Pneumatic Supply System Parameter Optimization for Soft Actuators

Abstract

Soft actuators using pressurized air are being widely used due to their inherent compliance, conformability, and customizability. These actuators are powered and controlled by pneumatic supply systems (PSSs) consisting of components such as compressors, valves, tubing, and reservoirs. Regardless of the choice of actuator, the PSS critically affects overall performance of soft robots because it governs the soft actuator pressure dynamics, and thereby, the general dynamic behavior. While selecting and controlling PSS components for meeting desired soft actuator performance, specifications such as PSS mass, volume, and duration of operation must also be considered. Currently, there is no comprehensive study on PSS optimization for meeting dynamic performance and PSS specifications, due to limited understanding of soft actuator pressure dynamics, large solution space for PSSs, and variability in soft actuators. By considering critical parameters of PSS and soft actuators, we introduce and demonstrate PSS parameter optimization. We propose a normalized model for soft actuator pressure dynamics and quantify the relationship between PSS parameters, soft actuator design parameters, and dynamic performance metrics of rise time, fall time, and actuation frequency. After experimental validation, we applied these results and optimally select and control PSS components to meet desired soft actuator performance for a soft exosuit, while minimizing mass of selected components. The measured pressure response with this prototype agrees well with simulations, with root mean square errors <5.2%. This work is a step toward furthering the scope of soft robotics, as it enables PSS modeling and optimization, for meeting the desired soft actuator performance while also addressing PSS specifications.

Introduction

Soft robotics^1,2 provides customizable solutions to a range of applications such as wearable devices,^3–9 mobile robots,^10–14 and grippers^2,15,16 due to unique characteristics such as compliance, conformability, and versatility. A significant number of soft robots generate motion through soft pneumatic actuators (SPAs) (Fig. 1B) powered by pneumatic supply systems (PSSs).^{4–13,17–21} PSSs (Fig. 1A) typically consist of three main components: source for generating pressurized air, pneumatic line for connection, and valves for controlling flow direction.²² By virtue of the pneumatic flow, PSSs govern the soft actuator pressure dynamics, and thus the overall soft robot dynamic behavior.^1,2,22 To meet dynamic performance specifications such as rise time, fall time, or actuation frequency, PSSs have to be tailored, that is, the PSS components and their control parameters such as source pressure, valve duty cycle, and compressor duty cycle must be selected carefully. Regardless of the soft actuator used, improper selection or control of PSS components can lead to either too slow or too fast dynamic response, or can even damage the soft actuator.

FIG. 1.

A PSS powering SPAs. (A) Schematic of a PSS showing the main components. (B) Different actuation patterns achieved by soft actuators when pressurized (P $>$ 0). (C) General trend depicting how improved soft actuator dynamic performance leads to bulkier PSSs. PSS, pneumatic supply system; SPA, soft pneumatic actuator. Color images are available online.

Accordingly, researchers have studied some aspects of PSSs for soft robotic applications. Wehner et al.²² compared portability and relative merits and demerits of different choices for the pneumatic source such as battery-powered compressors, liquid $C O_{2}$ , and peroxide decomposition. However, this study does not include the role of valve, pneumatic lines, and soft actuator design parameters such as internal volume, operating pressure, and maximum expansion. In addition, it does not investigate soft actuator dynamic behavior. In another study,²³ researchers heuristically tuned source pressure and valve duty cycle to increase maximum actuation frequency from 2 to 4 Hz. Although effective for tuning PSS control parameters, such a heuristic approach is insufficient for PSS component selection. Furthermore, it cannot predict how dynamic behavior will be affected if the soft actuator or a PSS component is changed. By employing finite elemental analysis (FEA), Marchese et al.¹³ calculated the source pressure required to achieve high-speed inflation response in a soft robotic fish. However, their results were limited to the specific application and do not lead to a general understanding of PSSs' influence on the soft actuator dynamic behavior. As of yet, the relationship between PSS components, soft actuator design parameters, and dynamic performance is not well quantified.

Although there are numerous studies on soft actuator kinematics, deformation and force interactions using analytical modeling,^14,24–26 rigid-kinematic approximations,^27–29 or FEA,^30,31 soft actuator pressure dynamics are relatively less explored. The major reasons limiting study of soft actuator pressure dynamics are (i) nonlinear nature of pneumatic flow, (ii) large variability in PSSs with respect to components and their control parameters, (iii) nonlinear nature of soft actuator behavior due to material properties, design geometry, and large deformations, and (iv) large variability in soft actuator design with respect to design, materials, and operating pressures.^1,32 As a result, soft actuator dynamic response is most commonly modeled by fitting the experimentally measured response to first or second order linear systems.^{6,12–14,29,33,34} Such models do not relay information about the underlying flow mechanics, and are insufficient for customizing PSSs to meet performance requirements. In addition to performance, applications often dictate specifications on the PSS in terms of its mass, volume, tubing length, and duration of operation; and a trade-off has to be made in PSS design, as improved dynamic performance necessitates a bulkier PSS as shown in Figure 1C.²² Currently there is no systematic method for PSS component selection and control to simultaneously meet specifications of dynamic performance and the PSS.

