Series Elastic Actuators for Small-Scale Robotic Applications

A number of robotic applications demand a miniature actuation system that can deliver precise control of force or torque. Examples include an exoskeleton for hand rehabilitation, a haptic interface, and small-scale robotic manipulator. In most small-scale robots,² electromagnetic actuators (e.g., DC motors) are the most suitable given the small form factor, ease of use, and clean design. For electromagnetic actuators, precise control of torque (or force) can be achieved either with a torque (or force) sensor and feedback control, or with an arrangement of a spring in series with the actuator and hierarchical control with position feedback. Adding a torque (or force sensor), e.g., a load cell, is costly and also requires significant space, which makes this option infeasible, especially for small-scale robots, when compared to the SEAs. An SEA consists of a motor, an elastic element, and accurate position sensors on either side of the spring [1]. Torque (or force) in an SEA is estimated by measuring the deflection of the spring. An SEA also adds an inherent mechanical compliance in the system which is shown to be advantageous in the robotic systems that physically interact with the human body [2]. While SEAs are suitable for actuation with precise force control, there is a lack of miniature, compact, and bidirectional SEAs for the robotic applications.

For the small-scale robotic applications, we set the requirements on SEAs stiffness to be small (<1 N·m/rad) and peak torque to be ∼0.5 N·m, and we want the actuator to fit in the space of 45 × 35 × 20 mm³. The requirements are decided by the hand requirements in therapy settings. The peak torque requirement is chosen based on the fact that experienced therapists apply a bidirectional peak torque of at least 0.3 N·m on human digit joints as measured through a torque measuring device [3]. The requirement on the dimensions of the SEA is governed by the fact that two such SEAs should fit within the length of a finger to actuate two of the proximal finger joints independently and that each SEA should be accommodated within the width of a finger for these to be used for multiple digits adjacent to each other [4]. Also, the torque bandwidth of at least 2 Hz is required based on the fact that the bandwidth of the human force compliance control loop is 1–2 Hz [5,6]. In addition, an appreciable angular deflection (∼15 deg) is required for the elastic element at the maximum torque to provide sufficient torque resolution under the noisy angle sensor measurements. We aim for our hand exoskeleton to have a weight of under 300 g and a total of 10 SEAs to actuate the joints of all the digits. To meet this criterion on weight, each SEA is required to weigh less than 30 g excluding the Bowden cables.

We presented the design of a LC-SEA and preliminary testing of the SEA including torque tracking performance, performance at different peak torques, and torque bandwidth using a second-order model in Ref. [4]. In this work, we make the following novel contributions: (i) present the design of a miniature Bowden-cable-based HT-SEA that by design ensures symmetrical torque versus deflection profile bidirectionally; (ii) carry out a thorough evaluation of LC-SEA and HT-SEA including estimation of effective stiffness, dynamic range, performance under disturbance, results from interaction with an environment using the hand exoskeleton; and (iii) compare and contrast the performance of the two SEAs (Fig. 1) through experiments. This work makes contributions over our previous work by presenting a new SEA design, thorough experimentation of the two designs and insights on the trade-offs between the two designs.

There have been five main types of actuation mechanism used for hand exoskeletons: (i) locally situated motor with tendon-based actuation [7–9], (ii) locally situated motor with linkage-based actuation [3,10–12], (iii) remotely situated motor with cable and sheath transmission [13–20], (iv) remotely situated motor with flexible shaft transmission [21], and (v) pneumatic actuation [22–25]. These mechanisms do not allow for accurate and stable torque control of the exoskeleton. Furthermore, these have poor back-drivability and result in high reflected inertia. In addition, there is nonlinear friction and stiction in the transmission and actuator gearing of some of these mechanisms, which makes it difficult to control actuator force or torque accurately.

