Electrohydraulic System Analysis of Variable Recruitment Fluidic Artificial Muscle Bundles With Interaction Effects

McKibben actuators, often referred to as FAMs are inherently compliant actuators known for their high power-to-weight ratio [1,2]. These actuators were created by Joseph McKibben in 1958, motivated to aid his daughter whose hands had been paralyzed due to polio. As its moniker suggests, FAMs were created to mimic the contractile motion of the human muscle, which still motivates researchers today to investigate its application to exoskeletons, orthotics and prosthetics [3–11]. A FAM consists of a double-helical fiber mesh wrapped around an inner elastic bladder. While one end of the bladder is blocked off, the other end is connected to a fitting through which pressure is applied. The FAM is activated by applying pressure, either pneumatically or hydraulically, causing the mesh to constrain the bladder to expand radially and contract axially.

Although originally intended as a contractile actuator, modifications can be made to FAMs to produce other modes of actuation. Increasing the angle at which the mesh fibers are helically wrapped around the elastic bladder can alter the FAM to produce extensile motion [12]. Adding inelastic layers to the mesh allows the actuator to increase contraction, bend, or twist [13,14]. Other researchers have focused on the arrangement of multiple FAMs in different configurations. Conventional use of FAMs considers them as individual single-acting actuators with a single FAM applying unidirectional force to a joint or an antagonistic pair of FAMs producing bidirectional actuation. Others, however, have taken inspiration from parallel or pennate human muscle tissue topologies to view each FAM as a subcomponent of a single multi-unit actuator, much like how each natural muscle organ consists of multiple motor units (MUs) that act together to produce unidirectional force and displacement on a joint [15]. Kurumaya et al. developed a multifilament muscle actuator by bundling multiple thin FAMs to actuate a musculoskeletal robot [16]. Bryant et al. created a FAM bundle using multiple FAMs in parallel with added VR functionality becoming a an active field of research [17–19]. The FAMs within the bundle can be grouped into independently-activated units of actuation, which are equivalent to the MUs in biological muscle tissue. According to Henneman's size principle, MUs are sequentially recruited from smallest to largest in size and the subsequent MU is only activated when the previous MU has reached its peak force output [20]. Inspired by this recruitment scheme, the MUs within the FAM bundle are sequentially activated to meet the load demand. Compared to a single larger FAM with equal total cross-sectional area, this VR bundle can increase the average efficiency over the force–strain working space by reducing the energy losses due to pressure throttling during submaximal pressure actuation [17]. The performance increase of VR bundles has been experimentally validated for both pneumatic and hydraulic power sources [17,21,22]. Different schemes of MU activation have also been investigated. In “batch” recruitment, a single pressure control valve and multiple on-off valves are used to control the pressure of MUs, while in “orderly” recruitment, the pressure of each MU is controlled separately and activated sequentially. This scheme can either be implemented by using a pressure valve for each MU, or by using a single orderly recruitment valve, which decreases fluid circuit complexity by using fewer valves [23]. In addition to such “active” recruitment schemes, the bladder elastic properties can be separately tailored for each MU to passively recruit them in a desired sequence [24]. Real-time control schemes for variable recruitment have been developed and tested [21,25,26].

Previous studies have mainly focused on the downstream actuator efficiency comparing the mechanical work output to the fluid energy input into the pressure control valve [17,19]. Later studies have modeled the entire electrohydraulic system including the motor, pump and accumulator required to supply hydraulic power [27,28]. Such studies compare the mechanical work output of the actuator to the electrical energy input to the motor during a set period of time, providing a more holistic perspective of the performance when VR is applied to a robotic system. Chapman et al. showed that a FAM operating at its peak efficiency does not necessarily guarantee the peak efficiency of the overall system by building and modeling a quadrupedal wall-climbing robot and its fully coupled electrohydraulic system [28]. A subsequent study compared system-level performance metrics such as overall efficiency and bandwidth of a FAM with equal cross-sectional area to that of a VR bundle during continuous and intermittent pump electrohydraulic configurations [27]. In these studies, a modified version of the ideal FAM model that takes into account the nonlinear properties of the inner elastic bladder were used to simulate the overall bundle force and strain [29,30].

