicates sensing success.
In the third closed-loop control experiment ( Fig. 6 and Supplementary Video 6), we detect the ripeness of tomatoes by applying the method mentioned in Fig. 2f and g to estimate the indentation depth. We demonstrate that the proposed sensing approach can provide feedback for automated sorting of an overripe tomato from ripe tomatoes. For versatility, we select the commercial vacuum gripper for this demonstration, also as it has the largest sensing resolution ( Table 1). The demonstration includes one cycle of calibration and five separate cycles of sensing to determine repeatability. The positions of the four tomatoes (including one overripe tomato) and one dummy are shuffled randomly between sensing cycles. All size predictions are based on one calibration process with 3D-printed rigid dummies, where the gripper pressure P gripper is compared to a reference response P ref to determine ∆P = P gripper − P ref over time for different diameters of the object that is being gripped (Fig. 6b and c). As explained by the stiffness sensing method in Fig. 2f and g, the size of the tomato upon gripping can be inferred by the time of first contact t c (Fig. 6d), and the size at equilibrium can be inferred by the ∆P at equilibrium (Fig. 6e).
It is important to note that the method that uses the time of first contact is strongly affected by the alignment of the tomato inside the gripper, leading to an early rise of ∆P -time curve and inaccurate size predictions of the tomato upon gripping and sensing success rates of 40% and 45% ( Fig. 6f and g) for the rigid dummy and tomatoes, respectively. To increase the sensing success rate, we can perform the gripping event twice, the first to center the object inside the gripper, and the second gripping event to extract sensing feedback ( Fig. 6h and i). Alternatively, we can choose t c at higher ∆P values for both calibration and sensing ( Fig. 6j and k), so that the object has been effectively centered during sensing. With t c at ∆P = 2kPa, sensing during either the first or second gripping event gives 100% success rates for both rigid dummy and tomatoes ( Fig. 6j-m). We can also infer the initial size of the tomato by ∆P max (Fig. S19 and Supplementary Video 6) instead of t c to avoid the influence of misalignment, which gives 100% and 90% success rate for rigid dummy and tomatoes, respectively. While we were able to pick out the rotten tomato consistently, since the gripper squeezes tomatoes for ripeness detection in this method, post-harvest studies should be performed in the future to avoid extra damage to the produce when applying the method in practical applications.
It should be noted that all the closed-loop control demonstrations above were performed under quasistatic conditions. We tested the sensing strategy at different actuation speeds (Fig. S20) and find that the effectiveness of the sensing strategy is not affected by the actuation speed, as long as the sensing feedback is collected after the actuator comes into contact with the environment. To speed up the sensing process for real-world applications, it is important to ensure an actuation speed that is high enough for the interaction with the environment to happen before

Conclusion
In conclusion, we present a versatile fluidic sensing strategy that relies on measuring the fluidic input response instead of embedded sensing elements into the soft actuators. The soft robotenvironment interaction can be accurately interpreted with one pressure sensor that is connected remotely. We show that the proposed strategy can be retrofitted to a wide range of soft devices, and implemented in closed-loop control of gripping applications. We believe that this relatively straightforward integration of sensing capabilities makes it readily available for other soft robotic devices and applications, including wearable assistive devices (54) and soft locomotive robots (48, 55), without the need to alter the design of the soft device itself.
While we were able to retrofit our sensing approach to a range of soft actuators and grippers,

