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Abstract

Soft robots have morphological characteristics that make them preferred candidates, over their traditionally rigid counterparts, for executing physical interaction tasks with the environment. Therefore, equipping them with force sensing is essential for ensuring safety, enhancing their controllability, and adding autonomy. At the same time, it is necessary to preserve their inherent flexibility when integrating sensory units. Soft-fluidic actuators (SFAs) with hydraulic actuation address some of the challenges posed by the compressibility of pneumatic actuation while maintaining system compliance. This research further investigates the feasibility of utilizing the incompressible actuation fluid as the means of actuation and of multiaxial sensing. We have developed a hyperelastic model for the actuation pressure, acting as a baseline pressure. Any disparities from the baseline have been mapped to external forces, using the principle of pressure-based fluidic soft sensor. Computed tomography imaging has been used to examine inner deformation and validate the analytically derived actuation-pressure model. The induced stresses within the SFA are examined using COMSOL simulations, contributing to the development of a calibration algorithm, which accounts for geometric and cross-sectional nonlinearities and maps pressure variations with tip forces. Two force types (concentrated and distributed) acting on our SFA under different configurations are examined, using two experimental setups described as “Point Load” and “Distributed Force.” The force sensing algorithm achieves high accuracy with a maximum absolute error of 0.32N for forces with a magnitude of up to 6N.

Introduction

Soft robots made from pliable materials and incorporating flexible mechanisms offer advantages over conventional rigid robots.¹ Their physical attributes enable the safe handling of delicate objects and facilitate tasks requiring physical interaction with complex anatomical systems. For these reasons, soft robots are utilized across various fields, ranging from soft industrial grippers and manipulators^2–4 to biomedical devices such as wearables and implantables,⁵ tools for minimally invasive surgery,⁶ medical tool guidance,⁷ and endoscopic applications.^8,9 However, to unleash their full potential and enable high-precision controllability, the body of the soft robots must act as sophisticated systems for both actuation and sensing,¹⁰ resembling monolithically integrated synthetic organisms.^11,12 Thus, it is necessary to deepen our comprehension from an engineering perspective using analytical modeling.

Among the several actuation techniques used for soft robots,¹³ tendons and/or pressure actuation, known as soft fluidic actuators (SFAs), are widely used.¹⁴ The former technique stimulates the robot by controlling the length of the tendons, whereas the latter uses inflatable cavities. An example of a solely tendon-driven robot is described in the work of Kato et al.,¹⁵ whereas Tao et al. developed a soft actuator that combines both actuation techniques.¹⁶ Nonetheless, tendon rigidity and friction create difficulties in the control of soft robots, making SFAs pneumatic^17,18 or hydraulic,^19,20 the most versatile choice, particularly in the field of medicine.²¹ One limitation of pneumatics is the compressibility of the actuation fluid, which significantly complicates the development of analytical models and precise control. Consequently, recent designs utilizing hydraulic actuation leverage the morphological advantages of soft robots,²² without being impeded by fluid compressibility.

The endeavor to overcome the lack of force feedback by artificially replicating it poses a substantial engineering hurdle requiring multimodal sensory systems.²³ The present cutting-edge tactile sensing technology relies on five distinct sensing principles as follows: resistive, piezoresistive, optical, inductive, and capacitive.²⁴ The sensing mechanisms can either surround the body of the soft robot, smart skin,²⁵ and flexible electrodes,²⁶ or be embedded within structural cavities, conductive liquids,²⁷ or hydrogels.²⁸ However, it is undesirable to integrate many rigid sensors in soft robots, because of the manufacturing challenges in miniaturizing electronics²³ and the fear of compromising their ultraflexibility. These challenges led to the development of the pressure-based fluidic soft sensor (PBFSS).

The sensing principle of the PBFSS is to map variations of fluidic pressure against the force stimuli. To date, only a limited number of studies have demonstrated attempts to achieve this, by mostly utilizing pneumatic systems.^29–32 While most sensors are used to detect one-dimensional forces, three-dimensional (3D) force sensing is also demonstrated.³³ Because hydraulic sensors respond faster than pneumatics, due to higher stiffness,^34,35 we have developed a 3D hydraulic force sensor by coupling three 1-degree-of-freedom (DOF) actuators to form a parallel robot.³⁶ However, the linear intrinsic force sensing and control method are limited to small motion ranges and rely on empirically calibrated variables for validation. In addition, we have achieved intrinsic force sensing using a stiffness-based nonlinear finite element (FE) model.³⁷ To account for geometric nonlinearities and to estimate total deflection and orientation changes, the beam was discretized into several serial linkages. Nonetheless, this approach resulted in high-rank matrices that are complex to calibrate. Due to intricate geometric and material nonlinearities, calibration is necessary when using soft robots.^38,39 These nonlinearities affect the neutral axis (NA) location, which is crucial for analyzing the kinematic and stress distribution profile.

