Experimentally Identified Models of McKibben Soft Actuators as Primary Movers and Passive Structures

Soft robots made of fluid-driven actuators join body and actuation, relying on the same soft actuators that deform the robot to support external loads. This combination of structure and actuation is inherent to the function of many soft robots, such as soft grippers or arms, which rely on compliance to disperse force, conform to surfaces, and handle delicate objects safely [1,2]. Fluid-driven actuators have an actuation region or a set of pressures and strains that produce force in the intended direction. McKibben actuators, for example, contract and produce a tensile force in their actuation region. Actuation regions are well studied, and experimental and analytical models exist to predict behavior [3–5]. However, there exists no guarantee that soft actuators will always be used in their actuation region. Actuators under external load may be pushed or pulled beyond their maximum stroke and into a state of pressurized compression or extension. Multidirectional soft arms combine two or more actuators [6–9], and arm behavior is dictated by the combination of the pressurized and passive (i.e., unpressurized) actuator reaction forces. This unification of robot body and primary mover requires an actuator model that covers a more complete set of strain–pressure states in order to accurately predict soft robot behavior under internal and external loads.

Models of soft robots commonly employ an actuator stiffness characterization, such as stiffness parameters in rod and beam models of soft arms. Existing techniques rely on actuator behavior simplifications focused on the actuation region [10] or empirically determine stiffness from a complete soft arm [11]. These techniques, while sufficient in limited cases, are not generalizable across designs and cannot predict when the actuator ceases to function as an actuator. Many arm and gripper bending models consider only uniaxial forces [10,12–14], but even a uniaxial force model must consider the possible strain and pressure permutations. Prior work applied a limited passive and actuated piecewise stiffness characterization to soft arm bending under no load [15] but did not consider all strain states.

A fluid-driven actuator’s uniaxial response space encompasses the reaction force at all feasible combinations of uniaxial strain and pressure. Figure 1(a) illustrates a McKibben actuator’s uniaxial response space. Force, though not shown, is measured perpendicular to the strain ɛ and pressure P axes, with a value at each (ɛ, P) combination. Passive lines, when the actuator is unpressurized, lie along the x-axis. External loads, from the environment or other actuators in the robot, cause passive deformations. Pressurized compression may occur when a compressive load is placed on a maximally contracted actuator, while pressurized extension can occur when a tensile load greater than the blocked force is applied.

A methodology for experimentally identifying the actuator reaction force in each region of the response space is demonstrated. The methodology is applied to McKibben actuators, a subset of fiber-reinforced, fluid-driven actuators. McKibben actuators are representative of general fluid-driven actuators, which all have an actuation region, a passive region, and can be deformed opposite their actuation direction. Variants may extend, twist, or bend, and their particular actuation direction determines the response space layout. Standard McKibben actuators are studied because they produce higher forces than extending variants [16] and remain popular in soft arms and grippers [14,17]. McKibben actuators are composed of a cylindrical elastomeric enclosure (e.g., silicone, rubber, polyurethane) that is wrapped by a braided set of fibers with a helix angle below 54.7 deg, as measured from the helix axis [3]. Each actuator is a heterogeneous structure, where the geometry and properties of the tube and sheath determine the location of features in the response space, such as the maximum contraction strain ɛ_limit or the maximum extension strain ɛ_max.

Analytical models of McKibben actuators focus on actuation and neglect regions such as pressurized compression and extension [3–5,18–20]. Experimental models of McKibben actuators also exist and similarly focus on actuation [9,16,20]. Neglecting non-actuation deformations, particularly those caused by high external loads, makes these models ill-suited to analyze soft robots interacting with the world.

The primary contribution of this article is to extend experimental identification to passive extension, pressurized extension, passive compression, and pressurized compression. These separate strain–pressure states are unified by considering each region as one piece of a single response space. The developed models are intended for use in models of more complex soft robots, specifically those that deform constituent actuators out of the actuation region (e.g., soft arms). Experimental identification provides an inherent measure of accuracy and simplicity. Elastomer and sheath material properties are not required nor are friction coefficients or sheath braid parameters. While experimental actuator characterizations are specific to each actuator design, the characterizations can be combined with bending models to allow design studies of soft arms with any number or arrangement of actuators. Seven McKibben designs with varying wall thickness, tube material durometer, and actuator diameter are examined. Each design is fit to the same model forms. The developed model forms assume length independence, which is explored experimentally and are continuous at region boundaries.

Section 2 defines and describes a McKibben actuator’s response space, and Sec. 3 reviews existing models in the context of the defined regions. Section 4 describes experimental methods. Section 5 quantifies potential variances from time, history, and length dependencies. Region fits and design comparisons are reported in Sec. 6. Section 7 discusses applicability and future work. Appendix  A describes actuator manufacturing.