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it should be noted that the air tank method in Fig. 1b is not directly applicable to soft actuators driven by incompressible liquid, as the method depends on the compressibility of air and the final pressure balance between the tank and actuator. This could be solved by using a flexible tank (e.g., a balloon), such that the compressibility of the air is replaced by the elasticity of the tank. A simpler approach could instead be to use the pressure control ( Fig. 1c) or flow control ( Fig. 1d) method to obtain sensory feedback from the liquid-driven actuator's interaction with the environment.
According to equation 2, the sensing resolution is influenced by the final internal volume v 1 of the actuator after the interaction with the environment and its derivative over the sensing target dv 1 /dξ. Yet, there is no trivial relationship between the initial volume of the actuator and the sensing resolution. Even though we demonstrate that we can retrofit sensing to pneumatic actuators, optimizing the sensing resolution by for example changing the tank size and the initial tank pressure should be done on a case by case basis, and can best be done by experimentally obtaining the relation between pressure and volume for specific interactions with the environment. For example, if we want to apply our strategy to pneumatic actuators that can generate complex motions with multiple degrees of freedom (56), the sensing resolution depends on how the interaction of the actuator with the environment affects v 1 and dv 1 /dξ in equation 2, which is not straightforward to predict beforehand and depends on the application.
Moreover, we did find relatively small variations over time during cyclic gripping. These variations may be due to the performance of the soft actuator, pressure regulator and pressure sensor, or environment variations like the temperature. Comparing the sensing response to a reference helps reduce the influence of long-term system and environment variations, as also demonstrated in the closed-loop control examples. Still, any non-unique pressure-volume response of a soft actuator would cause inaccurate sensing. Moreover, since the proposed sensing principle uses the soft actuator itself as a sensor, any fragility or unreliability of the soft actuator would have a direct influence on the sensing performance. Additionally, the soft actuation of more complex devices should be designed such that it achieves sensitivity to the sensing task.
Moving forward, machine learning can potentially be applied to read the higher-order difference in the pressure-time curves of the soft robot for various interactions with the environment to achieve more complex sensing applications (12,57). Although our sensing strategy removes the need of embedding or attaching sensors to the soft actuator, the hardware of the system is still bulky and may not be suitable (yet) for small-scale untethered soft robotic system (58).
To further reduce the bulkiness of the system, the fluidic sensory feedback from the soft robotenvironment interaction can be potentially read out by soft pneumatic valves (59-61) to build electronic-free soft robots (60, 62, 63) that can sense and respond to their environment. The basic yet powerful principles studied in this work make it possible to bring (some) sensing capabilities to most soft fluidic devices without the need for design changes, and paves the way towards new functionalities in soft interactive devices and systems for real world applications.

Methods
Details on the methods are provided in Supplementary Materials.

Data Availability
All experimental data that support the findings of this work and computer algorithms necessary for running the analysis will be uploaded to Zenodo before publication.

Code availability
The numerical data in the Methods section Modeling the Fluidic Sensing Approach and computer algorithms necessary for running the model will be uploaded to Zenodo before publica-24 tion.

Fabrication of PneuNet actuators and soft gripper
The PneuNet actuator was molded in a two-step process (Fig. S22). In the first step, the extensi- with a span of 20 data points to remove high frequency noise before any further analysis.
To obtain the pressure-volume curves in Fig. 1a, we measured both the flow input and pressure response over time as the actuator was inflated onto the aluminium bar from a distance h.
The actuation system (Fig. S23)  Pressure control with pressure measurement The actuation system (Fig. S24)  Pressure control with flow measurement The actuation system (Fig. S25)  Flow control with pressure measurement The actuation system (Fig. S23) was the same as the one for the characterization of pressurevolume curves mentioned above, except that we used a ± 34.5 kPa pressure sensor to measure the pressure in the actuator. A total of 8 actuation cycles were performed at each h by programming the custom software for the data acquisition system.

TPU bending actuator
The TPU bending actuator was designed based on dimensions described in reference (43) and printed with TPU filament (Filaflex 82A) on a Fused Filament Fabrication 3D printer (Felix Tec). The actuation system (Fig. S29)  deviation was used to plot the error bar in Fig. 3i. Note that the sensor offset of the ± 206.8 kPa pressure sensor was recalculated for each actuation cycle based on the average of the last 1 s measurements before switching on the second valve.

Filament actuator
The filament actuator was fabricated with the gravity-assisted molding approach described in reference (44). A 400 mm long steel rod with a diameter of 1.5 mm was coated with liquid silicone (Elite Double 32 fast) and suspended at a 5 degree angle relative to the vertical direction during curing. The filament actuator was cut into a length of 150 mm after curing. One end of the filament actuator was dip coated with extra silicone to form a thicker part that could be mounted on the robotic forearm by piercing with a plastic pin (Fig. S30). HSGM GmbH). The actuation system (Fig. S31) was the same as that for the filament actuator.