Assuming quasistatic conditions and small deformations, mechanical modeling⁴⁰ proved effective in characterizing the behavior of soft actuators, especially in the bending.^41,42 However, hyperelastic models are typically used for actuation pressure, accounting for significant and nonlinear deformations of the silicone body, similar to inflatable balloons.^43,44 This study aims to analytically model the biaxial internal deformations and the actuation pressure of unconstrained multichambered SFAs, known as cavity actuators,⁴⁵ and the stress distribution under the influence of external force. This will allow intrinsic multiaxial force sensing based on fluidic pressure variations. Although constrained fluidic chambers offer higher actuation consistency,⁴⁶ constraining them adds additional manufacturing steps and increases operating pressures even further, affecting structural compliance. The algorithms derived from this investigation are experimentally validated to ensure their accuracy and reliability.

The main contributions of this study to the field of soft robotics are as follows: •

Validation, for the first time, of the biaxial deformation of unconstrained fluidic chambers with computed tomography (CT) imaging, allowing the development of an actuation-pressure hyperelastic model.

•

Illustration of the NA shift due to geometric and cross-sectional nonlinearities, improving kinematic modeling.

•

Derivation of an algorithm for intrinsic 3D force sensing, for different force types and configurations, using fluid as both the actuation and sensing medium.

Materials and Methods

The SFA used (Fig. 1) is part of MorphGI,⁴⁷ a hydraulically actuated endoscope that uses its tip to anchor and autonomously propel. The main body of the segments has a cylindrical shape of diameter, $D_{seg} =$ 17.5 mm and length $L_{0}$ =70 mm, manufactured with silicone rubber (SilbioneRTV 4420A&B, Elkem Silicone Inc). Multidirectional movement is achieved by introducing fluidic volume, via a syringe pump, into the three unconstrained actuation chambers. The chambers are arranged symmetrically in a circular pattern at an off-axis distance ( $y_{d}$ ). Upon applying pressure, the actuation chambers expand biaxially. A woven PET mesh sleeve is integrated into the rubber core to control radial expansion and facilitate bending. The segment elongates axially if an equal fluid volume is injected into all chambers. Conversely, asymmetrical actuation induces bending. Conduits, reinforced by springs, are incorporated to allow the passage of tubes, cables, and biopsy forceps. Nylon 3D-printed caps are adhered to each end of the segment.

FIG. 1.

Design illustration. (a) The assembled MorphGI probe, featuring three segments for inchworm-like locomotion.⁴⁷ The tip includes a camera and biopsy channel for diagnostic and intervention purposes. (b) The front segment of MorphGI has male and female end caps for segment stacking. The tubing length is approximately 2 m with a 1.25 mm inner and 1.85 mm outer diameter. (c) Cross-sectional view showcasing outer diameter, actuation chamber, and conduit arrangement. It also illustrates how springs support the four conduits and how the external constraint is embedded in the silicone body.

A. Mechanical modeling and stresses

The analysis aligns with the coordinate system depicted in Figure 2. The sensing principle of PBFSS is to map pressure variations (ΔP) of sensory pressure readings $(P_{s})$ from the chambers actuation-pressure model ( $P_{m})$ , to the 3D external force stimuli ( $F)$ through a stress coefficient matrix ( $G)$ as shown below: $Δ P = P_{m} - P_{s} = G \cdot F .$ (1)

FIG. 2.

Front segment coordinate system used to derive (1)–(9) and (15)–(24). (a) Side view: $e_{1}$ represents the shift of the neutral axis (NA) due to area nonlinearities. Actuation chambers are located $y_{d} = 4.2$ mm away from the geometric center with r = 8.75 mm. (b) Front view: $e_{2}$ represents the NA shift caused by geometric nonlinearities resulting from the flexibility of the segment. (c) Curved configuration: According to curved beam theory, as the segment bends at an angle θ with a bending radius ρ, the NA further shifts by a distance $e_{3}$ . (d) Twisting effect: A curved beam experiences twisting γ due to an out-of-plane force.

Hence, to achieve our objective of utilizing fluid as both the actuation and sensing medium, a $P_{m}$ with respect to (w.r.t) injected fluidic volume ( $V_{w})$ in free space is required. Since our SFA has 3 actuation chambers, $Δ P$ are related to the sum of stresses ( $σ_{i})$ within the silicone body as follows: $[\begin{matrix} d p_{1} \\ d p_{2} \\ d p_{3} \end{matrix}] = \sum σ_{i} = G \cdot [\begin{matrix} F_{x} \\ F_{y} \\ F_{z} \end{matrix}]$ (2)