By studying the effect of critical parameters of PSSs and soft actuators on dynamic performance, here, we introduce parameter optimization. Based on ideal gas laws and the ISO 6358 standards,^35,36 we propose a comprehensive model for pressure and flow dynamics of soft actuators. This normalized nonlinear model is applicable to a wide range of PSSs with respect to source, valve, and pneumatic line, as well as to soft actuators with varying sizes, operating pressures, and amount of deformation. After validating the results experimentally, we apply them to the design and testing of a portable PSS for a wearable robot, which meets desired dynamic performance while minimizing mass of selected components. In addition to improving the understanding of soft actuator pressure dynamics, this study enables optimal design through selection and control of PSS components to simultaneously address specifications of PSS and soft actuator performance. The main contributions of this study are

a comprehensive model for soft actuator pressure dynamics,

investigation of the impact of PSS parameters and soft actuator design parameters on actuators' dynamic performance, and

validation of PSS parameter optimization through novel soft exosuit prototype design and testing.

Proposed Approach for PSS Parameter Optimization

The source of the PSS generates the air pressure, which drives the flow through the valve and pneumatic line to the soft actuator. Based on the pressure generation mechanism, PSSs in academia and industry can be largely categorized into two configurations as shown in Figure 2.

FIG. 2.

The two PSS configurations: (A) Constant pressure PSSs: PSS with pressure regulator at the source.^2,13,23 (B) Constant flow PSSs: compressor-powered PSS without pressure regulator.^10,15,37 Color images are available online.

Configuration A: PSSs with a pressure regulator at the outlet of either large off-board compressors, high pressure gas cylinders (20 or 30 MPaG), or liquid CO₂ cylinders (5.5 MPaG) as the source (Fig. 2A).^2,13,23 Assuming perfect regulation, we model these as constant pressure PSSs.

Configuration B: PSSs consisting of an on-board compressor and a reservoir as the source (Fig. 2B).^10,15,37 We model these as constant flow PSSs.

For the flow propagation through the PSS, we use the ISO 6358 standard model,^35,36 which is applicable to a diverse range of commercially available components for the valves and pneumatic line. After deriving the governing equations of pressure and flow dynamics, we identify critical parameters of these PSS components affecting pneumatic flow, and classify them into two types as follows:

Flow parameters, f: These include nonvarying pneumatic parameters such as internal volumes of reservoir, tubing, and fittings; and flow capacities of regulators, valves, compressors, tubing, and fittings. These parameters form a discrete set of values.

Control parameters, c: These include actively controlled parameters such as source pressure, and duty cycles of compressor and valves, which can take continuously varying values.

Next, we identify the soft actuator design parameters affecting its pressure dynamics, and denote them as $α$ : operating pressure, volume, and extent of deformation. Table 1 lists the flow, control, and soft actuator design parameters considered for this study.

Table 1.

Considered Parameters of Soft Pneumatic Actuator and Pneumatic Supply System

Classification		Symbol	Description
f	PSS flow parameters	$C_{e q}$	Equivalent sonic conductance
		$b_{e q}$	Equivalent critical ratio
		$Q_{c o m}$	Compressor flow output^a
		$V_{r e s}$	Reservoir volume^a
c	PSS control parameters	$P_{s r c}$	Source pressure
		D_V	Valve duty cycle
		D_C	Compressor duty cycle^a
$α$	Soft actuator design parameters	$P_{o p}$	Operating pressure
		$V_{s p a}^{0}$	Internal volume
		$K_{e x p}$	Expansion ratio

Only applicable for configuration B.

PSS, pneumatic supply system.

Lastly, we categorize design requirements as:

Dynamic performance specifications: These include the soft actuator fall time, rise time, and maximum actuation frequency.

PSS specifications: These include the PSS volume, mass, tubing length, number of outputs, and total air delivery.

For optimizing PSSs, the relationship between PSS parameters, soft actuator design parameters, and the mentioned specifications must be quantified. In the subsequent sections, we derive the model for soft actuator pressure dynamics, and simulate its pressure response while varying $f,$ c, and $α$ , to create a direct relationship between these parameters and dynamic performance metrics. Using this relationship, we calculate the range of PSS parameters f and c that meet performance requirements and compare commercially available options for source, valve, and pneumatic lines to find the optimal set of PSS components and parameters.

Modeling Pressure and Flow Dynamics

To study pressure dynamics of soft actuators systematically, we use existing established models for air flow through PSS.^36,38–42 In addition, we propose a new quasi-static pressure–volume relationship for modeling soft actuator expansion. The commonly used assumptions in pneumatic studies^36,38–42 are as follows: (i) there is no air leakage; (ii) air follows ideal gas equations; (iii) the ambient is at standard pressure and temperature, P₀ = 0.1 MPaA, $T_{0} = 293.15 K$ ; (iv) air compression and expansion occurs isothermally at T₀; (v) internal pressure for pressurized volumes at any instant is uniform, that is, a lumped model. We apply the mentioned assumptions in our study.

Modeling pressure dynamics of PSSs

As described in Figure 2, we classify PSSs into two types based on the source used. For constant pressure PSSs (Fig. 2A), we have $Ṗ_{s r c} = 0 .$ (1)

For constant flow PSSs (Fig. 2B), $P_{s r c}$ is affected by inflow from the compressor and outflow to the soft actuator, $Q_{s p a}$ , where spa stands for soft pneumatic actuator. Applying ideal gas equations, the pressure dynamics of the reservoir are given by (Supplementary Data S1): $Ṗ_{s r c} = \frac{P_{0}}{V_{r e s}} (Q_{c o m} - p o s (Q_{s p a})),$ (2)

where $P_{s r c}$ is the instantaneous reservoir pressure, $V_{r e s}$ is the reservoir volume including dead volume of tubing and fittings between compressor and valve, and $Q_{c o m}$ is the instantaneous mass flow output of the compressor expressed in standard L/s. $p o s (Q_{s p a})$ is the positive part of mass flow to soft actuator, that is, from reservoir to soft actuator.