Vacuum-powered commercial soft gripper
We mounted the vacuum-powered commercial soft gripper (SG.S74DS70.SDS70.F.G14F.00, Piab) on the robotic arm and kept the robotic arm stationary throughout the gripping test. The actuation system (Fig. S32) was similar to that described in section Size Sensing with Soft Gripper, except that a vacuum pressure source and 0.1 L air tank were used. We added a bidirectional flow sensor (HAFBLF0750CAAX5, Honeywell) to measure the flow between the air tank and the soft gripper and a flow resistor to restrict the flow within the range of the flow sensor. A total of 8 actuation cycles were performed for each test. Each actuation cycle started by setting the proportional pressure regulator at -60 kPa and waiting for 10 s, switching on the first valve to pressurize the air pipe for 20 s, switching off the first valve and waiting for 10 s, then switching on the second valve to inflate the gripper for 30 s, finally switching off the second valve to deflate the filament actuator and waiting for 30 s to fully discharge. For each actuation cycle, the equilibrium pressure was averaged between 24 s and 25 s after switching on the second valve. For each gripping object, the equilibrium pressure was averaged over the last 5 actuation cycles and the standard deviation was used to plot the error bar in Fig. 4h. Note that the sensor offset of the ± 103.4 kPa pressure sensor was recalculated for each actuation cycle based on the average of the last 1 s measurements before switching on the second valve.

Commercial soft PneuNet gripper
We mounted the commercial soft PneuNet gripper (SFG-FNC3-N5087-S, Soft Robot Technology) on the robotic arm and kept the robotic arm stationary throughout the gripping test.
The actuation system (Fig. S33) was similar to that described in section Size Sensing with Soft Gripper, and we added a bidirectional flow sensor (HAFBLF0750CAAX5, Honeywell) to measure the flow between the air tank and the soft gripper and a flow resistor to restrict the flow within the range of the flow sensor. A total of 8 actuation cycles were performed for each test. Each actuation cycle started by setting the proportional pressure regulator at 70 kPa and waiting for 10 s, switching on the first valve to pressurize the air pipe for 20 s, switching off the first valve and waiting for 10 s, then switching on the second valve to inflate the gripper for 60 s, finally switching off the second valve to deflate the filament actuator and waiting for 60 s to fully discharge. For each actuation cycle, the equilibrium pressure was averaged between 50 s and 51 s after switching on the second valve. For each gripping object, the equilibrium pressure was averaged over the last 5 actuation cycles and the standard deviation was used to plot the error bar in Fig. 4k. Note that the sensor offset of the ± 103.4 kPa pressure sensor was recalculated for each actuation cycle based on the average of the last 1 s measurements before switching on the second valve.