The relationship between stress ( $σ_{i})$ and strain ( $ε_{i})$ in silicone follows a nonlinear behavior.⁴⁸ However, small external forces induce small strains that are approximated by linear elastic models. Hooke’s law (3) describes the linear relationship within the elastic region of a straight beam as follows: $σ_{i} = E \cdot ε_{i}$ (3)where E is the modulus of elasticity. If the beam is under the influence of a force along its longitudinal axis ( $F_{z}$ ), the equations for axial stress ( $σ_{z}$ ), strain ( $ε_{z}$ ), and axial stretch ratio (λ), which is the ratio between the final $(L ’$ ) and initial $(L_{o})$ length,⁴⁹ are as follows: $σ_{z} = \frac{F_{z}}{A_{eff}}$ (4) $ε_{z} = \frac{L’ - L_{o}}{L_{o}} = \frac{Δ L}{L_{o}} = λ - 1$ (5)with $A_{eff}$ being the effective cross-sectional area of the beam and $Δ L$ the elongation. Under lateral forces, beam theory pertains to the analysis. Stresses that occur within the body, namely bending stresses ( $σ_{x y})$ , are expressed via the Flexure formula (6) as follows: $σ_{x y} = \frac{M_{tot} \cdot y_{c}}{I}$ (6)where $M_{tot}$ , are bending moments, $y_{c}$ the orthogonal distance of a point from the NA and I the second moment of inertia. For such a compliant segment, $M_{tot}$ are generated both from the weight of the SFA ( $F_{m})$ and lateral external forces ( $F_{x y})$ . Approximating both as a point load, acting at the middle and tip of the segment, respectively, results in the following: $M_{tot} = F_{m} \cdot \frac{L’}{2} + F_{x y} \cdot L’ .$ (7)

For a curved beam at an angle θ, the stress distribution deviates from linearity and follows a hyperbolic function instead. A force in-line with the bending plane (along ${F'}_{x}$ or ${F'}_{z}$ as shown in Fig. 2c) will induce both bending circumferential stresses ( $σ_{θ}$ ) and axial stresses ( $σ_{z}$ ). Using the Winkler-Bach method,⁵⁰ the expression of $σ_{θ}$ is given by the following: $σ_{θ} = \frac{M_{tot} \cdot y_{c}}{A_{eff} \cdot e \cdot (ρ - y_{c})}$ (8)where $ρ$ is the radius of curvature and e is the eccentricity (difference of $ρ$ between centerline and NA). In SFA, the centerline is not fixed and moves according to the center of mass equation and the amount of volume within each chamber. Finally, out-of-plane external forces ( ${F'}_{y}$ in Fig. 2d), superimpose $σ_{z}$ and torsional stress ( $τ_{θ}$ ) as developed by Wahl^51,52: $τ_{θ} = K_{c} \cdot \frac{32 \cdot M_{tot} \cdot ρ \cdot y_{c}}{π \cdot D_{seg}^{4} \cdot (ρ - \frac{D_{seg}^{2}}{16 \cdot ρ} - y_{c})}$ (9)with $K_{c}$ being Whal’s correction factor. For curved configurations, $M_{tot}$ = $ρ \cdot F_{x, y, z} \cdot \sin (θ) .$

B. NA position

NA is the imaginary line that divides the segmental cross section into two hemispherical sectors. At the NA, stresses and strains are zero (dashed line in Fig. 2). The cross-sectional design and mechanical properties of our segment create a complex variation of NA location, affected by the following three factors: (1) force (or bending) direction ( $ψ$ ), (2) force magnitude, and (3) precurved configuration. The force direction is important because the cross section is equal every 120° and along our x-axis. This implies that NA shifts from point O, the origin of reference, by a direction-dependent distance $e_{1} (ψ)$ . Due to the compliant nature of the segment, force magnitude can induce large deformations resulting in geometric nonlinearities, shifting further the NA by a distance, $e_{2}$ , toward the compression side of the beam.⁵³ As such, $e_{2} (F, ψ)$ is a factor dependent on the force vector. Finally, in a precurved beam, NA shifts even further toward the inner $ρ$ denoted as $e_{3} (θ$ ). These shifts will also affect the second moment of area calculation in accordance with the parallel axis theorem,⁵⁴ and can be summarized to $e_{4} = e_{3} + e_{2}$ + $e_{1}$ (Supplementary Data S1).

C. Chamber axial and radial kinematics

Unconstrained chambers experience significant biaxial and nonlinear deformations during actuation. Using conservation of volume, the axial elongation ( $Δ L)$ w.r.t the total injected fluidic volume ( $Δ V_{w})$ is as follows⁴⁷: $Δ L = \frac{1}{\frac{π}{4} \cdot D_{c}^{2} - \sum a_{c}} \cdot Δ V_{w} = \frac{1}{A_{x s}} \cdot Δ V_{w} .$ (10)where $D_{c}$ is the diameter of the mesh sleeve and $\sum a_{c}$ is the sum of the conduit area. Assuming uniform mechanical properties, the geometry of the chamber is expected to be cylindrical. Knowing the final length of the segment and the injected volume in the $i^{t h}$ -chamber ( $V_{w, i})$ , the radial expansion can be estimated as follows: $\frac{π}{4} \cdot D_{c h}^{2} \cdot (L_{o} + Δ L) = V_{o} + V_{w, i}$ (11)where $D_{c h}$ is the diameter and $V_{o}$ the initial volume at the unactuated state of the $i^{t h}$ -chamber. Rearranging (11) results in the following: $D_{c h} = \sqrt{\frac{4 \cdot (V_{o} + V_{w, i})}{π \cdot (L_{o} + Δ L)}}$ (12)to predict the radial kinematics.