During soft actuator inflation and deflation, air flows through the valve, tubing, and fittings. We model this flow using ISO 6358 standard,^35,36 which calculates the flow through a component using the upstream and downstream pressures, and two parameters, sonic conductance C and critical ratio b, ( $0 < b < 1$ ), as shown hereunder: $Q = C Ψ P_{H i g h} .$ (3) $Ψ = (\begin{matrix} \sqrt{1 - {(\frac{\frac{P_{L o w}}{P_{H i g h}} - b}{1 - b})}^{2}}, & \frac{P_{L o w}}{P_{H i g h}} \geq b \\ 1, & \frac{P_{L o w}}{P_{H i g h}} < b \end{matrix},$ (4)

where $P_{H i g h}$ and $P_{L o w}$ are the absolute upstream and downstream pressures, respectively,and $Ψ$ is the nonlinear flow function.

This model, an approximation of flow through an orifice, is applicable to a wide range of commercial components and has been used previously in PSSs with positive pressure^36,42–45 as well as vacuum.^46–48 To model the combined effect of tubing, fitting, and valves, we calculate the equivalent $C_{e q}$ and $b_{e q}$ ³⁶ of the PSS, using C and b of each component and applying appropriate relations for series and parallel combinations [Eqs. (S5)–(S10)].

Modeling soft actuator pressure dynamics

We model the soft actuator as a chamber with variable internal volume, $V_{s p a}$ . Similar to Equation (2), we can express its pressure dynamics as (Supplementary Data S2): $Ṗ_{s p a} = \frac{P_{0}}{V_{s p a}} (Q_{s p a}) - \frac{P_{s p a} {\dot{V}}_{s p a}}{V_{s p a}},$ (5)

where $P_{s p a}$ is the instantaneous soft actuator pressure, $V_{s p a}$ is the soft actuator internal volume including dead volume of tubing and fittings between valve and soft actuator, and $Q_{s p a}$ is the mass flow to the soft actuator expressed in standard L/s.

$V_{s p a}$ is governed by $P_{s p a}$ , soft actuator design, external forces, and inertial forces. Neglecting effect of the external and inertial forces, we express the freely expanding/contracting soft actuator volume quasi-statically as a function of its pressure. Although a nonlinear relation, $V_{s p a} = f (P_{s p a})$ (Supplementary Data S2) can be modeled, we model a linear relationship based on^{13,23,27,32,49}: $V_{s p a} = V_{s p a}^{0} (1 + K_{e x p} (P_{s p a} - P_{0})),$ (6)

where we define the expansion ratio, $K_{e x p}$ , as a constant relating $P_{s p a}$ to $V_{s p a}$ , and $V_{s p a}^{0}$ is the volume of actuator at $P_{s p a} = P_{0}$ .

The soft actuator pressure dynamics, while implicitly considering internal volume dynamics, are then described by $Ṗ_{s p a} = \frac{P_{0} Q_{s p a}}{V_{s p a}^{0} (1 + K_{e x p} (2 P_{s p a} - P_{0}))} .$ (7)

Normalization

We normalize soft actuator and PSS pressure dynamics with soft actuator internal volume and the equivalent sonic conductance. We first define normalized sonic conductance as $\bar{C} = \frac{C_{e q}}{V_{s p a}^{0}} .$ (8)

For constant flow PSSs, we additionally define normalized reservoir volume, $\bar{V}$ , and compressor output flow, $\bar{Q}$ : $\bar{V} = \frac{V_{r e s}}{V_{s p a}^{0}}, \bar{Q} = \frac{Q_{c o m}}{V_{s p a}^{0}} .$ (9)

Next, we normalize soft actuator pressure dynamics by $\bar{C}$ as $Ṗ_{s p a} |_{I n f l} = (\frac{P_{0} Ψ_{I n f} P_{s r c}}{1 + K_{e x p} (2 P_{s p a} - P_{0})}) \bar{C},$ (10)

Ṗ_{s p a} |_{D e f l} = (\frac{- P_{0} Ψ_{D e f} P_{s p a}}{1 + K_{e x p} (2 P_{s p a} - P_{0})}) \bar{C},

(11)

where $Ψ_{I n f}$ and $Ψ_{D e f}$ correspond to the flow function, $Ψ$ , during inflation and deflation, respectively.

For constant pressure PSSs, $Ṗ_{s r c} = 0 .$ (12)

And for constant flow PSSs, the normalized source pressure dynamics are given by $Ṗ_{s r c} = (\frac{P_{0}}{\bar{V}} (\bar{Q} ∕ \bar{C} - Ψ_{I n f} P_{s r c})) \bar{C} .$ (13)

Using the above, the normalized soft actuator pressure dynamics are given by Equations (3), (4), and (10)–(13). For constant pressure PSSs, the normalized dynamics depend on $P_{s r c}$ , $P_{s p a}$ , $K_{e x p}$ , and $b_{e q}$ ; whereas for constant flow PSSs, they additionally depend on $\bar{Q}$ and $\bar{V}$ . The effect of $V_{s p a}^{0}$ and $\bar{C}$ is seen as scaling of the dynamics in magnitude and time, respectively.