A fluid-driven actuator’s reaction force depends on the actuator’s strain and pressure. Each actuator’s response space is divided into strain–pressure regions. The size, shape, and layout of these regions depend on the actuator type. McKibben actuators have actuation regions to the left of zero strain, while extending actuators have actuation regions to the right of zero strain. Figure 1(a) shows a McKibben actuator’s response space, with examples of actuators in key regions shown in Fig. 1(b). Solid lines represent physical, measurable features (e.g., the free contraction line) of the response space, while dotted lines represent practical or imposed limits (e.g., user-imposed maximum strain).

Practical and user-imposed limits determine the response space’s outer shape. Empirical methods rely on copious data, and improperly selected limits will damage specimens before testing concludes. Limits can be identified with destructive testing, but they can be more quickly estimated via observation, consideration of system practicalities, and engineering judgement. McKibben actuator limits are

• P_limit, P_burst: The actuator’s maximum pressure is divided into a true burst pressure P_burst and a user-selected maximum operation pressure P_limit. P_burst varies by strain and, while measurable, is an impractical limit because each characterization test would be destructive. P_limit is ideally lower than P_burst at any strain. The ideal P_limit for a given design can be determined through destructive testing or engineering judgement. Stiffer elastomers generally require higher pressures; most of the tested design variants were limited to 68.9 kPa (10 psi), but variants with elastomeric tubes made from Dragon Skin 10, a stiffer elastomer, were limited to 110 kPa (16 psi). Higher pressures occasionally caused separation between actuator caps and tubes.

• ɛ_min: The user-selected largest compressive strain, which was set to a constant 55%. Larger compressive strains wrinkled some sheaths.

• ɛ_max: The user-selected maximum strain is the largest allowable extension strain. McKibben actuators are compliant when initially extended, and they stiffen as the strain increases. A low maximum strain will fail to capture the change in stiffness, but repeated, large, extension strains (or forces) may damage an actuator at strains lower than the break strain. ɛ_max was set as the strain that produced a 12.5 N tensile reaction force, which was sufficient to observe the change in stiffness for all tested actuators.

The response space’s shape and features are driven by the actuator’s physical design, and region boundaries were placed where the dominant physics changed (e.g., during actuation, the tube contacts the sheath, resulting in contraction and a tensile force, but during pressurized compression, the tube may not contact the sheath and the force is compressive). The features and regions are

• ɛ_limit: The limit strain represents the maximum actuation strain achievable by P_limit. ɛ_limit is not necessarily the nominal maximum stroke of an ideal McKibben actuator, which may only be reached at pressures over P_limit. The limit strain ɛ_limit divides region I into region I(a) and region I(b).

• ɛ_crit: The critical strain represents the theoretical maximum stroke, and it may be estimated by calculating the strain that produces a fiber angle of 54.7 deg [3]. When pushed beyond the critical strain, a standard, contracting, McKibben actuator will behave like an extending McKibben actuator.

• Region I(a): The actuator is pressurized but compressed beyond ɛ_limit, which is referred to as pressurized compression. The actuator force in this region is compressive and, for ɛ < ɛ_crit, pressure increases force magnitude.

• Region I(b): The actuator is contracted less than the limit strain, but the pressure is too low to overcome the force required to deform the sheath and elastomeric tube. Like region I(a), region I(b) is a form of pressurized compression, but the actuator is under-inflated. The actuator force is compressive, but pressurization decreases force magnitude.

• Region II(a): A McKibben actuator’s actuation region is the region where the actuator produces a tensile force at a contraction strain (ɛ ≤ 0). McKibben actuators are limited physically by a maximum actuation, but the actuation region is further limited to pressures above those required to achieve a given contraction (line IV).

• Region II(b): The actuator is pressurized but extended, which is termed pressurized extension. The reaction force and strain are tensile (normal actuation is contractile). The force is a combination of actuation and passive extension, which requires that the force be greater than at zero strain and identical pressures.

• Line III(a): The actuator is unpressurized and passively compressed. The region I(a) and I(b) models, evaluated at zero pressure, must equal line III(a) for consistency.

• Line III(b): The actuator is unpressurized and passively extended. The region II(b) model, evaluated at zero pressure, must equal line III(b) for consistency.

• Line IV: The free contraction line is defined the pressure required to contract the actuator to a given strain under no load. The reaction force is zero along the line.

Existing force–displacement–pressure relationships for McKibben actuators focus on the actuation region (region II(a) in Fig. 1). Analytical, numerical, and experimental modeling techniques have been used, but passive behavior is rarely studied, and no examples of models validated in pressurized compression were identified.

Kinematic models have been used to explore actuator design, but do not predict reaction force for a given pressure and strain [22,23]. Simple analytical models neglect losses to the elastomeric tube and sheath and are often derived from principles of virtual work or energy conservation [3,24,25]. Neglecting losses to the tube and sheath means that these models overpredict actuation force and are unable to predict passive force (line III in Fig. 1) or the free contraction line (line IV in Fig. 1)). Accurately predicting force in pressurized compression and extension similarly requires including the tube and sheath stiffness.