D. Actuation-pressure modeling

To develop the baseline pressure $P_{m}$ , the chambers are approximated as “cylindrical balloons” using the nonlinear Mooney-Rivlin formulation.⁵⁵ We introduce another stretch ratio in the radial direction ( $μ$ ), which relates to radial strain as follows: $ε_{x} = \frac{D_{c h} - D_{o}}{D_{o}} = μ - 1$ (13)where $D_{o}$ is the initial diameter of the chambers. $P_{m}$ can then be obtained as derived in⁵⁵: $P_{m} = 4 \cdot \frac{H_{o}}{D_{o}} \cdot C_{10} \cdot [(\frac{1}{λ \cdot μ^{2}}) \cdot (λ^{2} - \frac{1}{λ^{2} \cdot μ^{2}}) \cdot (C_{01} + μ^{2})]$ (14)where $H_{o}$ is the original wall thickness of the chambers and $C_{10}, C_{01}$ are material parameters.

E. Pressure variation and intrinsic force sensing

The analytically derived pressure stiffness $(K_{p})$ links pressure changes to induced stresses via the bulk modulus equation (Supplementary Data S2) as follows: $Δ P = \frac{L_{o} \cdot A_{x s} \cdot β_{v}}{E} \cdot σ_{i} = K_{p} \cdot σ_{i}$ (15)where $β_{v}$ is an empirical value of actuation fluid bulk modulus. For a straight beam, combining (2), (4), (6), and (15) into a matrix format, we evaluate the following: $[\begin{matrix} d p_{1} \\ d p_{2} \\ d p_{3} \end{matrix}] = [\begin{matrix} N_{11} & N_{12} & C_{z} \\ N_{21} & N_{22} & C_{z} \\ N_{31} & N_{32} & C_{z} \end{matrix}] \cdot [\begin{matrix} F_{x} \\ F_{y} \\ F_{z} \end{matrix}]$ (16)where $N_{i, 1} = \frac{K_{p} \cdot L \cdot (y_{d \cdot} \cos (δ_{i}) - e_{4})}{I_{x}}$ , $N_{i, 2} = \frac{K_{p} \cdot L \cdot (y_{d \cdot} \sin (δ_{i}) - e_{4})}{I_{y}}$ , $δ_{i}$ = [180°, 60°, 300°], and $C_{z}$ = $\frac{K_{p}}{A_{x s}} .$ Since $e_{4}$ depends on the force vector, the $N_{ij}$ parameters can be assumed as follows: ${\begin{matrix} \begin{matrix} N_{11} = C_{11} + K_{11} \cdot F_{x} \\ N_{12} = C_{12} + K_{12} \cdot F_{y} \end{matrix} \\ N_{21} = C_{21} + K_{21} \cdot F_{x} \\ N_{22} = C_{22} + K_{22} \cdot F_{y} \\ N_{31} = C_{31} + K_{31} \cdot F_{x} \\ N_{32} = C_{32} + K_{32} \cdot F_{y} \end{matrix}$ (17)

Given the 180° symmetry of the segment: $d p_{1}^{ψ} = d p_{1}^{2 π - ψ}$ (18) $d p_{2}^{ψ} = d p_{2}^{\frac{4 π}{3} - ψ}$ (19) $d p_{3}^{ψ} = d p_{3}^{\frac{2 π}{3} - ψ}$ (20)with $ψ$ representing the shear force direction. In addition, using the 120° symmetry: $d p_{1}^{ψ} = d p_{2}^{\frac{2 π}{3} - ψ}$ (21) $d p_{1}^{ψ} = d p_{3}^{\frac{4 π}{3} - ψ}$ (22)

Expanding and simplifying (18–22) prove that $N_{ij}$ only depends on $C_{11}$ and $K_{11}$ . Thus, we get the calibration matrix $(M F)$ : $[\begin{matrix} d p_{1} \\ d p_{2} \\ d p_{3} \end{matrix}] = [\begin{matrix} F_{x} & F_{x}^{2} + F_{y}^{2} & F_{z} \\ \frac{- F_{x} + \sqrt{3} F_{y}}{2} & F_{x}^{2} + F_{y}^{2} & F_{z} \\ \frac{- F_{x} - \sqrt{3} F_{y}}{2} & F_{x}^{2} + F_{y}^{2} & F_{z} \end{matrix}] \cdot [\begin{matrix} C_{11} \\ K_{11} \\ C_{z} \end{matrix}] = MF \cdot CK$ (23)

The parameters CK can be found for a given $Δ P$ and MF data set using the least-squares method. Evaluating them into (17) and inverting (16) allows force sensing such as: $[\begin{matrix} F_{x} \\ F_{y} \\ F_{z} \end{matrix}] = {\vec{G}}^{- 1} \cdot [\begin{matrix} d p_{1} \\ d p_{2} \\ d p_{3} \end{matrix}]$ (24)

More details in Supplementary Data S3.