Generating Mapping from PSS Parameters and Soft Actuator Properties to Dynamic Performance

In this study, we simulate the normalized soft actuator pressure dynamics and use the predicted pressure response to quantify the relationship between dynamic performance and the parameters f, c, and $α$ , listed in Table 1. We then validate these simulation results by measuring the inflation and deflation response of a pneunet actuator²³ in different operating conditions.

We first quantify soft actuator dynamic performance using the following metrics:

$T_{f a l l}$ : Time required for $P_{s p a}$ to drop from 98% to 2% of $P_{o p}^{g}$

$T_{r i s e}$ : Time required for $P_{s p a}$ to rise from 2% to 98% of $P_{o p}^{g}$

$f_{m a x}$ : Maximum frequency at which soft actuator can be cyclically inflated at deflated between 2% and 98% of $P_{o p}^{g}$ ,

where $P_{o p}^{g}$ is the soft actuator operating pressure, $P_{o p}$ , expressed in gauge ( $P_{o p}^{g} = P_{o p} - P_{0}$ ), and $P_{0} \leq P_{s p a} \leq P_{o p}$ .

Methods

1. Fall time and rise time: We simulated soft actuator pressure response of inflation and deflation between 0 and 0.2 MPaG, through a flow path of $\bar{C} = 10$ $V_{s p a}^{0}$ /s/MPa. We carried out the simulations at a sampling rate of 10 kHz, for the following set of parameters:

a. Source pressure, $P_{s r c}$ : 0.2–0.6 MPaG, in steps of 0.025 MPa.

b. Expansion ratio, $K_{e x p}$ : 0–15 $V_{s p a}^{0}$ /MPa, in steps of 0.5 $V_{s p a}^{0}$ /MPa.

c. Critical ratio, $b_{e q}$ : 0.1–1, in steps of 0.1.

For constant flow PSSs, we further carried out simulations for

d. normalized compressor flow, $\bar{Q} ∕ \bar{C}$ : 0.025–0.25 MPa in steps of 0.025 MPa and

e. normalized reservoir volume, $\bar{V}$ : in five values ranging from 0.1 to 24 $V_{s p a}^{0}$ , as shown in Figure 5. $\bar{V} = 0.1$ represents constant flow PSSs without a reservoir. We chose this small but nonzero value to account for internal dead volume of tubing and fittings.

By linear interpolation of the pressure response data, we calculated $T_{f a l l}$ and $T_{r i s e}$ for different operating pressures, $P_{o p}$ : 0.005–0.2 MPaG, in steps of 0.005 MPa.

Maximum actuation frequency: For constant pressure PSSs, we calculated $f_{m a x}$ by using results from $T_{f a l l}$ and $T_{r i s e}$ . For constant flow PSSs, we used principle of conservation of mass to calculate $f_{m a x}$ .

Results

Fall time, $T_{f a l l}$ : As air flows from the soft actuator to the ambient, soft actuator pressure dynamics do not depend on type of source, that is, they are unaffected by $P_{s r c}$ , $\bar{Q}$ , $\bar{V}$ , and D_C. $T_{f a l l}$ for different $P_{o p}$ , $K_{e x p}$ , and $b_{e q}$ is shown in Figure 3. The labels depict the fall time scaled by $\bar{C}$ . As shown in the figure, fall time increases with increasing $P_{o p}$ and $K_{e x p}$ and decreases with $b_{e q}$ and $\bar{C}$ .

Rise time, $T_{r i s e}$ : As air flows from the source to the soft actuator, constant pressure PSSs and constant flow PSSs exhibit different behavior.

FIG. 3.

Fall time, $T_{f a l l}$ , at varying values of actuator operating pressure, $P_{o p}$ , expansion ratio, $K_{e x p}$ , and equivalent critical pressure ratio, $b_{e q}$ . An axis of increasing $b_{e q}$ is displayed showing the nature of $K_{e x p}$ versus $P_{o p}$ versus $T_{f a l l}$ for three values of $b_{e q}$ . The fall time in seconds is given by the labels on the contour lines, with $\bar{C}$ expressed in $V_{s p a}^{0}$ /s/MPa. We observe that $T_{f a l l}$ increases with $P_{o p}$ and $K_{e x p}$ and decreases with $\bar{C}$ and $b_{e q}$ . The highlighted point corresponds to the SPA-pack (Fig. 9A), with a fall time of $16.4 ∕ \bar{C}$ . At the optimized PSS parameters $\bar{C}$ = 26.89 and $b_{e q}$ = 0.49, the predicted fall time is 0.6 s. Color images are available online.

a. Constant pressure PSSs: During inflation, a higher source pressure leads to higher flow in the PSS and, therefore, a faster response. Figure 4 shows rise time for different $P_{s r c}$ , $K_{e x p}$ , $b_{e q}$ , and $P_{o p}$ . The labels on each curve depict the rise time scaled by $\bar{C}$ . As shown in the figures, $T_{r i s e}$ increases with increasing operating pressure, $P_{o p}$ , expansion ratio, $K_{e x p}$ , and decreases with source pressure, $P_{s r c}$ , normalized conductance, $\bar{C}$ , and critical ratio, $b_{e q}$ .