SEAs have been successfully implemented in a number of robotic applications to precisely control torque, including in a few exoskeletons for the lower extremity [26,27] and upper extremity [28], humanoids [29], and bipeds [30]. Table 1 presents a survey of important SEAs for robotic applications. None of the existing SEAs satisfy our requirements for small-scale robotic applications. SEAs could be categorized based on two main criteria: (i) type of elastic element and (ii) location of the actuator. Within the first category, an SEA can have three different types of elastic elements: (i) linear compression spring, (ii) helical torsion spring, and (iii) structural torsion spring. In the second category, an SEA can have either a locally situated or remotely located actuator.

Pratt and Williamson introduced the idea of an SEA with a structural torsion spring and a locally situated geared motor to allow for accurate and stable force control, lower reflected inertia, and greater shock tolerance [1]. Several other SEAs have been presented for different applications that fall under the same categories of a structural torsion spring as elastic element and a locally situated motor (electric or hydraulic) [33–39]. A limitation of such a design is that structural torsion springs need to be custom designed for each application, and therefore, an explicit characterization is needed experimentally to validate the torque–deflection behavior for such springs. Furthermore, designing a structural torsion spring with a certain stiffness characteristics requires accurate finite element analysis. A few other designs with more standard off-the-shelf linear tension or compression or helical torsion springs have been presented that do not require custom design and analysis of springs [32,34,35]. However, locally situated actuator in all these designs makes the SEA bulky.

To reduce the bulkiness of the SEA, Veneman et al. presented an SEA with a remotely located actuator and a Bowden-cable-based transmission to transfer mechanical power to the SEA joint [26,40]. However, in all such designs an explicit sensor was used for measuring the deflection of the elastic element [40]. In applications with limited space, incorporating an extra sensor at each joint with additional wiring becomes infeasible. In our designs, the deflection of the spring is estimated from the joint and motor position. This simplifies the design as the need of an extra sensor is eliminated. However, from a control perspective, it adversely affects the stability of the system and requires estimation of the effective stiffness of the compression spring and the Bowden cable.

The existing Bowden-cable-based SEA design employing helical torsion spring used only one spring to generate only unidirectional torques [27]. Our helical torsion spring SEA, on the other hand, has two torsion springs and could generate bidirectional torques, which is needed for the application of a hand exoskeleton for rehabilitation. The symmetric bidirectional linear nature of torque with respect to spring deflection is preserved in such a design as the springs are prestressed and arranged so as to generate opposing torques. This arrangement ensures that both the springs operate in the correct direction of winding and results in same stiffness in both clockwise and counter-clockwise directions of the torque. This is critical for a design with Bowden-cable-based SEA, which does not employ an additional sensor for deflection of the elastic element, as the effective stiffness needs to be estimated experimentally. The structural torsion spring designs presented in the past becomes infeasible for an application which requires low torques in a small space.

Bowden cable introduce elasticity and nonlinear friction in the transmission [16,43,44]. To achieve good torque control with Bowden-cable transmission, estimation of the effective stiffness of the Bowden cable and sheath combination is necessary. In addition, the effective stiffness also changes with the configuration of the design, which affects how the load is transferred through the sheath. This requires explicit testing of SEAs having different configurations even when the stiffness of the metallic elastic element is known a priori. In the large-scale robotic applications, the problems due to Bowden-cable elasticity and friction are overcome with explicit sensors to measure the deflection of the elastic element. In the small-scale robotic applications these problems lead to more severe effects and the small size makes it difficult to mount additional sensors.

A system which is required to be accommodated in a very small space requires minimum number of parts. We designed complex shaped parts serving multiple functions that are difficult to be machined using conventional processes and are, therefore, 3D printed. The design of the elastic elements within the limited space itself poses distinct challenges for each configuration: (a) for linear compression springs—how to achieve high stiffness while avoiding buckling of the spring, (b) for helical torsion springs—how to achieve high stiffness in limited outer diameter and width while maintaining linear bidirectional nature of the torque, and (c) for structural torsion spring—how to achieve large angular deflection with relatively high stiffness of such springs without material failure [45]. Also, to ensure compact design a small joint angle sensor is required, but the resolution of small angle sensors is typically poor. This requires the angular deflection of the elastic element to be sufficiently large for it to be properly resolved by the sensor and in turn achieve a desirable torque resolution. A stiffer spring allows to achieve higher peak torques and torque bandwidths, however, at a poorer resolution. Thus, a tradeoff is involved in choosing the stiffness of the spring while addressing all the aforementioned challenges.