F. Experiments

FE analysis in COMSOL⁵⁶ was performed to visualize the NA shifts. Initially, a constant-magnitude force (0.1 and 2N) of varying direction $ψ \in [0 °, 360 °]$ was applied at the tip of a straight segment ( $θ = 0 °$ ). Next, a force with a varying magnitude but constant direction (ψ = 90°) was applied to a prebent beam (θ = 90°). The built-in functions of COMSOL were used to plot the bending stresses across the cross-sectional area.

To verify the validity of 10 and 14, the SFA was mounted on a linear guide, allowing frictionless elongation. Deionized water, 9.0 mL, was injected inside all three chambers (3.0 mL/chamber) in steps of 1.5 mL, using a syringe pump (PHD Ultra, Harvard Apparatus). At every interval, both during infusion and withdrawal, the pressure and length of the segment were recorded using a pressure sensor (PX190, Omega Inc) and a caliper, respectively. The experiment was repeated three times for three different segments, since the diameter of PET mesh sleeve was optimized from our previous characterization.⁴⁷

To evaluate (12), the SFA was placed inside an 8W NanoPET/CT scanner (Mediso Ltd.), while injecting fluidic volume, as described above. Image feedback was captured at each step. To sharpen the distinction between the fluidic chambers and soft material, contrast enhancement (Omnipaque) was used as the sole actuation fluid, since it provides high attenuation of X-rays.⁵⁷ A 3D reconstructed model was produced by manual segmentation using a 3D slicer (https://www.slicer.org/).

To verify (15)-(24), two experimental setups were used. To examine multidirectional “point loads,” the SFA was horizontally fixed on a robot arm (MECA500, Mecademic), while calibration weights (0–200 g) were hung from its tip. Due to the compliance of SFAs, even low forces can cause them to deflect, which is why they are generally tested for low-force applications.^47,58 After adding each weight, the end-effector was rotated (5°/s) for $ψ \in [0 °, 180 °]$ . This rotational movement is sufficient to generalize behavior, given the symmetry of the segment, and results in constant axial, but varying shear forces since deflection angle remains fixed (Supplementary Data S5). For “distributed force” investigation, the SFA was connected from its tip to a robotic arm (Elfin3, HansRobot) equipped with a 6-DOF load cell (M3815D, Srisensor). By adjusting chamber volumes, the segment was bent before being constrained at different bending angles $θ \in [0 °, 30 °, 45 °, 60 °, 90 °]$ . A cylindrical rigid connector (Connector A) and a universal joint (Connector B) were used for couplings, both 10 mm in diameter. With the end-effector being perpendicular to the tip, predefined waypoints were traced, always returning to the initial position in between. At each waypoint, the velocity of the robot was momentarily zero. The waypoints were selected along $φ \in [- 180 °, 0 °]$ (push-pull) for axial force- and $ψ \in [180 °, 0 °, 270 °, 90 °]$ (cross) for shear force transmission. For both experimental setups, pressures within the chambers and forces, when required, were recorded at 150 Hz. Pressure sensors were placed near the SFA and solenoid valves controlled the hydraulic circuit.

In all experiments, we allowed 3 min for viscoelastic effects to diminish before measuring fluidic pressure, since their modeling is beyond the scope of this study.^59,60

Results

NA location

Figure 3 displays excerpts from Supplementary Videos S1, S2, S3, S4, showcasing positive bending stresses ( $σ_{x y}$ ≥ 0 Pa) from COMSOL simulations across the different force vectors and configurations. By comparing Figure 3a and c with Figure 3b and d, respectively, it is evident that the displacement of the NA depends on the area profile it encounters. In addition, as the force magnitude increases, the NA tends to shift toward the compression side of the beam. Furthermore, even when subjected to a 0.1N force, a curved beam exhibits a shift of the NA toward the center of curvature (Fig. 3e). This occurs despite the absence of any area nonlinearities since the bending plane is along a symmetric area profile. Generally, the complex geometric and material nonlinearities introduced by the NA result in a curved profile in all studied scenarios.

FIG. 3.

Stress distribution results from COMSOL. Panels (a–e) correspond to snapshots from the Supplementary Videos S1, S2, S3, S4. The red arrow specifies the vector of the force. Only positive stresses are indicated to aid locating the position of the NA for the experiment with (a) F = 0.1N, ψ = 0°, θ = 0°, (b) F = 2.0N, ψ = 0°, θ = 0°, (c) F = 0.1N, ψ = 90°, θ = 0°, (d) F = 2.0N, ψ = 90°, θ = 0°, and (e) F = 2.0N, ψ = 90°, θ = 90°.

Chamber axial and radial kinematics

The experimental setup, average recorded data across the three segments, and estimated model (10) are shown in Figure 4. Our SFA shows high repeatability. Initially, the rate of $Δ L$ per unit of $Δ V_{w}$ is low ( $Δ V_{w} < 1.5 m L)$ , presumably due to the presence of trapped air within the actuation system. Pressure built-up first compresses trapped air before inducing any noticeable $Δ L$ . We call this the Preactuation state and thereafter Actuation state. In our model, it is assumed that trapped air causes an offset ( $V_{off}$ ) in the elongation data, defined as $V_{w} = Δ V_{w} - V_{off}$ . Setting $V_{off} = 1 m l$ yields a root-mean-squared error (RMSE) of 0.79 mm and a mean relative error (MRE) of 2.17% in the Actuation state.