b. Constant flow PSSs: During inflation, air flows from the reservoir to the actuator, leading to a reduction in reservoir pressure, $P_{s r c}$ . In this study, the reservoir acts similar to a large capacitor. Therefore, for a large reservoir ( $\bar{V} > > 1$ ), the drop in $P_{s r c}$ is small, leading to quasi-static source pressure conditions. This is shown in Figure 5 where $T_{r i s e}$ for constant flow PSSs is similar to that of constant pressure PSSs (dotted curve), for certain pressures and reservoir volumes. Thus, the reservoir permits rapid bursts of flow to the soft actuator, and can compensate for small compressors with low flow capacity $\bar{Q}$ . However, this is valid only for intermittent operation, as the flow to the soft actuator has to be replenished by the compressor. We show this in Figure 5, where at some values of $P_{s r c}$ and $\bar{V}$ , $T_{r i s e}$ increases significantly. This occurs when $P_{s r c}$ drops below $P_{o p}$ during inflation and, therefore, the compressor has to pressurize both reservoir and actuator. The point at which this occurs can be found using Figure 5 or, alternatively, by using conservation of mass, as described hereunder.

FIG. 4.

Rise time, $T_{r i s e}$ , for constant pressure PSSs at varying values of actuator operating pressure, $P_{o p}$ , source pressure, $P_{s r c}$ , expansion ratio, $K_{e x p}$ , and equivalent critical pressure ratio, $b_{e q}$ . Two axes of increasing $b_{e q}$ and $K_{e x p}$ are displayed showing the nature of $P_{s r c}$ versus $P_{o p}$ versus. $T_{r i s e}$ for three values each of $b_{e q}$ and $K_{e x p}$ . The rise time in seconds is given by the labels on the contour lines, with $\bar{C}$ expressed in $V_{s p a}^{0}$ /s/MPa. We observe that $T_{r i s e}$ increases with $P_{o p}$ and $K_{e x p}$ and decreases with $P_{s r c}$ , $\bar{C}$ , and $b_{e q}$ . The highlighted point corresponds to our prototype, powering an SPA-pack with $P_{o p}$ = 0.2 MPaG and $K_{e x p}$ = 1.51 $V_{s p a}^{0}$ . At the optimized PSS parameters $P_{s r c}$ = 0.22 MPaG, $\bar{C}$ = 26.89, and $b_{e q}$ = 0.49, the predicted rise time is 0.39 s. Color images are available online.

FIG. 5.

Rise time, $T_{r i s e}$ , for constant flow PSSs at varying values of source pressure, $P_{s r c}$ , compressor flow output, $\bar{Q}$ , and reservoir volumes, $\bar{V}$ , for $P_{o p} = 0.2 M P a G$ , $K_{e x p} = 1.51 V_{s p a}^{0} ∕ M P a$ , and $b_{e q} = 0.49$ . An axis of increasing $\bar{Q} ∕ \bar{C}$ is displayed showing the nature of $P_{s r c}$ versus $T_{r i s e}$ for three values of $\bar{Q} ∕ \bar{C}$ . The rise time in seconds is given by the labels on the Y-axis, with $\bar{C}$ expressed in $V_{s p a}^{0}$ /s/MPa. We observe that $T_{r i s e}$ for constant flow PSSs is very close to that of constant pressure PSSs (dashed curve), for certain values of $P_{s r c}$ and $\bar{V}$ . This is because the pressure in the reservoir does not vary much during inflation, leading to a quasi-constant pressure. However, at other values of $P_{s r c}$ and $\bar{V}$ , $T_{r i s e}$ is substantially higher. This occurs when $P_{s r c}$ drops below $P_{s p a}$ , and the compressor has to pressurize both, reservoir and soft actuator. The transition point at which this occurs can be found using the figure, or from Equation (14). Here, the highlighted point corresponds to our prototype, with predicted rise time of $10.5 ∕ \bar{C}$ that gives $T_{r i s e} = 0.39 s$ for out selected valve and tubing. Color images are available online.

Soft actuator pressure and normalized volume before inflation are P₀ and 1, respectively. After inflation, they become $P_{o p}$ and $1 + K_{e x p} (P_{o p} - P_{0})$ . For the reservoir with volume $\bar{V}$ , let pressure before and after inflation be $P_{s r c}$ and $P'_{s r c}$ , respectively. The air entering through the compressor is $P_{0} \bar{Q} T_{r i s e}$ . In the limiting condition, $P_{s r c}$ drops to $P_{o p}$ , that is, $P'_{s r c} = P_{o p}$ . Let $T^{*}$ be the rise time for constant pressure PSSs at the same $P_{s r c}$ . The condition that $P_{s r c}$ in a constant flow PSS does not drop below $P_{o p}$ during inflation is then given by

P_{s r c} \bar{V} + P_{0} \bar{Q} T^{*} + P_{0} > P_{o p} (\bar{V} + 1 + K_{e x p} (P_{o p} - P_{0})) .

(14)

As the average $P_{s r c}$ in constant flow PSSs is always lower than that in constant pressure PSSs, $T_{r i s e} > T^{*}$ . Hence, if the mentioned condition is satisfied, it is always ensured that $P_{s r c}$ after inflation does not drop below $P_{o p}$ and, therefore, $T_{r i s e}$ will be similar to that of constant pressure PSSs. However, if Equation (14) is not satisfied, $T_{r i s e}$ increases greatly.