More complex analytical models include losses to the tube and sheath [3,18–20,26] and end effects [4,5,21]. These models may be derived using a modified virtual work formulation, energy methods, or static analysis. However, these models are also focused on the actuation region (region II(a) in Fig. 1). Passive stiffness is predicted by evaluating the models at pressure P = 0 kPa, but the analytical models generally make domain-specific assumptions (e.g., the tube is in contact with the sheath) that are violated in other regions, such as passive compression (line III(a) in Fig. 1) [4,18,19]. The models can be evaluated at strains and pressures within the pressurized compression and extension regions, but analytical models have not been validated in those regions [3–5,18–21]. One study considered negative contraction ratios (i.e., pressurized extension), but the samples had initial braid angles near neutral and pressurized extension behaved similarly to actuation [26]. The actuator shape may change considerably when highly compressed or extended, violating cylindrical shape assumptions.

Finite element, or numerical, models also focus on the actuation region. Finite element models require detailed material data, can be slow to solve, and often also assume that the sheath is in perfect contact with tube [27,28]. Such models are nominally capable of including every region but have only been developed and validated for the actuation region, including the free contraction line boundary.

Experimental fits are not as informative as physics-based models, but they can be more accurate because they are derived by measuring actuator force directly. Empirical fits generally cannot be extrapolated to different actuator designs or beyond the tested domain. The simplest experimental characterizations test free contraction, blocked force, or stiffness at a single pressure [9,16]. Multiple force–displacement–pressure models have measured forces within the actuation region [3,20,29,30]. A hybrid approach combined an analytical model form with experimentally determined model parameters [31], while implicit actuator models have been developed through experimental identification of soft systems [11]. Experimental methods are not inherently limited to the actuation region, but there exists no systematic experimental identification of a McKibben actuator’s complete uniaxial response space.

The presented method is demonstrated with McKibben actuators, but the fundamental approach can be adapted to other fluid-driven soft actuators. Each design has an actuation region, but depending on the design, that region may encompass contraction, extension, twist, etc. Similarly, most soft actuators can be deformed outside of their actuation region by external loads, and this deformation may be either in the direction of actuation or opposed to it.

A response space encompasses these possible states and forms the set of inputs that determine an actuator’s reaction force F. The response spaces considered are limited to a single deformation measure (e.g., contraction, extension). Note that reaction force is equivalent to the external load that must be applied to reach a particular point in the response space. Measuring or modeling the reaction force across the space can be simplified by dividing the space into regions that align with dominant physics or similar deformations (e.g., a McKibben actuator’s internal tube is in contact with its sheath during actuation, but not during passive compression). The process for identifying an actuator’s response space is defined as:

1. Identify the minimum and maximum deformations, which create the x-axis boundaries. These deformations need not be uniaxial strains: a twisting actuator may use torsional strain, and a bending actuator may use curvature. Extrema are design dependent but may be selected as either the likely range of operation or the limits beyond which the actuator will fail. The passive deformation line runs between the extrema.

2. Identify the maximum operational pressure, which creates the y-axis boundary. Burst pressure is likely dependent on deformation, and selecting a maximum operational pressure, P_limit, that is less than the burst pressure at any deformation simplifies testing.

3. Identify the actuation region’s boundaries. The actuation region will most commonly be bounded by the line from (ɛ = 0, P = 0) to (0, P_limit) and the free deformation line. The free deformation line is the set of deformations reached when the actuator is pressurized under no external load (e.g., the free contraction line for McKibben actuators).

4. Verify the opposing-actuation region’s boundaries. The opposing-actuation region encompasses deformations opposite the actuation direction. The opposing-actuation region for McKibben actuators is pressurized extension, but an extending actuator’s opposing-actuation region is pressurized compression. The x-axis boundaries are generally defined by the line from (ɛ = 0, P = 0) to (0, P_limit) and the line from (ɛ_max/min, 0) to (ɛ_min/max, P_limit). Certain actuators may have physical limits that merit further divisions; long, thin extensile actuators, for instance, are likely to buckle under compression.

5. Identify the region of under-inflated actuation. The free deformation line relates actuation deformation to the minimum pressure required to achieve that deformation and pressures less than the minimum under-inflate the actuator. The under-inflated region lies under the free deformation line and above the passive deformation line (region I(b) in McKibben actuators). The remaining x-axis boundary is located by evaluating the free deformation line at P_limit.

6. Verify the beyond-actuation region’s boundaries. The beyond-actuation region encompasses deformations greater than the free deformation at P_limit, in the direction of actuation (region I(a) for McKibben actuators, which is pressurized compression). This region stretches from 0 kPa to P_limit and from the free deformation at P_limit to deformation extrema in the actuation direction. Certain actuator behaviors, such as buckling or a sudden stiffness change, may require subdividing the region.