Recently, a FAM model that better captures the interaction effects between FAMs within a bundle has been proposed [31]. Due to the sequential manner in which bundle MUs are recruited, inactive or partially active FAMs within a bundle experience compression beyond their free strains, exerting “resistive forces” acting against the overall force output of the bundle. In the physical system, the inactive/partially active FAMs tend to buckle outward, increasing the spatial envelope of the bundle. Several mitigation strategies have been implemented to prevent the FAMs from buckling. Embedding the bundle in silicone proved effective in inhibiting FAMs from buckling but decreased strain output [17]. Using tendons in series with the FAMs prevented buckling and thus increased overall bundle volumetric energy density while increasing efficiency in some portions of the bundle force–strain space at the expense of others [32]. Although such mitigation strategies can prevent motor unit buckling, they all require additional components and material as compared to a plain FAM bundle and come with tradeoffs in maximum bundle contraction and efficiency. This study first aims to address the challenges posed by bundle FAM interaction effects without taking such measures. Using a FAM model that accounts for resistive effects gives insight into the overall force–strain space of a VR bundle that is not evident when using simpler models. The purpose of this study is to investigate the implications of these inter-FAM effects on the recruitment control strategy required to track a desired trajectory and to evaluate the full electrohydraulic system efficiency of a variable recruitment FAM bundle in the presence of inter-FAM effects with several pump operation architectures. The specific contributions of this paper are to (1) demonstrate the effects resistive force and electrohydraulic system dynamics on recruitment state transition, (2) integrate pressure control valve dynamics and the effects of a variable displacement pump into a fully coupled system model, and (3) evaluate the system efficiency and bandwidth of a VR bundle actuator with inter-FAM effects and under various electrohydraulic configurations.

The subsequent sections of the paper are organized in the following way. In Sec. 2, the FAM model used in this study and the electrohydraulic subsystem models are presented. The implications of resistive forces and electrohydraulic system dynamics during recruitment state transition are discussed in Sec. 3. In Sec. 4, the metrics for which the system performance is evaluated in presented, followed by the three different simulation electrohydraulic configurations. The simulation results from the different configurations are presented and compared. The conclusions of the paper are presented in the final section.

where P is the pressure applied, $r0$ is the initial radius of the braid, $α0$ is the initial braid angle, and ε is the strain of the FAM. Researchers have further developed this model to include the tapered geometry of the bladder during contraction, wall thickness of the bladder, and bladder elasticity [2,30,33]. Rather than using a virtual work balance approach, some FAM models use a force balance approach to model a FAM's force output [29,30].

where $Fc$ is the axial force of the bladder in its collapsed shape and $εc$ is the strain at collapse. The transition from buckled to collapsed regions is characterized by $β$ which is tuned based on parameter optimization. The Kim et al. resistive force FAM model captures the behaviors of and predicts the forces exerted by the FAM in both tensile and compressive regimes. For applications in which FAMs are used as stand-alone actuators, the axial force behavior past free strain may not be of interest. However, in predicting the overall force output of a VR FAM bundle, these negative forces must be considered. Depending on the recruitment state, FAMs within a bundle may be in a combination of fully active (maximally pressurized), partially active (submaximally pressurized), or inactive (unpressurized) states. In a situation when fully active FAMs and inactive/partially active FAMs coexist, the latter FAMs are compressed beyond their free strain exerting a force opposing the force output of the active FAMs. For this reason, the negative force is also referred to as the resistive force. Figure 1 illustrates a bundle with two FAMs, at an instance when one FAM is fully active (right) while the other is only partially active (left). The fully active FAM generates force while the partially active FAM contributes resistive force.

Figure 2 shows the isobaric force–strain curve of a FAM including the resistive forces. At the blocked force condition, the FAM force is at its maximum. As strain increases, the force output decreases, eventually reaching zero at its free strain condition according to Eq. (2). As strain is increased past free strain and the FAM begins to buckle outward, the FAM exerts a resistive force shown by the negative region. Further compression causes the FAM to collapse, which is when the magnitude of resistive force is largest. Immediately following collapse, the magnitude of resistive force decreases and eventually approaches the axial force of the collapsed force $Fc$ from Eq. (4).