In HT-SEA, two helical torsion springs were introduced between the Bowden-cable pulley at the joint side and the output link (Figs. 2(a) and 3(a)). However, since the cable pulley and output link need to be separated in this design, it had more components and was wider (24 mm) than the LC-SEA (17 mm) (Figs. 2(b) and 3(b)). The torsional springs were installed in a prestressed state and arranged so as to generate opposing torques. This arrangement ensures that both the springs operate in the correct direction of winding and results in same stiffness in both clockwise and counter-clockwise directions of the torque.

Typically, SEAs employ explicit sensors (e.g., potentiometer) to directly measure the deflection of the elastic element. However, the considerable size of off-the-shelf sensors makes it difficult to mount these in the limited space on a hand exoskeleton. Instead, we estimated this deflection by using the motor and joint position measurements for both the SEAs.

$β=eμθ$ is the Bowden-cable coefficient for the selected friction model [46], where μ is the coefficient of friction between the Bowden cable and sheath, and θ is the total wrapping angle of the cable. The sign function (⁠$σ(.)$⁠) is used to capture the direction of friction based on the rate of change of the spring length, which signifies the direction of movement of the cable inside the sheath. We model the motor as a position source as the motor bandwidth is much higher (>ten times) than the achievable SEA torque bandwidth due to the much lower stiffness of the SEA needed for our application. In addition, the position control loop at the actuator level runs at several kHz as compared to the torque control loop, which runs at 500 Hz. Furthermore, typically the rehabilitation exercises for the hand digits are carried out at full range of motion frequencies below 0.5 Hz [47]. Considering all these factors, we do not take into account the motor dynamics explicitly and model the motor as a position source.

It can be seen that the feed-forward terms in Eqs. (3) and (6) are mathematically equivalent when $kj=krj2$⁠. However, in practice the effective stiffness and friction acting in the system is significantly affected by the design configuration, and hence, the performance of the two SEAs needs to be experimentally validated.

Once a model of the torque output from each SEA is derived, a controller is designed for the SEA. The goal of this controller is to track the reference torque at the SEA joint using feed-forward proportional–intergral–derivative (PID) control. The controller consists of an inner position control loop at the actuator level and an outer force control loop at the SEA level (Fig. 5). The output of the system is the torque generated at the output joint through SEA. The PID controller with the corresponding feed-forward term is then given by Eq. (7). The feed-forward term is introduced to obtain a model-based input to the system for the reference output and a subsequent correction is done using PID control based on the error between the reference and estimated torque. The feed-forward term helps in improving the control accuracy and stability [48]

where T is the sampling time of the overall control loop. The outer force control loop runs at 500 Hz and the inner position control loop runs at several kHz. Since inner position control loop runs at several kHz, we do not account for the transport delay in that loop.

The following values for the various parameters in the system model are used for the simulation: I_j = 3.4 $×10−5$ kg m², r_j = 12 mm, r_m = 28 mm, k = 1103 N/m, $Ti0=14N$⁠, β = 1.1, τ_A = 0.3 N·m, f = 0.5 Hz, and $ϕ$  = 0 deg. Since the SEA joint has a bearing, we do not consider the damping at the joint for the simulation. We simulated the system for different values of spring stiffness.