Empirical characterization requires that each region be tested to determine the reaction forces at a large number of deformation-pressure combinations, but test details, such as setup and sample length, will vary by design. Test data can be gathered by holding an actuator at a deformation and pressurizing it (strain, or deformation, isolines), by holding an actuator at a pressure and deforming it (pressure isolines), or by combining a changing pressure and deformation. Strain isolines were used because precise pressure control is not required. The final step is to produce model forms based on experimental results. Model forms are strongly dependent on the actuator design and type (e.g., extending, contracting, twisting), and while models can be piecewise, they must be continuous at region boundaries.

All regions and features of the response space, except the true burst pressure, were characterized. However, characterizing a set of individual regions with experimental fits is insufficient to produce a usable response space model. Three constraints are placed on the experimental models.

1. Models are to be consistent with the anticipated response beyond tested values, in order to avoid false solutions at unexpected strains when solving system models.

2. Models in adjacent regions must be consistent at shared boundaries, such that the predicted force at the boundary is identical, regardless of which region’s characterization is used to calculate it. This requirement ensures identical solutions, regardless of whether a numerical solver approaches from a higher or lower strain.

3. Models must be length independent in order to most closely mimic the material stiffnesses they are intended to replace. The validity of assuming length independence was examined (Sec. 5.2).

Seven McKibben actuator designs were tested. Each design was fit to the same set of experimental models, in order to determine the extent to which the proposed fit forms may generalize. The variants’ characterization data are further used to determine the relative effects of selected design variables. The seven designs range in tube material, tube wall thickness, and actuator diameter (Table 1). The manufacturing method (Appendix  A) and capping procedure were refined from accepted manufacturing techniques and were the same for each variant. All actuator sheaths were made from a commercial polyester plastic expandable sleeving (McMaster #9284K611 for 4 mm ID samples and #9284K612 for all others) and were annealed to the indicated diameters.

Sample length differed necessarily between regions with tensile and compressive reaction forces. Actuators can be expected to experience local (i.e., over part of the length) compressive forces during use that would lead to buckling if experienced over longer lengths. The goal is to replace simplified material models, which means the actuator model must predict the reaction force independent of length. Standard practice in material testing is to determine tensile properties on long, thin samples (e.g., dumbbell samples) and compressive properties on short, wide samples (e.g., buttons). A similar practice was adopted. Passive extension (line III(b)), the actuation region (region II(a)), and pressurized extension (region II(b)) were tested with nominally 200 mm long actuators, while pressurized compression (regions I(a) and I(b)) and passive compression (line III(a)) were tested with nominally 100 mm long actuators for all but one variant. The 2 mm inner diameter, 1 mm wall thickness actuator compression test specimens were nominally 75 mm due to their reduced critical buckling load.

Sample lengths were selected to minimize end effects, while maintaining a pure strain state. Actuator ends constrain the sheath and tube, while the rest of the length has an identical, unconstrained cross section. The EF30-4-1 variant was tested at six lengths, ranging from 75 mm to 200 mm (results in Sec. 5.2), which showed good agreement in the tensile results for longer samples and some sensitivity to length in the compressive results. Samples for compressive force regions were selected to have the longest length that did not buckle, while samples for tensile force regions were selected to have the longest available length.

All force–displacement tests were conducted on a Mark-10 ESM1500 tension-compression test stand, which has a position resolution of 0.02 mm. Actuators were connected via end caps with attachment points (Fig. 1(b)). The actuator’s bottom, the end with the pressure inlet, was connected to the base of the test stand. The actuator’s top was connected via a bracket to a 50 N load cell (model MR03-10). The maximum value of the load cell was selected based on the largest expected force, which occurs in pressurized extension (region II(b)). The load cell had a resolution of 0.02 N and an accuracy of 0.075 N.

Correlated pressure–force–position data were gathered by reading actuator pressure from a Honeywell TruStability 30 psi (200 kPa) pressure sensor, through an Arduino. The pressure sensor has an accuracy of ±0.08 psi (±0.55 kPa) and, in the test setup, a resolution of 0.03 psi (0.2 kPa).

Each region was characterized by displacing an unpressurized actuator to a set of pre-selected strain before pressurizing the actuator with air. Pressure and force were measured continuously during pressurization, at an interval of approximately 0.02 s. The actuation region (region II(a)) was tested in strain intervals of approximately 2% (4 mm), while the typically smaller pressurized extension region was tested in strain intervals of 0.5% (1 mm) or 1% (2 mm). Pressurized compression was tested in intervals of approximately 2%, which resulted in smaller displacements due to the shorter actuators. Actuators were pressurized manually via a syringe. The pressurization rate was not specifically controlled, but pressurizations generally took 2–4 s. Note that, except where stated, displacements were tested consecutively and without rest time between each run. Passive behavior (line III) was tested via traditional tension and compression tests.