Figure 3 illustrates the effects of resistive forces on the force–strain space of a VR bundle with two recruitment states. The bundle consists of two FAMs of equal dimensions, each representing one MU. The isobaric force–strain curves at various pressures are plotted as predicted by the model with resistive forces included and compared to the model prediction with resistive forces neglected. In recruitment state 1 (RS1), only one MU is either partially or fully activated. In recruitment state 2 (RS2), the first MU is fully activated while the second MU is partially or fully activated. The main difference between the force–strain space of the VR bundle when resistive forces are included and when they are neglected is shown by an overlapping region between RS1 and RS2. When resistive effects are neglected, the force–strain space of RS1 and RS2 do not overlap. During the transition from RS1 to RS2 when the first MU has reached its maximum pressure, any additional pressure supplied to the second MU either maintains or increases the force output of the bundle. However, in a more realistic case when resistive forces are considered, an initial increase in the pressure of MU2 results in a decrease in bundle force output. Due to the negative forces exerted by the second MU, which is in compression, the force generated in RS2 at low MU2 pressures falls below that generated in RS1. A detailed discussion of this overlapping region has been presented in the paper by Kim et al. [25]. The present study presents the implications of the electrohydraulic system dynamics, resistive forces, and the resulting overlapping recruitment state region on controlling bundle strain and force to track a desired motion trajectory under VR.

While the mathematical model itself provides qualitative information and shows similar trends to experimental measurements of FAM behavior past free strain, more refined modeling matching is required for model-based control applications. Isobaric force–strain curves for the FAMs used in this study have been measured and compared to the model. Figure 4 shows the result of using empirically-based correction factors to improve model matching to actual measurements. The method for model tuning using experimental data is presented in the paper by Kim et al. [31] and the FAM parameters and correction factors used are summarized in Table 1.

It should be noted that the model characterizes the quasi-static axial force output with respect to the strain. Other researcher groups have explored the dynamic response of FAMs both theoretically and experimentally [40–42]. However, for the amplitude, frequency, and loading conditions of the prescribe sinusoid trajectory used in the simulations of this paper, a quasi-static force model has been shown to be sufficient as the purpose of this study is to demonstrate the system-level performance gains of variable recruitment and various electrohydraulic configurations [21,33,43].

where $Pacc,0$ and $Vacc,0$ are the initial pressure and volume of the accumulator, respectively. $Vdown$ is the amount of fluid volume downstream of the servovalve while $Vup$ is the upstream fluid volume. $Vm$ denotes the fluid volume in the FAMs, which is a function of its strain. Lastly, $Vp$ is the amount of fluid volume delivered by the pump/motor.

where $Pacc$ is the accumulator pressure and $D$ is the pump displacement. The motor parameters and operating characteristics were adapted from a 20 W Maxon motor (BRx42-40) with a nominal voltage rating of 12 $V$ [27,33] and are shown in Table 2.

Previous studies have both mathematically and experimentally investigated the force and strain generation of a VR bundle and its transition between recruitment states. Jenkins et al. implemented a model-based recruitment state switching controller and discusses the discontinuities in force and strain that result from modeling errors during transitions [21]. In his study, an empirically based tuning method was used to smooth out discontinuities in bundle force that exist when transitioning to a higher recruitment state. Chapman et al. implemented a strategy in which the recruitment state transition occurs when the MUs in the lower recruitment state reaches a preset pressure equal to 90% of the maximum pressure [27]. Meller et al. implements a recruitment logic state machine to switch between recruitment states with separate thresholds for recruitment and derecruitment [43]. This paper continues to investigate the transition between states in a variable recruitment FAM bundle by introducing two contributing factors: resistive forces and electrohydraulic system dynamics. In this section, the way these factors affect the force and strain during transition are discussed and demonstrated through experiments. Experiments were performed on an in-house built hydraulic testing platform called the linear hydraulic actuator characterization device (LHACD) as illustrated in Fig. 5. More details about the characterization setup are provided in the paper by Chipka et al. [45].