The simulation results show that the PID control with a feed-forward term is able to track the desired torque trajectory with a root mean square error (RMSE) of 2.67% (0.008 N·m). Small deviation from the desired torque was observed at the peaks of the sinusoid (Fig. 6(a)). The motor angle stays within safe bounds for the system to achieve bidirectional torque control with a peak torque of 0.3 N·m (Fig. 6(b)). The tension in the cable on joint side shows a sudden change due to the nature of the model that captures the change in the direction of friction as the motion reverses (Fig. 6(d)). The peak tension in the Bowden cable is observed to be 30 N, which ensures that the stretch in the cable is negligible and that good spring deflection estimates could be obtained using the joint and motor position measurement. We chose a commercially available low friction polytetrafluoroethylene (PTFE) coated sheath (1 mm inner diameter, 3 mm outer diameter) and fluorinated ethylene propylene (FEP) coated cable (0.66 mm diameter) with a rated load capacity of 35.6 N and design safety factor of 5:1 compared to breaking strength for the SEAs. A quick test by hanging 10 kg load on the cable showed that it stretched less than 1 mm. The simulations also showed that an effective spring stiffness of around 2000 N/m or 0.3 N·m/rad is needed to achieve the required peak torque with a motor angle of above 20 deg.

A test rig was developed for testing the performance of the two SEAs (Fig. 7). On the joint side, a six-axis load cell was mounted to measure the output joint torque. The same magnetoresistive angle sensor (KMA210, NXP Semiconductors) as was installed on the hand exoskeleton was used to measure the joint angle on the test rig. This ensures that the torque tracking performance that could be achieved with the sensor noise on the actual device is captured on the test rig. A noise of around 0.5 deg peak magnitude was observed in the sensor measurements. Detailed characterization of the sensor measurement error is available in the datasheet.³ We used a geared brushed DC motor (22 W) from Maxon, Inc. (Detroit, MI) as the actuator for the SEAs.

We have developed a torque controlled index finger exoskeleton for rehabilitation (Fig. 8) [4]. The device has two actuated degrees-of-freedom (DOF) to assist the motion at the metacarpophalangeal (MCP) and proximal interphalangeal (PIP) joints of the index finger. The kinematics of the device is designed to induce low finger joint reaction forces during the operation of the device. Each finger joint is actuated using a four-bar mechanism to avoid finger-exoskeleton axes misalignment.

The experiments with the SEA test rig and the developed exoskeleton were aimed at characterizing the following: (i) accuracy and fidelity of torque tracking; (ii) torque bandwidth (iii) dynamic range; (iv) performance at different peak torque magnitudes; (v) performance under disturbance; and (vi) rendering low and high joint stiffness on index finger exoskeleton. Experiment (v), which is carried out for detailed testing of the SEA, is conducted with LC-SEA only, since HT-SEA showed much lower dynamic range, more nonlinear scaling of torque at different peak torque magnitudes and limited peak torque, making LC-SEA more suitable for our application. Also, experiment (vi) is carried out on the LC-SEA implemented on the index finger exoskeleton.

The effective stiffness acting in the system differs from the catalog value of the stiffness of the off-the-shelf springs. This is due to the fact that there are several factors that contribute to the effective stiffness in the system. These include the compression of the sheath and the interface in which the spring rests for LC-SEA, which is designed to have a tight fit with the spring for it to act as a built-in support for reducing sideways bending of the spring, leading to reduction in the effective length of the spring and friction. This requires explicit estimation of the effective stiffness acting in the system. Also, for the miniature SEA with 3D printed parts, the joint inertia is very small. Furthermore, for our application of hand rehabilitation the system is required to operate at relatively low accelerations and velocities. In addition, the damping at the joint is small as the joints are supported with miniature ball bearings. Thus, the contribution of the inertial and damping terms is much smaller and the torque applied using the Bowden cables is transmitted to the joint without significant reduction.

The system is then run with the identified stiffness for the same desired torque trajectory and the process is iterated a few times to ensure that the identified effective stiffness does not vary significantly. This identified stiffness is kept constant throughout the experiments for each SEA. Finally, the PID gains of the torque controller are manually tuned to achieve a stable and accurate control of the joint torque.

The goal of this experiment was to track a sinusoidal joint torque on the SEA test rig and verify the tracking accuracy and fidelity [26,50,51] of the output torque using the measurements obtained through the load cell.