Five samples of each variant were tested in order to characterize regions II(a), II(b), line IV, and line III(b), while three samples of each variant were tested to characterize regions I(a), I(b), and line III(a). More samples were tested in the predominantly tensile regions due to higher variations in passive extension compared to passive compression.

Simple fit forms are preferred over more complex ones. Though more complex fits can be more accurate, they risk overfitting. Other models of the complete uniaxial response space do not exist for comparison. This work focused on consistency at strain–pressure region boundaries and easily implemented fits. The number of samples makes it space prohibitive to show visual comparisons for all samples and regions, but the EF30-4-1 fits are compared visually to the experimental data in Sec. 6.5, and error metrics are used to quantify fit quality elsewhere. The fit quality metrics were also used to determine coefficients through minimization. Each quality metric is based on root mean squared error (RMSE), but select regions require a weighting factor because of an unequal measurement distribution.

The fit quality is affected by the fit form and the underlying variance in the measured samples. Multiple samples were tested for each variant and the final characterizations were determined using all samples. Errors are reported for all samples as σ_all and for averages of individual samples as $σ¯ind$⁠. Higher σ_all values are accepted for fit forms that produce low $σ¯ind$ values, because the error is caused by normal manufacturing and testing variance.

The potential variance from hysteresis, stress relaxation, and sample length are presented. McKibben actuators are heterogeneous combinations of materials (polyester plastic and silicone elastomers) with known viscoelastic properties that are convenient to model as time-, history-, and length-independent materials with homogenized properties. While hysteresis and stress relaxation are understood to occur in soft actuators, the models that do not include these effects are simpler and faster to create and compute. This section evaluates the possible variance caused by factors not captured in the presented empirical models.

Hysteresis tests were conducted using the EF30-4-1 variant for three passive extension and three passive compression strains (2%, 4%, and 6% in extension, 15%, 30%, and 45% in compression). The stress relaxation of the EF30-4-1 variant was tested at the same three strains with a 15 min hold at the designated strain. The same tests were conducted on sheath-only test samples of otherwise identical length and diameter, in order to investigate the sheath’s contribution to each effect. The reaction force of the elastomeric tube alone was estimated to be <0.1 N, and it was not tested. The hysteresis and stress relaxation results are plotted in Fig. 2.

The actuator relaxed to 89.7% of its initial force after 15 min at 45% compression and to 80.5% at 6% extension. Hysteresis increases with strain magnitude, from a maximum difference between loading and unloading of 0.1 N at 15% compression to 0.31 N at 45%. The maximum difference between loading and unloading in extension was 2.9 N. The sheath is responsible for the majority of the actuator’s compressive stiffness and hysteresis and stress relaxation in both directions. The sheath and actuator passive compression force were within 1% at 15% strain, and they differed by only 15% at 45% strain. Passively extending a sheath took 25% of the force required for an actuator, but the sheath’s hysteresis was 40% that of the actuator (1.16 N versus the actuator’s 2.9 N). The reaction force of the elastomeric tube is negligible at 6% strain, which suggests that actuator force is caused by interactions between the tube and sheath.

Hysteresis and stress relaxation produce the largest percent variance at high extension strains (ɛ > 3%). Extension strains increase the sheath thread angle and place threads more directly under tension, which increases relaxation. The largest variations are expected in pressurized extension, because pressurization further increases thread tension.

An ideal McKibben actuator, with a constant cross section for its entire length, will have the same reaction force at the same strains, regardless of length. The actuator ends, however, introduce non-uniform behavior. The response of the EF30-4-1 variant for several lengths was compared in passive compression, passive extension, the actuation region, and for the free contraction line.

Six lengths were compared for passive extension, ranging from 200 mm to 75 mm (Fig. 3(a)). The three shortest lengths are also compared in passive compression (Fig. 3(c)). Passive extension reaction forces were similar for 150–200 mm long samples, but sample variance increased as length decreased. The most likely cause of variance is uneven clamping of the sheath to the end cap. Actuator ends are a larger fraction of the overall length in shorter actuators, and variance in the ends would have a greater effect. Shorter lengths led to visibly larger passive compression forces, but the cause is more difficult to isolate. Shorter lengths are more sensitive to actuator end stiffness, but shorter lengths are also less likely to buckle. The 125 mm long sample was observed to buckle slightly near 50% compression, but the 100 mm and 75 mm samples were not.

The free contraction lines (line IV) for all six lengths are plotted in Fig. 3(b). Strain isolines from the actuation region (region II(a)) for 75 mm, 125 mm, and 200 mm samples are compared in Fig. 3(d). The free contraction line and actuation region show less effect from actuator length than the passive results. The largest difference occurs at the shortest length. During characterization, tensile regions were tested at 200 mm in order to minimize end effects, and compressive regions were tested at the longest length with no visible buckling in order to minimize end effects to the extent possible while maintaining a pure compressive strain.