Modeling the fluidic sensing approach
We develop a basic model based on the interaction of an extension actuator (with a linear stiffness k) with a rigid wall (Fig. S12), to characterize and better understand the dynamics between the air tank and actuator in the proposed sensing strategy. The linear actuator is initially at atmosphere pressure p act = p atm , while the tank with compressed air starts at p tank = p 0 . We take the sensing target as the initial distance L between the tip of the actuator and the wall. The valve between the air tank and actuator is opened at t = 0, so that the system will reach the equilibrium pressure p 1 . Assuming incompressible and laminar flow between the air tank and actuator, the Hagen-Poiseuille equation that characterizes the flow rate between air tank and actuator can be written as where V represents the volume of air that transfers from the air tank to the actuator, r represents the flow resistance between the air tank and actuator, and p tank and p act indicate the absolute pressure in the air tank and actuator, respectively. According to the ideal gas law, we have and p act v act = n act RT, where R represents the ideal gas constant, T represents the absolute temperature of the air, and v tank and v act represent the internal geometrical volume of the air tank and actuator, respectively.
The volume v tank is constant, while v act depends on the interaction of the actuator with its environment. Assuming the internal volume of the actuator does not change anymore after the actuator comes into contact with the wall, we have where A and l 0 represent the cross section and initial length of the actuator, respectively. By substituting equation 6 into equation 5, we have Combining equations 3, 4 and 5, we obtain two differential equations that describe the change of the amount of air mass n tank in the air tank and n act in the actuator over time where M and ρ represent the molar mass and density of air, respectively, and p act is given For the parameter values presented in Table S3, we determine the pressure-time and pressurevolume responses, calibration and sensing resolution points as shown in Fig. S12b-e. Note that for the basic interaction of the linear extension actuator with a rigid wall, assuming that the actuator comes into contact with the wall at the equilibrium state, the initial and the equilibrium internal volume of the actuator are known under different environment settings, i.e., v 0 = Al 0 , v 1 = A(l 0 +L), then we also have dv 1 /dL = A. Therefore, we can also determine the analytical solution of the equilibrium pressure p 1 and sensing resolution dp 1 /dL from equation 1 and 2 (ξ = L) With the numerical simulation and analytical solution, we can qualitatively study the influence of each individual system parameter on the sensing resolution. The initial internal volume of the actuator can be changed by varying either the initial length l 0 (Fig. S12f), or the cross section A (Fig. S12g), which interestingly have different effects on the sensing resolution. To understand this difference, we can first look at an extreme case when the volume of the actuator is infinitely large. In this extreme case, the air mass in the air tank is too small to extend the actuator, then the sensing resolution is dp 1 /dL = 0. This vanishing of sensing resolution with an extremely large actuator can be seen from the numerical simulation results when varying either l 0 or A (dp 1 /dL becomes zero after l 0 and A exceed a certain value in Fig. S12f and g), but not from the analytical solutions. This is because the analytical solution assumes that the actuator always comes into contact with the wall at the equilibrium state. However, the occurrence of this contact also depends on the initial conditions. Therefore, the numerical simulation results agree with the analytical solution in Fig. S12f and g until the actuator can not reach the wall anymore at given initial conditions. Still, the analytical solution in equation 11 provides theoretical insights on how the sensing resolution is affected by l 0 and A.
Interestingly, the analytical solution also indicates that the stiffness k (Fig. S12h) does not influence the sensing resolution, because k does not affect any of the parameters in equation 11. However, when k is extremely large, the actuator becomes too stiff to extend a distance of L with given initial conditions, such that the sensing resolution becomes dp 1 /dL = 0 as no contact occurs, which can be seen from the numerical simulation results in Fig. S12h. Therefore, a higher stiffness does require a higher tank and actuation pressure. On the other hand, the magnitude of sensing resolution increases with the initial tank pressure p 0 when the other parameters in equation 11 stay constant (Fig. S12i).
It should be noted that, in this simplified model with the linear extension actuator, we assume that the actuator volume stops changing after it hits the wall. However, in reality, the interaction with the environment does not fully constrain the deformation and thus the internal volume of the actuator, which can complicate the analysis and needs to be analyzed case by case. We believe that, despite the simplifications made in our model, it provides a framework for choosing available parameters for improving the sensing resolution.  Table S1: Comparison of the two commercial grippers. The dv 1 /dD is obtained from the linear fits in Fig. S11c and f. The calculated dp 1 /dD is based on equation 2. The measured dp 1 /dD is obtained from the linear fits of the data in Fig. 4h and k.     