As presented in Sec. 2.1, inactive/low-pressure FAMs under compression exert resistive forces that act against fully active ones. A transition to a higher recruitment state is required when the MU of the current recruitment state is saturated and force and/or strain is required from the subsequent MU. Thus, at the initiation of the recruitment state transition, the current MU has fully contracted and the FAMs in the subsequent MU are buckled outwards. In order for the subsequent MU to generate additional bundle force and/or strain, the subsequent MU must reach a deslack pressure, at which the subsequent MU is no longer buckled and its strain has caught up to the strain of the current MU. Figure 6 shows the deslack pressures of MU2 calculated from the model presented in Sec. 2.1 and measured from experiments. Using the LHACD, two MUs were bundled in parallel, and the force and strain of the bundle were measured. Figures 6(a) and 6(b) show analytical results given by the model. The pressure of MU1 is held constant at 1379.0 kPa (200 psi) in Fig. 6(a) and 1723.7 kPa (250 psi) in Fig. 6(b). Two values of MU1 pressure are shown to demonstrate that the deslack pressure is dependent on the strain of MU1 and independent of its pressure. For both MU1 pressures, the strain is held constant at three different values: 0.2 (blue), 0.225 (red), and 0.25 (magenta). The overall bundle force is plotted for different values of MU2 pressures. When MU2 pressure is zero, MU2 is in the buckled state. As pressure increases, MU2 remains buckled until it has reached the deslack pressure indicated by a circle marker. The deslack pressure is plotted for a range of bundle strain and shown in Fig. 6(c) along with experimental data for the three different strain values.

For MU2 pressures that require additional MU1 pressures that are greater than the instantaneous pressure margin, the pressure of MU1 becomes saturated by the source pressure, and MU1 is unable to provide enough force to counteract the resistive force of MU2.

Electrohydraulic system dynamics as modeled in Sec. 2, limit the fluid flowrate and the rate of increase of pressure. This results in a lag between when pressure is first applied to a MU and when it begins to provide positive force and strain to the bundle. Figure 8 plots an example of model predictions and experimental measurements for the pressure history of a FAM with parameters specified in Table 1, connected to the LHACD. Initially, the servovalve is fully closed, and at $t=0$⁠, the servovalve is given a step command to fully open.

The model-predicted pressure growth is shown in Fig. 8(a) while experimental measurements are shown in Fig. 8(b). The model-predicted and experiment pressures upstream of the servovalve are shown as well. As the valve is opened, the upstream pressure immediately drops to equalize the upstream and downstream pressures. However, the fluid stored in the accumulator and additional fluid input from the motor/pump allows both the upstream and downstream pressures to increase according to pressure dynamic equations given by Eq. (8). The discrepancy between the model and experiment pressure growth is due to fluid inertia that is not modeled in this study. However, the model is able to capture the trend of fluid flow between upstream and downstream circuits and the initial pressure growth rate. In a recruitment state transition, these effects result in a recruitment lag, which we define as the time from initial MU activation until the MU reaches its deslack pressure and is able to begin contributing force output to the bundle. Therefore, to prevent this recruitment lag from causing tracking error during recruitment state transitions (which has been reported in prior implementations of variable recruitment [43]), the MU being recruited must be activated sufficiently prior to when additional bundle force is required to track the desired joint trajectory. The recruitment lag, and therefore the minimum amount of time that MU activation must begin before the MU force contribution is needed, is not constant and varies with the instantaneous states of the electrohydraulic system and the strain in the bundle. In this study, inverse dynamics is used to determine the required MU pressures for prescribed sinusoidal trajectories of various amplitudes and frequencies at every instant of time. Then, the electrohydraulic system dynamics required to generate those required pressures are simulated for the purpose of evaluating and comparing system metrics such as efficiency and bandwidth. Because the trajectories are prescribed, the time at which force and strain from MU1 becomes saturated is known. In other words, the required time to saturation is known at every instant in time. Making use of this information and calculating the required pressure margin and recruitment lag at each instant of time, the time at which MU2 should be activated to mitigate resistive effects and flowrate restrictions due to the electrohydraulic system dynamics is determined. In summary, we set MU activation to begin when at least one of the following two criteria is met: (1) when the additional pressure required for compensation becomes greater than the instantaneous pressure margin, or (2) the recruitment lag becomes greater than time to saturation.

where the amplitude, $xmax$⁠, and frequency, $ω$⁠, are both prescribed values. Simulations are performed for various amplitude, frequency, and loading scenarios to compare the overall system performance metrics such as efficiency and bandwidth.

To ensure that the system efficiencies are comparable across different electrohydraulic configurations and due to the difference in actuator types, the aforementioned motor size in Sec. 2.2.2 was selected as to operate as close to its maximum efficiency as possible. While the motor efficiency depended on the amplitude and frequency of the sinusoidal trajectory, the differences were minute, and the efficiencies were determined to be ∼68%. Throughout one cycle of sinusoidal motion, the motor operates at different instantaneous power efficiencies and suffers from losses due to inertial effects. Therefore it was deemed close enough to the reported 77% maximum efficiency according to the motor specification sheet.