Comparative results are presented for passive extension, passive compression, and the actuation region, and fit results are presented for all tested regions. Fit forms, which were developed for each region based on behavior, are presented and discussed. Physics-based models were used occasionally to guide fit form, but emphasis was placed on forms that are numerically similar to the experimental results.

Each region was fit using individual actuators to evaluate appropriateness, but final coefficients were fit using all samples for a given variant. Regions were fit in the following order, in order to reduce inaccuracy from boundary conditions in earlier regions: line III, line IV, region II(a), region II(b), region I(b), and region I(a).

Passive extension and compression (line III) are analogous to uniform tension and compression material testing, and like many composite structures, actuators can have different stiffnesses in tension and compression. The tensile stiffness of all actuator variants was drastically higher than the compressive stiffness, and both directions were nonlinear (Fig. 4). McKibben actuators are known to have limited passive extension [3], and the amount depends on weave density and the thread angle, which is related to diameter by sheath annealing (Appendix  A). Passive compression is less affected by material and wall thickness because the response is sheath-dominated.

The response space boundary strains, ɛ_min and ɛ_max, were selected to be as high as possible, while minimizing the chance of damaging the actuator (Table 2). ɛ_max was selected as the strain required to generate an extension force of 12.5 N. The maximum strain was defined in terms of a force, because force corresponded to a rapid increase in stiffness in all tested actuators, while the maximum strain varied considerably. This force visually corresponded to a braid angle near the shear lock angle, where deformation can no longer be accommodated by geometric means. ɛ_min was chosen as 55%, because initial tests to higher compressive strains wrinkled and permanently deformed the sheath.

The fit coefficients and qualities are provided in Tables 3 and 4. The higher σ_all values (indicating a poorer fit quality) are driven by sample variation. The only variant not well fit (⁠$σ¯ind>0.5$⁠) by the proposed forms is the thinnest wall actuator. The cause of the irregular behavior of the thinnest wall variant during extension is unknown, but the response was similar for all samples.

Lower order polynomial fits were also considered. The average error for all samples across all variants for a quadratic fit of the extension response was 0.35 N and 0.15 N for cubic fits. The average error for all samples of all variants for a cubic fit of compressive response was 0.057 N, while the selected fourth-order fit had an average error of 0.041 N.

The behavior of each passive extension fit beyond the tested region was checked, and only the DS10-4-1 fit changed concavity in the extrapolated domain. All compressive fits maintained the same concavity up to a strain of −0.8, and the EF30-4-1, EF30-2-1, EF30-6-1, EF30-6-0.075, and EF30-6-1.5 variants maintained concavity past −0.9 (note that −1.0 is equivalent to 100% compression or perfectly flat).

The actuation region (region II(a)) is bounded on the top by a user-selected maximum pressure P_limit and on left and bottom by the free contraction line (line IV). An ideal McKibben actuator, with no losses to the sheath or bladder, can contract until the maximum volume is reached, which nominally occurs when the sheath angle reaches 54.7 deg, measured from the helix axis [3]. However, energy is lost to the tube and sheath, which limits the amount of contraction possible at a given pressure. The free contraction line relates actuator pressure to contraction strain when there is no load on the actuator.

Third- and fourth-order polynomial fits of the free contraction line were investigated. The fit quality was calculated using Equation (1). Third-order fits had, across all samples of all variants, an average error of 1.42 kPa, while fourth-order fits had an error of 1.00 kPa. Maximum pressures ranged from 68.9 kPa (10 psi) to 110 kPa (16 psi), which means the improvement, as a percentage of the maximum pressure, is 0.6% and 0.4%, respectively. A third-order fit was selected as the simplest fit beyond which returns diminish. The fit coefficients and qualities are provided in Table 5.

The free contraction line data extracted from compression samples, gathered during pressurized compression testing, were not used to in line IV fits. The last column in Table 5, σ_c, gives the quality of the presented coefficients for the compression sample data. Mismatches are expected due to end effects and sample variance, but the extent of the mismatches is an indication of the deviations likely to be introduced by imposing continuity between regions I(b) and II(a).

The fit coefficients and quality metrics for region II(a) are given in Table 6. All fits use the free contraction line coefficients from Table 5. Improving free contraction line fits did not necessarily provide equivalent improvements in the actuation region fit. For instance, fitting P_IV to a fourth degree polynomial for variant EF30-6-1 improved σ_all, IV from 2.19 kPa² to 0.74 kPa², but only improved σ_all, II(a) from 0.13 N² to 0.12 N². Poorer fits occurred in actuators with nonlinear initial pressure responses (see Fig. 5(b), Dragon Skin 10 at ɛ = 0), which cannot be matched by the model.