FIG. 4.

Axial kinematics as derived in (10). (a) Illustration of the soft-fluidic actuators attached on the linear guide in a vertical configuration and (b) results of elongation $(Δ L)$ as a function of the total injected volume $Δ V_{w}$ . The gray area labeled as the “Preactuation” state is presumably due to trapped air within the hydraulic circuit.

For radial kinematics estimated by (12), the SFA was initially scanned for ${Δ V}_{w} = 7.5 & 9 m L$ and the results were utilized to generate a 3D representation (Fig. 5a and b). $D_{c h}$ experiences a trapezoidal distribution, reaching steady state after 30 mm from the female end cap. Thus, scanning the initial ∼45 mm for other volumes confirms steady-state $D_{c h}$ . Plots of average $D_{c h}$ across the three chambers for all experiments are shown in Figure 5c. Chambers maintain neither perfect circularity nor consistent expansion, with a maximum variation of 8.43% (Supplementary Figuer S8 in Supplementary Data S1), indicating material heterogeneity. CT data results and model have an RMSE of 0.15 mm when assuming only positive elongations (Fig. 6a). The initial diameter of the chambers ( $D_{o})$ is 2 mm but reaches $D_{c h} =$ 5.71 mm when ${Δ V}_{w} = 9 mL$ .

FIG. 5.

Radial kinematics as derived in (12) and results of computed tomography (CT) imaging. (a) Illustration and manual label segmentation of actuation chambers. The highlighted region was used to determine the diameter ${(D}_{c h}$ ) at steady state. (b) Single CT slice (top) and 3D reconstruction model (bottom) for $Δ V_{w}$ = 9 mL, (c) measured chamber diameter ${(D}_{c h}$ ) plotted as a function of length for the different injected volumes $(Δ V_{w})$ .

FIG. 6.

Experimental results with model estimation. (a) Diameter estimation $(D_{ch})$ as expressed in (12) and (b) actuation pressure ( $P_{m})$ as derived in (14). Hysteresis is visible in the pressure between the infusion and withdrawal State of the syringe pump.

Actuation pressure

The average of the experimental results along with model estimation is depicted in Figure 6b. To account for the expected hysteretic behavior of silicone segments,⁶¹ the material parameters in (14) are selected depending on the infusion/withdrawal state. The model accurately captures the pressure within the Actuation state with an RMSE of 0.12 bar and an MRE of 7.35%. For the maximum injected volume ( ${Δ V}_{w} = 9 mL$ ), pressure reaches 7.50 bar. The effects of viscoelastic relaxation are discussed in (Supplementary Data S6).

Pressure variations under external loading

The experimental setups, configurations, and traced waypoints are shown in Figure 7. For $θ$ = 0 $°$ and to transition from the “Preactuation state” each chamber was injected with $Δ V_{w, 1} = Δ V_{w, 2} = Δ V_{w, 3} =$ 0.50 mL. For the remaining bending angles, $Δ V_{w, 2} = Δ V_{w, 3} =$ 0.50 mL while $Δ V_{w, 1} = [1.00,1.35,1.70,2.30] mL .$ Distances of waypoints are ${Δ x}_{push} = 10$ mm, ${Δ x}_{pull} = 5$ mm and ${Δ x}_{cross} = 15$ mm. The modulus (E) of the segment is determined through a tension test (Supplementary Data S4).

FIG. 7.

Experimental setups to examine pressure variation and verify equations (15–24). (a) Experimental setup for “Point Load” experiments. At this exact setup, the soft-fluidic actuators (SFA) is at a vertical orientation and has no prebend but is curved under 200 g calibration weights that are attached from the tip using a string. (b) Experimental setup for the “Distributed Force” experiments. In this specific configuration, the SFA is prebent at an angle of θ = 90° and linked to the force sensor through Connector A. (c) Tested configurations between the two experiments. (d) “Front” and “Side” view of the traced waypoints for the “Distributed Force” experiments, demonstrating the distance labels for each waypoint.

Figure 8a-(left) shows the sinusoidal pressure variations in each chamber as the end-effector rotates under a load of 200 g. The MRE of ${[\begin{matrix} {d p}_{1} & {d p}_{2} & {d p}_{3} \end{matrix}]}^{T}$ is 1.91%, 2.84%, and 1.55%, respectively. Material heterogeneity causes pressure variation lines to deviate from expected symmetry, intersecting at ψ≃70°&170° instead of ψ = 60°&150°. Figure 8a-(right) displays the ${[\begin{matrix} F_{x} & F_{y} & F_{z} \end{matrix}]}^{T}$ vectors and their estimation (x-axis). Despite minor hysteresis, the maximum absolute error (MAE) is 0.16N (all results in Supplementary Data S7).

FIG. 8.