In addition to system efficiency, the bandwidth limit, which is the maximum operating frequency of the system, is evaluated for a single equivalent cross-sectional area motor unit (SEMU) and VR bundle with constant and variable pump displacements. A SEMU is defined as a single FAM for which the dimensions are chosen such that the blocked force and free strain are the same as a VR bundle at its highest recruitment state. The actuator bandwidth is limited by the flowrate and pressure dynamics of the valve. For example, a FAM may be able to contract for a single cycle at a high frequency using the fluid stored in the accumulator. However, if the flowrate into the accumulator is not able to recharge the accumulator by replacing the depleted fluid and pressure in time for the next actuation cycle, the system cannot maintain cyclic operation. Therefore, the deciding factor in determining the bandwidth of an electrohydraulic system is the ability of the pump to replenish the accumulator volume output within the timespan of a cycle.

Simulations are performed for various amplitude, frequency, and loading scenarios to compare the overall system performance metrics such as efficiency and bandwidth. The cases considered can be categorized by the type of actuator and the pump operating configuration. The two actuator types are the SEMU and the VR bundle. Both actuator types are simulated for two different pump operating configurations: constant pump displacement or variable pump displacement. After all the results for each case are presented, a comparison of the efficiencies is discussed in Sec. 4.2.4.

The subsystem models presented in Sec. 2 are used to perform fully-coupled dynamic simulations of a VR bundle and electrohydraulic power system producing sinusoidal actuation under different electrohydraulic configurations. To demonstrate the capabilities of VR, the results are compared to that of a SEMU. FAM dimensions for the VR bundle are identical to that of Table 1, and dimensions of the SEMU are summarized in Table 3 and remain the same for all cases.

In the first electrohydraulic configuration case considered, the pump displacement remains constant while the motor runs continuously. The electrohydraulic system circuit containing the subsystems modeled in Sec. 2 is illustrated in Fig. 9.

where $Qp$ is the pump flowrate output and $Dconst$ is the constant pump displacement. The pump is decoupled from the volume requirement of the bundle during actuation. When the accumulator is full and has reached its maximum pressure, the excess fluid out of the pump returns back to the reservoir through a relief valve. The initial conditions are presented in Table 4, and the simulated results for both a SEMU and a VR bundle are shown in Fig. 10.

A prescribed sinusoidal contraction with frequency of $0.25 Hz$ and an amplitude of $x/L0=0.232$ is tracked for both SEMU and VR bundle cases. For the VR bundle, the sinusoid is tracked using the method presented in Sec. 3.1. In case of the SEMU, the required pressure or volume is continuous and thus, does not require the anticipation and compensation methods previously discussed. A hanging mass weighing 20% $(mg/Fbf=0.2)$ of the overall actuator blocked force attached to the end of the actuator. Initially, the accumulator is at its maximum pressure as shown in Fig. 10(c). While the volume output from the pump into the accumulator is greater than that of the actuator, the accumulator pressure remains constant at maximum pressure. However, as the volume required by the actuator exceeds the pump output capability, the fluid stored in the accumulator is used and thus, the accumulator pressure decreases. Consequently, the electrical power required by the motor decreases as the torque required by the pump decreases while motor speed increases. The total fluid volumes required for one cycle of contraction are equal for both actuators. However, the fluid required by the VR bundle during recruitment state 1 is lower than that of a SEMU. A sharp increase in volume is observed for the VR bundle when the transition from recruitment state 1 to 2 occurs. This difference in required volume impacts the bandwidth performance of the actuation system.