Maximum actuation frequency, $f_{m a x}$ :

a. Constant pressure PSSs: We used $T_{r i s e}$ and $T_{f a l l}$ to calculate $f_{m a x}$ for different sets of $P_{s r c}$ , $K_{e x p}$ , $b_{e q}$ , and $P_{o p}$ . This frequency is given by

f_{m a x} = \frac{1}{T_{r i s e} + T_{f a l l}} H z = \frac{1}{T_{C y c l e}} H z .

(15)

From Equation (15) and Figures 3 and 4, we see that $f_{m a x}$ increases with $P_{s r c}$ , $b_{e q}$ , and $\bar{C}$ and decreases with $P_{o p}$ and $K_{e x p}$ . To achieve cyclic actuation, the valve must be controlled in a repeating and alternating pattern to inflate and deflate the actuator. The ratio of ON time to the OFF time is the valve duty cycle D_V (more rigorous definition described in Supplementary Data S3). To achieve $f_{m a x}$ given by Equation (15), the D_V required is given by $D_{V} = \frac{T_{r i s e}}{T_{C y c l e}} \times 100 % .$ (16)

At any frequency higher than that in Equation (15), the amplitude of pressure oscillation will reduce. In addition, at $f_{m a x}$ , duty cycles other than that in Equation (16) will either lead to overpressure or reduce the pressure amplitude.

b. Constant flow PSSs: In constant flow PSSs, as the source pressure, $P_{s r c}$ , is not constant, $f_{m a x}$ cannot be calculated directly from $T_{f a l l}$ and $T_{r i s e}$ . Although fast inflation can be achieved in intermittent operation, continuous cyclic actuation may not be sustained at the same speed as the compressor may not be able to replenish the air consumed. As $P_{s r c}$ drops, $T_{r i s e}$ will increase and the actuation frequency will reduce. However, after a while, steady state will be reached such that $T_{r i s e}$ does not increase further. At the steady state, $f_{m a x}$ can be found by applying conservation of mass to the air entering (through the compressor) and leaving (from the soft actuator) the system.

Soft actuator pressure and normalized volume before inflation are P₀ and 1, respectively. After inflation, they increase to $P_{o p}$ and $(1 + K_{e x p} (P_{o p} - P_{0}))$ , respectively. The amount of air entering the system during one actuation cycle is $P_{0} \bar{Q} T_{C y c l e} D_{C}$ , where D_C is the compressor duty cycle. By mass conservation, we have $P_{0} + P_{0} \bar{Q} D_{C} T_{C y c l e} = P_{o p} (1 + K_{e x p} (P_{o p} - P_{0})) .$ (17)

Rearranging, $f_{m a x} = \frac{1}{T_{C y c l e}} = \frac{P_{0} \bar{Q} D_{C}}{(P_{o p} - P_{0}) (1 + P_{o p} K_{e x p})} .$ (18)

If the system reaches steady state, Equation (18) predicts the sustainable $f_{m a x}$ for any compressor-based PSS powering any soft actuator, with a simple arithmetic expression containing only three terms: normalized average compressor flow, soft actuator operating pressure, and expansion ratio, irrespective of the other variables. Using $D_{C} = 1$ gives the theoretically maximum sustainable $f_{m a x}$ for a compressor-based PSS.

To achieve this $f_{m a x}$ , $\bar{C}$ should be sufficiently large enough to allow the desired flow rate, and D_V should be chosen appropriately to allow sufficient time for complete inflation and deflation. This condition is given by $D_{V} \geq \frac{T^{*}}{T_{C y c l e}} \times 100 %; D_{V} \leq 1 - \frac{T_{f a l l}}{T_{C y c l e}} \times 100 %,$ (19)

where $T^{*}$ is the rise time for constant pressure PSSs at the same initial $P_{s r c}$ .

If the normalized conductance, $\bar{C}$ , is insufficient to provide the desired flow rate, either one or both conditions in the mentioned equation will not be satisfied, giving an infeasible solution. In such a case, the steady state will not be reached, and the only solution to achieving the theoretically maximum $f_{m a x}$ would be to increase $\bar{C}$ .

For both types of PSSs, the air consumed per actuation cycle in standard L is given in Supplementary Data S4: $A i r C o n s u m e d = (P_{o p} - P_{0}) (1 + P_{o p} K_{e x p}) V_{s p a}^{0} .$ (20)

Validation

To validate the results from this parameter study, we powered a single pneunet actuator²³ with a pressure-regulated PSS (Fig. 2A), and compared the rise time, fall time, $f_{m a x}$ , and the inflation and deflation response with that predicted by the model. The PSS consisted of an external regulated pressure source, a solenoid valve (SMC VV100 series), and tubing (SMC OD: 4 mm, ID: 2.5 mm, length: 0.84 m between source and valve, 0.46 m between valve and pneunet). We measured the pressure response using a pressure sensor (MPX5500DP), and used two flow sensors (AWM5000 Series) to measure flow inlet and outlet to the pneunet. The inflation and deflation responses at 7 different source pressures (0.1 MPaG, 0.15 MPaG,…, 0.4 MPaG) were measured, 10 times each. We then simulated the pressure response in MATLAB to compare the results with our model by using the following parameters:

Actuator volume, $V_{s p a}^{0} = 0.0167 L$ , was calculated by adding the dead volume of tubing to the pneunet internal volume, obtained from CAD.

Operating pressure, $P_{o p}$ = 0.1 MPaG.

Expansion ratio, $K_{e x p} = 8.25 V_{s p a}^{0}$ /MPa, was calculated by measuring the amount of air consumed in one actuation cycle, and using Equation (20).