Reaction forces for two strain isolines and the free contraction line are compared in Fig. 5. Eco-flex 00–20 and Eco-flex 00–30 show nearly identical responses, as do variants with wall thicknesses of 0.75 mm and 1 mm. The response similarity suggests there is a point of diminishing returns on softening tubes to reduce lost energy. Dragon Skin 10 requires more than double the pressure to contract, which is not compensated for by the maximum achieved force.

Pressurized extension produces the highest reaction forces and is the region subject to the most stress relaxation and hysteresis. Samples were tested sequentially with no rests, due to the impracticality of testing all samples with a rest between each strain isoline. One EF30-4-1 sample was tested with a 30 min rest between each strain to investigate variance (Fig. 6(a)). The passive extension force was extracted from the pressurized extension results at P = 0 (Fig. 6(b)). The unrested test passive extension force was lower than the rested test and the tensile test, as expected. The rested test reaction force approaches the unrested test as pressure increases, but the initial difference lowers fit quality when the region III(b) boundary condition is applied.

The pressurized extension region was fit three times in order to demonstrate the effect of sample variance, exacerbated by stress relaxation and hysteresis, and boundary conditions on fit quality. Fit quality was calculated using Equation (2). The unconstrained column in Table 7 provides the average fit quality of individual fits with no boundary conditions. The constrained column also presents individual fits, but uses F_III(b) from Table 4 and enforces Equation (9). The third column, All Data, enforces the same boundary conditions and fits to all samples. Boundary constraints and sample variance are the largest drivers of poorer fits, and individual samples are well fit for five of the seven variants. Fit coefficients are given in Table 8.

Pressurized compression (region I) encompasses the largest range in behavior and the most complex mechanics. Region I is split into regions I(a) and I(b) at ɛ_limit, which is the strain at which an actuator pressurized to P_limit produces zero force. Region I(b) lies under the free contraction line, and the behavior transitions from passive stiffness to actuation. Region I(a) encompasses the rest of the strains, stretching left to ɛ_min. The actuator behaves like a McKibben actuator from ɛ_limit to ɛ_crit, and the strain isolines have a positive slope. The actuator reaches the true critical strain ɛ_crit, and the slope of the strain isoline becomes negative. Past ɛ_crit, the actuator behaves like an extending actuator (Fig. 7(a)). Existing analytical models predict the change from contraction to extension [3,18] but have not been formulated or validated for pressing a contracting actuator until it reaches a braid angle that generates extension. The tube does not remain in perfect contact with the sheath and does experience wrinkles, which violates one of the primary assumptions of existing analytical and numerical models.

The fit qualities and coefficients are given in Table 9. Fit quality was calculated with Equation (2). The boundary conditions limited fit quality, as they did in pressurized extension, but the greater limitation was the complexity of the mechanics. The reaction force at the edges of region I is well defined by linear pressure responses near P_IV or P_limit, and zero slope lines equal to the passive compression force at P = 0 kPa (Fig. 7(a)). The center of region I, however, is determined by interactions between the sheath and the wrinkled or folded tube, as the tube inflates. Folds likely affect inflation, but the extent of effect is different between an actuator with a 0.75 mm thick wall, made from Eco-flex 00-30, and one with a 1 mm thick Dragon Skin wall. The characterization assumes path independent, sheath-dominated behavior. Path-dependent models, while potentially more accurate, are also much more complex. The region I model forms were selected because they capture the dominant behaviors and meet the boundary conditions.

The fit coefficients from Table 3 were used for the line III(a) boundary. The passive compression data extracted from the pressurized compression test are compared to the line III(a) test for one variant in Fig. 7(b). Line III(a) was tested at a rate of 30 mm/min (total test time of 1.8 min), while pressurized compression testing took approximately 30 min to test all strains. The difference between the two lines is likely caused by stress relaxation and is a possible source of variance at high compressive strains. The fits in Table 5 were used for the free contraction line, for consistency with region II(a), which is another potential source of variance. Shorter samples are required for compression, but the larger end effects caused RMSEs of at least 2 kPa when the free contraction coefficients from tensile samples were evaluated with data from compression samples. The difference between the free contraction for one 200 mm and one 100 mm long EF30-4-1 actuator is shown in Fig. 7(c).

At the far left edge of region I(a), the compressed actuator is susceptible to a type of buckle. The compressive reaction force increases with strain and pressure and eventually reaches a strain where the lower energy state is extended and bowed left or right (Fig. 1(b), ɛ_min). This condition is not necessarily a failure but may be undesirable.