The system efficiency and bandwidth limit for a sinusoidal contraction operating with constant pump displacement values of $1.27×10−8 m3/rad (0.08 ml/rev)$ and $0.32×10−8 m3/rad(0.02 ml/rev)$ are evaluated by varying its amplitude and frequency and shown in Figs. 11 and 12, respectively. For the higher pump displacement case, the simulated range of amplitude is 0.01 $m$ to 0.023 $m$ and a frequency range of 0.2 $Hz$ to 0.8 $Hz$⁠. The load applied is $mg/Fbf=0.2$⁠. The color bar limits are equivalent across Figs. 11–18 in order to make the system efficiencies comparable. For both pump displacement values, the system efficiencies for SEMU and VR bundle within the limits of the bandwidth of the SEMU actuator are nearly identical with differences less than 0.1%. A similar result has been reported by Chapman et al. when using a FAM model neglecting resistive forces [27]. Given the same amplitude and frequency, a VR bundle operating in a recruitment state other than its highest recruitment state requires less volume relative to a SEMU. The use of less fluid volume translates to less energy input and thus increases the downstream actuator efficiency [17]. However, this does not lead to an increase in overall system efficiency for this configuration. As shown by the motor/pump subsystem Eqs. (11)–(13) and demonstrated in Fig. 10, the motor power is determined by the accumulator pressure, as it determines the torque load applied to the motor. The lower volume requirement of a VR bundle at lower recruitment states results in a smaller drop in accumulator pressure and thus a slightly lower motor load, to which the minute differences in efficiency (<0.1%) can be attributed. However, such differences in torque applied to the motor are negligible compared to effects of pump fluid volume loss through the relief valves. The fluid volume output from the pump remains nearly identical between a SEMU and VR bundle and is independent of the volume requirements of the actuator. Although the system efficiency is marginally affected by the low volume requirement of a VR bundle, it broadens the amplitude and frequency space in which the actuator can operate. The bandwidth limits for a SEMU and a VR bundle operating at its highest recruitment state are identical. However, the bandwidth limit is higher for a VR bundle when operating at a lower recruitment state as shown in Fig. 11. The tradeoff between efficiency and bandwidth limit is demonstrated in Figs. 11 and 12 showing the two pump displacement values. The frequency range for the low pump displacement case is shown for only 0.2 $Hz$ to 0.5 $Hz$⁠. While a higher pump displacement broadens the actuation frequency range, the efficiencies are higher for motions that are within the bandwidth limit of the lower pump displacement case.

In the second electrohydraulic configuration considered, the pump operates intermittently in response to the accumulator pressure. During continuous motor operation, the excess fluid from the pump flows through the relief valve and back to the reservoir, wasting energy. When the accumulator pressure is close to or at its maximum value, there is no need for the motor to keep running. Therefore, by intermittently turning the motor on and off according to the accumulator pressure has potential to reduce energy losses and increase system efficiency (Table 5). The fluid volumes into the actuators and out of the pump as well as the accumulator pressures for both SEMU and VR bundle are identical to case 1. The main difference is demonstrated by the power input to the motor. At time zero, a current spike is observed when the motor is turned on. As the accumulator pressure reaches its maximum value, the motor is switched off until the start of the next contraction cycle.

Simulations are performed for identical constant pump displacement, sinusoidal contraction amplitude and frequency ranges as case 1 but with intermittent motor operation. A significant increase in system efficiency can be observed for both SEMU and VR bundle actuators. Although the motor initially requires more power during startup, the energy conserved by turning off the motor is much greater. The effectiveness of intermittent motor operation is noticeable by comparing the SEMU efficiencies shown in Figs. 11(a) and 15(a). Also, the efficiencies for a VR bundle operating with intermittent motor operation are shown in Fig. 15(b). Compared to the continuous motor operation case in Fig. 11(b), an increase in efficiency is observed over the entire amplitude–frequency space.

The third electrohydraulic configuration case considered is when the pump displacement is able to adjust itself to the volume needs of the actuator. The system circuit is identical to case 1 as shown in Fig. 9 with the except of the constant displacement pump that is replaced with a variable displacement pump. In the constant pump displacement case with continuous motor operation, fluid output of the pump that exceeds the maximum accumulator volume must flow through a relief valve back to the reservoir, resulting in excess motor energy consumption. The purpose of variable pump displacement is to minimize this loss by using knowledge of the volume required by the actuator to instantaneously adjust the pump displacement. It should be noted that this study assumes that no energy is required to change the pump displacement and that the displacement is able to change instantaneously in response to volume demand.