Normalized conductance and critical ratio, $\bar{C} = 17.98 V_{s p a}^{0}$ /s/MPa, $b_{e q} = 0.19$ for inflation and $\bar{C} = 26.04 V_{s p a}^{0}$ /s/MPa, $b_{e q} = 0.29$ for deflation. These were found by using the sonic conductance and critical ratio of the valve from its datasheet, then calculating equivalent conductance of tubing and valve using Equations (S5)–(S10), and finally normalizing with actuator volume $V_{s p a}^{0}$ .

Source pressure, $P_{s r c}$ = 0.1, 0.15,…,0.4 MPaG.

Figure 6B, C shows the pressure response and Supplementary Table S1 shows the root mean squared (RMS) and peak errors between experiments and the model. In addition, Supplementary Table S2 compares the predicted versus achieved fall time, rise time, and maximum actuation frequencies. We see that the model closely captures the behavior of the pneunet with <5.2% RMS error for all conditions. We also see that with higher source pressures, the measured inflation response is faster than that predicted by the model. This is because the inertia of the soft actuator limits the peak displacement at high actuation speeds, which leads to a smaller actuation volume and faster response.

FIG. 6.

Validation of results from PSS parameter study by testing the inflation and deflation response with a pneunet actuator. (A) The pneunet used for testing. (B) Comparison of simulations (continuous line) with mean measured pressure response (dotted line) in deflation. (C) Comparison of simulations (continuous line) with mean measured pressure response (dotted line) in inflation. The different colored lines correspond to different source pressures ( $P_{s r c}$ ) during testing. Color images are available online.

Implementation to PSS Design

We implement the results from the previous section to select PSS components for an entirely portable soft exosuit through parameter optimization. The prototyped PSS was aimed to meet both dynamic performance and PSS specifications.

Design requirements

We have prototyped a soft exosuit as shown in Figure 8A for studying force interactions with the human torso. This exosuit consists of six soft actuators, called SPA-packs¹⁷ (Fig. 9A), capable of producing forces up to 40 N each. We calculated the dynamic performance requirements of this exosuit for an application to support the wearer's weight in awkward work postures, based on,^50–54 and a pilot study. These requirements must be met by a portable PSS, which should be compact and lightweight. Table 2 summarizes these performance requirements and PSS specifications.

Table 2.

Design Requirements and Corresponding Achieved Values of the Fabricated Pneumatic Supply System

Dynamic performance specifications^a		Achieved values
$T_{r i s e}$	<0.45 s	0.42 s
$T_{f a l l}$	<0.65 s	0.59 s
$f_{m a x}$	>0.9 Hz	0.99 Hz
PSS specifications		Achieved values
Overall mass	<3 kg	1.95 kg (optimized)
Overall volume	<0.4 × 0.3 × 0.2 m	0.3 × 0.2 × 0.15 m
Tubing length^b	0.5 m	0.5 m
Output channels	6	6
Number of actuations	>5000	$>$ 6000

Specifications for a single SPA-pack.

From valve to SPA-pack.

SPA, soft pneumatic actuator.

Parameter optimization and component selection

Using our mapping results, we calculated the range of PSS parameters f and c that meet performance requirements given in Table 2. Using these values, we compared commercially available options for source, valve, and pneumatic lines to find the optimal set of PSS components as described hereunder:

1. For tubing length 0.5 m (between valve and SPA-pack), we compared options for standard pneumatic tubing with internal diameters of 1.2, 2.5, 4, and 5 mm. Using Equation (S10), we calculated C and b for each, as well as $V_{s p a}^{0}$ and $K_{e x p}$ by adding the tubing dead volume to the SPA-pack internal volume. For our specific case, the tubing volume is much smaller with respect to the SPA-pack volume (0.079 L). Using a conservative approach, we considered the largest diameter to get $V_{s p a}^{0} = 0.083$ L and $K_{e x p} = 1.51$ $V_{s p a}^{0}$ /MPa.

2. For $P_{o p} = 0.2 M P a G$ and the above value of $K_{e x p}$ , we used Figures 4, 5, and 7 to find predicted values of $T_{r i s e}$ for a range of $P_{s r c}$ , $\bar{C}$ , and $b_{e q}$ . The regions above $T_{r i s e} = 0.45 s$ shown in Figure 7 correspond to values of $P_{s r c}$ , $\bar{C}$ , and $b_{e q}$ , which will satisfy the requirements of rise time. In addition, we predicted $T_{f a l l}$ for a range of $\bar{C}$ and $b_{e q}$ , using Figures 3 and 8E. Using the predicted $T_{f a l l}$ and $T_{r i s e}$ from the figures, we calculated the acceptable range of values of $P_{s r c}$ , $\bar{C}$ , and $b_{e q}$ .

3. Next, we compared available valves and fittings, and noted the combinations of valve, tubing, and fittings with $\bar{C}$ and $b_{e q}$ lying in the acceptable range.

4. From the desired number of full actuations, we calculated the total amount of air required as 897 standard L by applying Equation (20). Using this and acceptable range of $P_{s r c}$ , we compared available options for the source as follows:

a. Pressure regulated: We compared acceptable options among high-pressure cylinders and liquid $C O_{2}$ , which could meet the total air requirement, along with their regulators.

b. Compressor based: We compared acceptable compressors and battery combinations that could provide $P_{s r c}$ , the total air requirement, and $f_{m a x}$ [from Eq. (18)]. Then we calculated $\bar{V}$ required for simultaneously powering four SPA-packs, using Equation (14), and compared possible options for reservoir.