Figure S3: Accuracy of the various sensing approaches. The sensing accuracy is determined by the difference between the ground truth (set distance h) and the predicted value of h based on the calibration curves. The probability density function estimate f is shown in histograms with a normal distribution. a, Accuracy obtained when controlling the total mass using a pressurized air tank and measuring the pressure response. The h predict is calculated from the calibration curve in Fig. 1b at given pressure measurements. The sensing accuracy of this method is ±1.7 mm with a 95% confidence interval. b, Accuracy obtained when controlling the pressure and measuring the volume flow input. The h predict is calculated from the calibration curve in Fig. 1c at given volume measurements. The sensing accuracy of this method is ±4.7 mm with a 95% confidence interval. c, Accuracy obtained when controlling the volume flow input and measuring the pressure. The h predict is calculated from the calibration curve in Fig. 1d at given pressure measurements. The sensing accuracy of this method is ±3.4 mm with a 95% confidence interval. d, Accuracy obtained when controlling the total mass using a pressurized air tank and measure the pressure response over time. The h predict is calculated based on the interpolation of vertical displacement curve of actuator in the case of free actuation in Fig. 2c at given time of contact measurements. The sensing accuracy of this method is -2.9 to 3.8 mm with a 95% confidence interval.    Figure S7: Cyclic testing of the sensing strategy. We tested a total of 111 cycles of the soft PneuNet gripper (Fig. 1e) on the UR5 robotic arm. Each cycle contains two gripping events, the first one with nothing inside the gripper, the second one with a 80 mm diameter cylinder inside the gripper. In the second gripping event, after the gripper grasps the object, the robotic arm first moves up and holds the object in the air, then moves down and releases the object before going to the next cycle. For both gripping events, the equilibrium pressure inside the actuator is averaged between 8 s and 9 s after the opening of the valve, which corresponds to the time when the object was held in the air during the second gripping event. a, Pressuretime measurements of the actuator from 111 cycles superimposed on top of each other. b, Pressure-time measurements of the tank from 111 cycles superimposed on top of each other. c, Measurements of the TCP Z coordinate of the robotic arm from 111 cycles superimposed on top of each other. d, Equilibrium pressure of the two gripping events for all 111 cycles. e, Distribution of the equilibrium pressure of the first gripping event over 100 cycles. The mean value is 34.5951 ± 0.0824 kPa (95% confidence interval). f, Distribution of the equilibrium pressure of the second gripping event over 100 cycles. The mean value is 35.4834 ± 0.0706 kPa (95% confidence interval). Note that the soft gripper tested here is a different version from the one tested in Fig. 1 because the original one was accidentally damaged. We slightly changed the actuator design, resulting in a higher actuation pressure. g, The pressure difference ∆P = P act D80 − P act empty between the two gripping events over 111 cycles. Figure S8: Tuning the response time with different flow resistors in between the valve and actuator. The dashed curves represent the pressure in the 0.1 L air tank. The solid curves represent the pressure in the PneuNet actuator. The initial distance h between the actuator and rigid plate was set at 20 mm. Tests were done by switching the flow resistor shown in Fig. S24. The fluidic circuit still has some flow resistances due to the valve and tube connections, even when no flow resistor was added to the circuit. Therefore, the response time can be further improved by reducing these flow resistances in the circuit. Figure S9: Characterization of pressure-volume curves of the suction gripper attaching to soft objects with different shore hardness through controlled pressure input. a, Pressure of the suction gripper during the last three actuation cycles. b, Flow rate of the flow going out of (positive) and into (negative) the suction gripper during the last three actuation cycles. c, Pressure-volume curves of the suction gripper from the last actuation cycle.    Table S3: pressuretime (b) and pressure-volume (c) curves of the air tank and actuator; calibration curve of P eq over L (d); sensing resolution curve of dP eq /dL over L (e). f-i Simulation results of the effects of actuator's initial length l 0 (f), cross section A (g), stiffness k (h) and initial tank pressure p 0 (i) on the sensing resolution magnitude |dP eq /dL|. The makers and dashed lines in f-i represent numerical simulation results and analytical solutions, respectively. The circular, square and triangular markers in f-i represent L = 7.5, 17.5, 27.5 mm, respectively. The results in f-i are based on the input values in Table S3 except that an initial tank pressure of 300 kPa (absolute pressure) is used in g.       The error bars in c,d represent the standard deviation of three measurements. f-m, Sensing results. The solid line in e represents that the object diameter inferred by ∆P max equals that inferred by ∆P eq , the dashed line represents the object diameter inferred by ∆P eq is 0.5 mm smaller than that inferred by ∆P max . . For each measurement event, the equilibrium pressure was averaged over a 0.05 s period starting at 0.5 s after the actuation, as indicated by the red bands in the plot. The four pressure feedback measurements (average ± standard deviation) in a are 38.21 ± 0.11 kPa, 38.84 ± 0.07 kPa, 38.59 ± 0.07 kPa, 37.44 ± 0.07 kPa, respectively. The four pressure feedback measurements (average ± standard deviation) in b are 39.17 ± 0.11 kPa, 37.33 ± 0.06 kPa, 37.92 ± 0.07 kPa, 38.52 ± 0.06 kPa, respectively. Note that the initial tank pressure was set at 74 kPa here to ensure successful gripping of each object, especially the one with a diameter of 40 mm. Note that the coordinates of the robotic arm were intentionally not saved in these experiments in order to obtain a high data frequency (∼ 87 Hz) that allows the fast sensing over a 0.05s period.         Note that this setup contains more than the minimum number of required components for the sensing strategy to work. Solenoid valves 7) and 8) were arranged in such way to avoid leakage through the valves at high actuation pressure required by the filament actuator. Flow sensor 10) was used to obtain the full pressure-volume responses of both the pressurized air pipe and filament actuator and the flow resistor 9) was used to restrict the flow within the range of the flow sensor.