Results of ${[\begin{matrix} {d p}_{1} & {d p}_{2} & {d p}_{3} \end{matrix}]}^{T}$ and ${[\begin{matrix} F_{x} & F_{y} & F_{z} \end{matrix}]}^{T}$ vectors from the “Point Load” and “Distributed Force” experiments for θ = 0°. (a) Results of the 200 g case from the pressure variations and model estimations (left) and force vectors (right). $F_{i}$ represents the data estimation for the corresponding direction i = {x,y,z} (b) Results of the “Push/Pull” and (c) “Cross” experiments.

Figure 8b shows data obtained for θ = $0 °$ from the push-pull experiment using Connector A, whereas Figure 8c shows the cross movement using Connector B. Some force data noise is present due to vibrations of the end-effector. In addition, fluid dynamics introduce some delays, causing the force model prediction to lag the data. Using our calibration (23), the distributed forces in both experiments can be estimated with MAE in steady state of 0.15N, 0.15N, and 0.32N for ${[\begin{matrix} F_{x} & F_{y} & F_{z} \end{matrix}]}^{T}$ , respectively.

Figure 9 illustrates all force data and estimations for the remaining bending configurations. Data-driven calibration was conducted for each configuration due to (1) the fluctuating magnitudes of forces (up to 6N) across all different experiments (affecting NA position) and (2) the geometric complexities of our segments. Table 1 summarizes the MAEs for the various configurations, which are generally consistent. To examine the robustness of calibration, random forces were estimated for $θ = 90 °$ (Fig. 10). MAEs for the random movement are 0.15N, 0.16N, and 0.22N for ${[\begin{matrix} F_{x} & F_{y} & F_{z} \end{matrix}]}^{T}$ , respectively. Examining, Figures 9 and 10, becomes apparent that an increase in θ results in the generation of twisting, even under purely axial forces. This is evident by the growing magnitude $F_{y}$ and is attributed to material heterogeneity, causing stress imbalances.

FIG. 9.

Experimental and model estimation results of the force vector ${[\begin{matrix} F_{x} & F_{y} & F_{z} \end{matrix}]}^{T}$ obtained from the “Distributed Force” experiments for prebend configurations (θ ≠ 0°). The bending plane (α) is aligned with the blue chamber. The results for the “Push/Pull” and “Cross” trajectories are presented in pairs for the following prebend angles (a) θ = 30°, (b) θ = 45°, (c) θ = 60°, and (d) θ = 90°.

Table 1.

Steady-State Maximum Absolute Error of “Distributed Force” Prediction for Different Bending Configurations

Bending angle (°)	Maximum absolute error (N)
	Push/pull			Cross
	F_x	F_y	F_z	F_x	F_y	F_z
0	0.10	0.08	0.11	0.15	0.15	0.32
30	0.23	0.02	0.15	0.14	0.03	0.18
45	0.24	0.11	0.19	0.16	0.05	0.08
60	0.10	0.02	0.05	0.13	0.04	0.19
90	0.22	0.10	0.21	0.12	0.08	0.15

FIG. 10.

Experimental and model estimation results of the force vector ${[\begin{matrix} F_{x} & F_{y} & F_{z} \end{matrix}]}^{T}$ for the random movement at θ = 90°. These data are obtained using the “Distributed Force” experimental rig and Connector B.

Discussion

A. Neutral axis

Previous studies had effectively established mechanical and kinematic models for SFAs with an unconstrained fluidic chamber.^19,46,62 The geometrical center of the chambers had been shown to shift toward the NA during inflation, consequently impacting the net bending moment.¹⁹ Based on our findings, we postulate that the NA itself undergoes a shift. Even if the fluidic chambers are constrained and maintained fixed geometrical centers, the cross-sectional area and geometric nonlinearities still apply. Thus, the deflection angle cannot be assumed constant. This would affect our future force sensing characterization for configurations at bending planes along the $α \in [0 °, 120 °]$ . The shape of the SFA can be tracked using an RGB-D camera to geometrically identify the NA and account for possible deviations from the bending plane.³⁸

B. Chamber axial and radial kinematics

To the best of our knowledge, this is the first attempt to examine internal structure deformation of unconstrained fluidic chambers using CT imaging. The conservation of volume assumptions allows the development of (12) to estimate internal chamber expansion with relative accuracy. Chamber deformation should be uniform since we assume homogeneous mechanical properties. Yet, chamber expansion experiences a trapezoidal distribution and deforms out of a circular shape.

CT artifacts complicate the assessment of chamber deformation, but discrepancies in chamber expansion may be due to manual fabrication. The springs that reinforce the conduits are inserted postmolding and are plastically elongated toward the female end cap for enhanced gluing adhesion. Furthermore, the PET mesh sleeve has a nonuniform shape. Actuator fabrication and spring insertion will be revised to improve mechanical properties.

The trapezoidal distribution is probably also caused by trapped air inside the actuation chambers. Contrast agent and trapped air exhibit different buoyancy, causing air to rise toward the female end cap. Air has low X-ray attenuation, resembling the silicone body as a dark, low-density region making manual segmentation challenging. Furthermore, the presence of air violates volume constraint and incompressibility of actuation fluid. In general, air bubbles are a hurdle in microfluidic systems.⁶³ The tubes connecting the syringe pump and soft segment are long and have an inner diameter (ID) of 1.25 mm. Tubes with 3 mm > ID > 200 µm are classified as minichannels.⁶⁴ As fluid mechanics indicates, degassing these minichannels is exceptionally challenging, primarily due to the significant friction between the fluid and the tube wall. This high friction also creates pressure losses and is why the hydraulic circuit in our experiments was reduced using solenoid valves.