5. For every combination of PSS components that could meet performance requirements, we calculated the total PSS volume and mass. We selected the combination with minimum mass of components, as listed in Supplementary Table S3. It consists of a battery-powered compressor, two reservoirs, standard tubing and fittings, and six sets of proportional valves with custom manifolds.

6. For the selected components, we used Equations (18) and (19) to calculate optimal D_C and D_V.

With the optimally selected components, we prototyped and assembled the PSS in the form of a backpack as shown in Figure 8B. The total mass of the selected PSS components is 1.95 kg, and their optimal flow and control parameters are as listed in Figure 8D, and highlighted in Figures 3–5, 7, and 8E.

FIG. 7.

Predicted rise time for the SPA-pack with expansion ratio, $K_{e x p} = 1.51$ , and operating pressure, $P_{o p} = 0.2 M P a G$ , at varying values of $P_{s r c}$ , $\bar{C}$ , and $b_{e q}$ . Since the requirement is $T_{r i s e} < 0.45 s$ , the selected valve, tubing, and fittings should have $\bar{C}$ and $b_{e q}$ lying in the region above the curve $T_{r i s e} = 0.45 s$ . For our prototype, the selected components corresponding to $P_{s r c} = 0.22 M P a G$ , $\bar{C} = 26.89$ , and $b_{e q} = 0.49$ . At these values, we get $T_{r i s e} = 0.39 s$ as shown by the highlighted point. Color images are available online.

FIG. 8.

PSS design through parameter optimization (A) Soft exosuit, which is to be powered by the PSS. It consists of six SPA-packs marked 1–6. (B) Prototype of the optimized PSS, fabricated in the form of a backpack. (C) Schematic of the PSS for the exosuit powering six SPA-packs. (D) Optimized flow and control parameters required to meet requirements of Table 2. (E) Predicted fall time for $P_{o p} = 0.2 M P a G$ at a range of $\bar{C}$ and $b_{e q}$ . The highlighted point corresponds to the selected combination of valve, tubing, and fitting for the prototype, with predicted $T_{f a l l} = 0.6 s$ . Color images are available online.

Comparison with experimental testing

We tested the prototype while powering a single SPA-pack (Fig. 9A) in two conditions, unloaded and blocked. For both cases, we measured the pressure response 10 times on a bench top. To compare the observed dynamic behavior with that predicted by our model, we simulated the SPA-pack response in these conditions. For simulating blocked conditions, we took the expansion ratio, $K_{e x p} = 0$ .

FIG. 9.

Validation of PSS parameter optimization through off-board testing of the SPA-pack dynamic performance. (A) The SPA-pack used in the soft exosuit. (B) Comparison of simulations (continuous lines) with measured pressure response (dotted lines) in inflation and deflation, in blocked (black) and unloaded conditions (red). Each test was carried out 10 times. $P_{s r c}$ before inflation was 0.22 and 0.24 MPaG for unloaded and blocked conditions, respectively. Color images are available online.

From Figure 9B, we observe that the PSS meets desired performance requirements of $T_{r i s e}$ , $T_{f a l l}$ , and $f_{m a x}$ , and the measured data show good agreement with simulations. In addition, Table 3 gives the peak and RMS errors between simulation and experimental results. RMS errors for blocked conditions and deflation in unloaded conditions were found to be <3.1%. A slightly higher error (5.2% RMS, 13.2% peak) was observed for inflation during unloaded expansion (Fig. 9B), which can be attributed to the nonlinear behavior of the SPA-pack. A higher order model for the pressure–volume relationship along with inclusion of inertial and external forces will be carried out in future studies to better understand these rich dynamics exhibited by soft actuators.

Table 3.

Comparison of Measured Pressure Response of the Soft Pneumatic Actuator-Pack with Simulations

	RMS error (%)		Peak error (%)
	Unloaded	Blocked	Unloaded	Blocked
Inflation	5.2	1.6	13.2	6.0
Deflation	3.1	2.1	5.2	5.1

RMS, root mean squared.

Conclusion

In this study, we presented parameter optimization for design and control of PSSs. This is the first comprehensive study and validation of addressing the requirements of both soft actuator dynamic performance and PSS specifications. The model for soft actuator pressure dynamics proposed in this study goes beyond the scope of currently used fitted models as it captures the effects of the PSS components and soft actuator design parameters. Furthermore, it complements existing studies of soft actuator kinematics and deformation, and can be employed to study and dynamically control soft robot motion and forces. Using this model, we investigated and quantified the effect of 10 parameters on soft actuator dynamic performance. In addition, we identified physical limits of achievable dynamic performance for a given set of parameters. This mapping creates an improved understanding of the role of source, valve, pneumatic line, and soft actuator design parameters on dynamic performance. Experimental testing with a pneunet actuator showed high fidelity with the model, thus validating its accuracy and applicability to soft actuator pressure dynamics. Using results from the parameter study, we designed a PSS for powering an exosuit with six soft actuators, while minimizing mass of PSS components. Results from experimental testing with this device also show good agreement with simulations, thereby affirming the proposed approach. By enabling optimized PSSs for robotic applications, this study is a step toward customizing soft robotic systems and will help to broaden the scope of soft robots.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This study was supported by the Swiss National Science Foundation (SNSF) “START” Project 513956.

Supplementary Material

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Supplementary Material

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