Figure 8 compares the fits for all regions and features of the EF30-4-1 variant to the experimental data. The complete model uses 18 fitted coefficients (ɛ_crit and ɛ_limit are calculated directly, not fit through minimization), but each region averages three coefficients. The ratio of coefficients to data points in most regions is on the order of 1:100 or 1:1000. The feature most susceptible to potential overfitting is line IV, which has the fewest points. However, visual comparisons, like the one in Fig. 8(b), do not show evidence of overfitting when fit with a cubic polynomial.

Figure 9 shows the full EF30-4-1 response model, with the tensile regions shown in red and the compressive regions shown in blue. The boundary between tensile and compressive regions is, by definition, the free contraction line.

This article considered the complete strain–pressure response space of a McKibben actuator, as a representative of a larger group of fluid-driven soft actuators. Fluid-driven soft actuators have been well studied as primary movers, but soft systems are also used as support structures. Complex loadings, such as those that may be encountered in unstructured environments, are likely to push soft actuators out of the actuation region and into states of passive or pressurized extension and compression.

A McKibben actuator strain–pressure space was divided into regions, which were selected based on their core behavior (e.g., during actuation inflation expands the sheath to produce contraction with a tensile reaction force). Grouping like behavior allowed the development of simple, experimental reaction force models for each region. Fit coefficients and qualities were presented for seven variants. The proposed fits were more successful for actuators made from low durometers (e.g., Eco-flex 00-30), with relatively thin walls, than actuators made from stiffer elastomers (e.g., Dragonskin 10). Regions were fit in the following order to reduce inaccuracy from boundary conditions in earlier regions: region II(a), line IV, line III, region II(b), region I(b), and region I(a). The largest error occurred at high compression strains and pressures, due to tube buckling and boundary constraints from other regions. More complex fitting techniques, such as a surface fit or a machine learning, may produce for accurate models for pressurized compression, but any higher order model will be more susceptible to overfitting. The presented results reinforce the complexity of soft actuator mechanics and emphasize additionally the role of sample variance, hysteresis, and stress relaxation as a source of uncertainty. Future work may consider model extensions to include these effects.

While McKibben actuators were the focus of this work, they are not the only actuator that may be operated outside of their actuation region. Extending actuators, whether they use a braided sheath or reinforcing rings, may be compressed or pulled beyond their maximum stroke. Bending actuators may be curved further or backward. The region shapes of other actuators’ response spaces will vary, and the specific fits may not apply. However, actuators are expected to share certain characteristics, such as a free deformation line, and models will share common requirements, most notably that boundaries between regions be consistent. The general methodology (Sec. 4.1) may be used to identify regions and shapes within the response space, but work remains in determining appropriate test setups and model forms.

The presented experimental models are intended to improve soft roboticist’s ability to model, predict, and understand more complex soft structures built from constituent actuators. Soft arms, most commonly constructed from parallel combinations of contracting or extending actuators, are particularly targeted. Actuators within these arms will experience external loads while in use, which may push or pull the actuators out of the actuation region. The experimental models can replace more simplistic force expressions in arm models, including constant curvature, beam, or rod models. The presented results show highly variant and nonlinear actuator behavior across the response space and highlight the importance of considering all possible strain–pressure combinations. The technique presented is a foundation for building holistic and precise soft actuators models, and it can be extended to other actuators for use in grippers or locomoting soft robots.

Gina Olson is supported in part by the by the National Science Foundation (award IIS-1734627).

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtained from the corresponding author upon reasonable request.

Each elastomeric tube was manufactured with the following procedure:

1. Clean the mold halves, mandrel, and funnel with isopropyl alcohol and apply release agent. Allow 5 min for the release agent to dry (Fig. 10(a)).

2. Assemble the mold and place it in the degassing chamber (Fig. 10(b)).

3. Mix silicone (e.g., Eco-flex 00-30) Part A and B using a Thinky ARE-310 mixer.

4. Pour the mixed silicone into the funnel and immediately begin degassing. Degas for 2 min and 30 s.

5. Remove the mold from the degassing chamber and allow silicone to fall to the bottom by gravity. As needed, shake the mold on an orbital shaker table at 80–100 rpm.

6. When the silicone reaches the bottom, block the lower vent hole with clay.

7. Allow the silicone to cure and demold.

The sheaths were annealed to the desired configuration prior to actuator assembly. Sheaths for EF30-2-1 were initially nominally 1/8 in. in diameter. Sheaths for all other variants were initially nominally 1/4 in. in diameters.

The annealing procedure was

1. Slide the sheath onto a rod with the same diameter as the final desired inner diameter of the sheath.

2. Trim the sheath to the length of the rod and place the rod and sheath in the oven.

3. Heat to 140 °C at a rate of no more than 20 °C/h.

4. Dwell at 140 °C for 1 h.

5. Cool from 140 °C to room temperature at a rate of no more than 10 °C/h.

6. Remove the annealed sheath from rod.

The completed sheath and tube were cut to the desired length and then assembled to the caps with cable ties (Figs. 10(c) and 10(d)).