C. Actuation pressure

The performance of (14) in estimating $P_{m}$ confirms that hyperelastic models should be used for modeling the actuation pressure in soft robots. However, additional modeling should be integrated to account for viscoelasticity and hysteresis. According to our force sensing methodology (1), any inaccuracies in $P_{m}$ will affect pressure drop calculations ( $Δ P)$ , thereby influencing the estimation of external forces. Therefore, comprehensive characterization in the future is necessary. The Maxwell model is commonly used to describe viscoelasticity,⁵⁹ and the hysteretic behavior, also known as Mullins effect, can be modeled via the Prandtl-Ishlinskii framework.^65,66 Selection of sensory hardware to limit noise is also crucial. Equations (10 and 14) are only valid for $Δ V_{w} \in [0, 9]$ mL.

D. Force estimation

Our comprehensive modeling, which accounts for nonlinearities, allows intrinsic force sensing using fluidic pressure feedback with an MAE of 0.32N. Since geometric nonlinearities affect NA location, the algorithm derived here can be considered accurate only for the examined maximum bending and force magnitude ( $F < 6 N$ and $θ = 90 °$ ). Bending and force range is sufficient, given that anchoring in self-propelling endoscopes is achieved at $θ = 90 °$ and that maximum forces encountered during endoscopy, using the MorphGI probe, are below 2N.⁴⁷ Comparing the G matrix from the “Point Load” and “Distributed Force” experiments at θ = °0 shows that contact area influences shear force estimation, necessitating the integration of suitable contact point algorithms.⁶⁷ In addition, the sensor, which demonstrates comparable accuracy to previous studies,^36,37 estimates better concentrated rather than distributed forces.

The compliant PBFSS faces challenges transitioning to real-life applications due to the complex physical phenomena involved. Obtaining curved configuration kinematics relies on the assumption of constant curvature,^47,68 but curvature may vary due to the fabrication method or external forces.⁶⁹ Our future focus on force estimation involves integrating fluid flow sensors for dynamic capture and increased data collection. Machine learning techniques with analytical equations will then be utilized to enhance the robustness of our G matrix and identify trends across configurations.^70,71 In addition, we will explore the application of forces along the length of the silicone body.

Conclusion

This study leverages hydraulic fluid as a dual-role medium to meet the growing demand for highly flexible actuators and sensors. We introduce analytical methods to estimate the actuation pressure and the pressure variations caused by two external force types (concentrated and distributed) in various segment configurations. CT scan images are used to examine the internal deformation of the unconstrained actuation chambers—a novel approach that has not been previously undertaken. Furthermore, the research underlines the importance of accounting for cross-sectional and geometric nonlinearities, offering insights that enhance the design, manufacturing, and modeling of soft robots. Our proposed force sensing algorithm achieves high accuracy with an MAE of 0.32N for forces up to 6N.

Footnotes

Acknowledgments

The authors would like to acknowledge the Centre for Artificial Intelligence and Robotics (CAIR) in Hong Kong for providing facilities and equipment used in some of the experiments conducted.

Authors’ Contributions

D.M.: Conceptualization, methodology, software, formal analysis, investigation, data curation, writing—original draft, visualization. G.Z.: Methodology, investigation. S.W.: Investigation, resources; W.H.: Investigation, resources. L.L.: Conceptualization, writing—review and editing, supervision. B.Y.: Formal analysis. W.X.: Writing—review and editing, supervision. H.L.: Conceptualization, writing—review and editing, supervision, project administration, funding acquisition.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This research is funded by an internal King’s College London (KCL) grant from CDT in Surgical & Interventional Engineering, by the National High-Level Hospital Clinical Research Funding (2022-PUMCH-C-012) from Peking Union Medical College Hospital, by the CAMS Innovation Fund for Medical Sciences (2023-I2M-C&T-B-008), and the InnoHK program. The PET/CT scanning equipment is funded by a Wellcome Trust grand (WT 084052/Z/07/Z).

Supplementary Material

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Supplementary Material

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Fluidic Multichambered Actuator and Multiaxis Intrinsic Force Sensing

Abstract

Introduction

Materials and Methods

A. Mechanical modeling and stresses

B. NA position

C. Chamber axial and radial kinematics

D. Actuation-pressure modeling

E. Pressure variation and intrinsic force sensing

F. Experiments

Results

NA location

Chamber axial and radial kinematics

Actuation pressure

Pressure variations under external loading

Discussion

A. Neutral axis

B. Chamber axial and radial kinematics

C. Actuation pressure

D. Force estimation

Conclusion

Footnotes

Acknowledgments

Authors’ Contributions

Author Disclosure Statement

Funding Information

Supplementary Material

References

Supplementary Material