The change in pump displacement in response to the actuator volume needs is shown in Fig. 16(b). The change in volume and accumulator pressure remains the same as the constant displacement case. For the system to be comparable, the maximum pump displacement is set equal to that of the constant pump displacement case. During operation, the accumulator fluid volume is depleted when the volume required by the actuator is greater than the fluid volume supplied by the pump, which results in a decrease in accumulator pressure. In such circumstances, no fluid from the pump is wasted through the relief valve and the pump operates at maximum displacement. However, when the accumulator pressure is at its maximum, the pump displacement is adjusted to a submaximal value. The displacement range and the initial conditions are summarized in Table 6. The change in pump displacement does not affect the contraction as the volume supplied to the actuator remains the same as the constant pump displacement case. However, the current drawn by the motor at submaximal pump displacement values is lower than when the pump displacement is constant. Additionally, lower pump displacement results in faster motor speeds, which can allow the motor to operate more efficiently. As a result, a significant decrease in motor power can be observed as shown in Fig. 16(a) compared to cases 1 and 2 shown in Figs. 10(d) and 14(b), respectively.

The simulation results for variable pump displacement operation for applied load values of $mg/Fbf=0.2$ and $mg/Fbf=0.28$ are shown in Figs. 17 and 18. The simulations are conducted using identical sinusoid amplitude and frequency ranges as the constant pump displacement case. The effectiveness of variable pump displacement is noticeable by comparing the SEMU efficiencies shown in Figs. 11(a) and 17(a). The efficiencies for a VR bundle operating with variable pump displacement are shown in Fig. 17(b).

The amplitude–frequency efficiency map for a higher loading condition is shown in Fig. 18. It has been shown for isotonic contractile motions, the downstream actuator efficiency is higher for higher loads [32], during which there are less valve throttling losses. Although the simulation is not for isotonic loading conditions, the same principles apply that increase the downstream efficiency of the actuator and thus increasing overall system efficiency.

The percentage increases in average efficiency with respect to frequency is shown in Fig. 19 for the amplitude and frequency ranges considered. The average efficiency for a given frequency is computed for each case study using results from Figs. 11, 15, and 17. The average efficiency for the SEMU in case 1 is used as the baseline case to which other average efficiencies are compared. The lines in black compare the SEMU actuators of case 2 and case 3 to case 1. The blue lines compare the VR bundle actuators of cases 1, 2, and 3 to the SEMU actuator of case 1.

The efficiency increases due to the electrohydraulic configuration changes alone are demonstrated by the black lines that compare the average efficiencies of SEMUs. These results show that changing the electrohydraulic configuration to either and intermittently-operating constant displacement pump (case 2) or a variable displacement pump (case 3) can produce significant average efficiency improvements at lower frequencies, even with a SEMU actuator. For case 2, this can be attributed to a larger fraction of the cycle period with the motor turned off. Similarly for case 3, the variable displacement of the pump results in energy savings during a larger fraction of the cycle.

VRB cases 1, 2, and 3 in Fig. 19 illustrate the effects of combining the VR bundle actuator with each electrohydraulic configuration with respect to the first SEMU case. Compared to the SEMU, the VR bundle for case 1 shows a slight increase in average efficiency at higher frequencies which can be attributed to the functionality of the VR bundle that requires less fluid volume while operating in recruitment state 1. While the efficiencies of a VR bundle for case 1 may even decrease compared to a SEMU at lower frequency ranges due to the presence of resistive forces, the efficiencies of VR bundles for cases 2 and 3 show a significant efficiency increase for the entire frequency range considered. For case 3, this frequency-dependent behavior depends on the fraction of cycle period that is spent at maximum pump displacement. At higher frequencies, the pump is operating at its maximum displacement for a larger percentage of the contraction cycle. As the frequency decreases, the time during which the pump operates at its maximum displacement becomes shorter. At lower frequencies, the potential to conserve energy is greater due to the fact that the amount of fluid energy loss through the relief valve is greater than at higher frequencies. Using variable pump displacement minimizes the relief valve energy losses and thus shows a greater increase in average efficiency at lower frequencies. Due to the ability of the variable displacement pump to beneficially take advantage of reductions in the required fluid volume with the VR bundle, this combination shows the largest efficiency gains.

In addition to the percentage increase in efficiency with respect to frequency, the percentage increase averaged over the entire amplitude–frequency space is summarized in Table 7. As in Fig. 19, the baseline efficiency to which the average efficiencies are compared is that of case 1 SEMU. The effects of using electrohydraulic configuration cases 2 and 3 are evident from the first two rows that compare the amongst SEMU actuators. While switching to a VRB for case 1 shows a slight increase in average efficiency, more significant improvements can be gained from using different electrohydraulic configurations in tandem with the VRB actuator.

In this study, a FAM model that is able to capture the inter-FAM forces within a VR bundle is used to simulate actuation under load. Previous models that neglect the resistive forces of inactive/low-pressure FAMs were not able to fully convey the complicated force–strain space of a VR bundle, such as the overlapping region between recruitment states. By modeling the inter-FAM resistive forces and electrohydraulic system dynamics, this study more accurately captures the hydraulic pressure, flowrate, and motor current demand during the recruitment transition, thereby enabling analysis of system-level electromechanical efficiency for prescribed actuation trajectories. A system model including the motor/pump, accumulator and valve subsystems is presented and used to simulate sinusoid trajectories for a SEMU and a VR bundle with two recruitment states. For each type of actuator, dynamic simulations were run for continuous and intermittent motor operation and fixed and variable pump displacement configurations. The results from the continuous motor operation with constant pump displacement case showed that a SEMU and a VR bundle had nearly identical efficiencies. However, the low volume requirement of a VR bundle during lower recruitment states allowed the VR bundle actuator to operate in a broader range of frequencies. Simulation results from the intermittent motor operation with constant pump displacement showed increased efficiencies for both SEMU and VR bundles actuator types. Further increases in efficiency were shown to be possible by using variable pump displacement condition with continuous motor operation. The percentage increase in efficiency was dependent on the trajectory frequency, showing greater percentage increases at lower frequencies. The benefits of variable pump displacement were even greater for a VR bundle especially for lower recruitment states, indicating that variable displacement pump configurations may be well-suited to take advantage the reduced working fluid demand created by VR bundle actuators.

• The Faculty Early Career Development Program (CAREER) of the National Science Foundation (NSF) (Award No. 1845203 and Program Manager Irina Dolinskaya; Funder ID: 10.13039/100000001).

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

$Fideal$ =

FAM ideal model force $(N)$

$Fmesh$ =

FAM mesh force component $(N)$

$Fmesh′$ =

FAM mesh force component using arc length of buckled shape $(N)$

$Fbladder$ =

FAM bladder force component $(N)$

$W$ =

Mooney-Rivlin strain energy function

$εfree$ =

free strain of actuator

$Fb$ =

bladder force in the axial direction in buckled region $(N)$

$Fc$ =

bladder force in the axial direction in collapsed region $(N)$

$r0$ =

FAM initial inner radius $(m)$

$t0$ =

FAM initial thickness $(m)$

$α0$ =

FAM initial braid angle $(deg)$

$L0$ =

FAM initial length $(m)$

$β$ =

transition constant

$QN$ =

nominal flowrate $(l/m)$

$ΔpN$ =

nominal pressure drop $(kPa)$

$dv,max$ =

max. port diameter $(m)$

$Emax$ =

max. bulk modulus of hydraulic oil $(kPa)$

$Vmotor$ =

motor voltage $(V)$

$kb$ =

back EMF constant $(V·s/rad)$

$R$ =

terminal resistance $(Ω)$

$L$ =

motor inductance $(H)$

$ke$ =

motor torque constant $(Nm/A)$

$Bm$ =

frictional damping coefficient $(Nm/rad·s)$

$J$ =

rotor moment of inertia $(kgm2)$

$Pacc,0$ =

accumulator initial pressure $(kPa)$

$Pacc, max$ =

accumulator max. pressure $(kPa)$

$Vacc,0$ =

accumulator initial volume $(m3)$

$Vacc,max$ =

accumulator max. volume $(m3)$

$rc,down$ =

downstream conduit radius (m)

$lc,down$ =

downstream conduit length (m)

$rc,up$ =

upstream conduit radius (m)

$lc,up$ =

upstream conduit length (m)

$η$ =

oil dynamic viscosity $(Ns/m2)$

$x$ =

actuator position

$x˙$ =

actuator velocity

$Dp,const$ =

pump displacement $(ml/rev)$

$Dp,var,max$ =

variable pump max. displacement $(ml/rev)$

$Dp,var,min$ =

variable pump min. displacement $(ml/rev)$

$I$ =

motor current

$θ˙$ =

motor speed