Torque bandwidth of the two SEAs was evaluated by using a linear chirp signal as the desired torque for the system [4]. We evaluate the magnitude of the frequency response of the SEAs using the measured torque data before fitting any model to capture the response with the nonlinearities present in the system. We take fast-Fourier transform of the commanded and measured torque, which provides an amplitude spectrum of the system input and the output. The magnitude of the frequency response is then evaluated as the ratio of the amplitude of the output to the input in the frequency domain. Fast-Fourier transform has been used in the past in general for spectral analysis of nonlinear systems [52]. This helps in avoiding any approximation which is typically introduced when the response is obtained by using a linear model fitted to the data. We also identified the closed-loop system model, Eq. (9), from the sampled frequency response data by fitting continuous-time systems of different orders [49,53] to obtain the model that best describes our system and then used the identified models to obtain the frequency response.

Since the torque tracking accuracy and fidelity of the SEA vary with the change in the peak torque magnitude, we carry out the tests to verify the change in performance. We apply torques with different peak torque magnitudes and separately evaluate these metrics. In addition, we identify the effective stiffness that best explains the output torque.

Since the index finger exoskeleton will be moved around while the device is actuating a subject's hand, assessing the performance of the device under varying degree of disturbance was important. To assess the performance under disturbance, three different levels of disturbances were applied to the joint side setup while the torque tracking experiment is carried out. For mild disturbance, the joint side base is moved up and down in a plane. For medium disturbance, the joint side base was carried up and moved up and down as well as sideways such that the cable is bent by 90 deg both ways. For the severe case, the base is carried and moved so that the Bowden cable is bent by over 180 deg both ways. For our application, we anticipate a mild to medium degree of disturbance during the operation of the device. Moving the configuration of the cable during the experiment helped in determining if significant error would be introduced in torque tracking during the operation of the device.

where θ and θ_r are the exoskeleton current and reference joint angular position, respectively. A sinusoidal reference trajectory was chosen for the exoskeleton joint to achieve the finger flexion and extension motion at the PIP joint. First, a low stiffness (k_pip = 0.1 N·m/rad) was chosen for the controller and the subject was asked to occasionally impede the exoskeleton motion. Next, the experiment was repeated with a high stiffness (k_pip = 0.6 N·m/rad). The high stiffness values are chosen such that the exoskeleton is able to track the reference trajectory without any discomfort to the subject.

The controller was able to track the desired torque trajectory with high fidelity for a 0.2 N·m peak torque sinusoid for both the actuators (Fig. 9; Table 2). The torque measured using the load cell, torque obtained by fitting the best curve to the measured torque, desired torque trajectory as available to the real-time controller and the torque trajectory as estimated by the real-time controller using SEA are referred to as measured, best fit, desired, and estimated, respectively. All the trajectories show small errors (<11% of peak torque) with respect to the desired torque trajectory (Table 2). The load and configuration dependent effective backlash [43] in the Bowden cable leads to some deviations of the measured trajectory from the estimated one near the peak of the sinusoid where the motor changes the direction. Also, the effective stiffness was a result of the combined stiffness due to the springs and the Bowden cable sheath compliance.

The frequency response results showed that both the actuators satisfy the torque bandwidth criterion (>2 Hz). The LC-SEA and HT-SEA have closed-loop bandwidths (−3 dB magnitude) of 2.5 and 4.5 Hz, respectively, for a 0.2 N·m peak torque (Fig. 10). LC-SEA has a more damped frequency response as compared to HT-SEA. A resonant peak was also observed in the magnitude plot of HT-SEA, which increases its bandwidth. However, the system is sufficiently damped as the system does not exhibit a sharp resonance peak. The fitting results showed that while a fifth-order system best describes the LC-SEA, a fourth-order system provides a best fit response for HT-SEA (Table 3). A comparison of the system response from the identified model that best explains the system with the measured and the desired torque trajectory shows that there is low phase angle between the trajectories even at relatively higher frequencies of >2 Hz for LC-SEA and >3 Hz for HT-SEA (Fig. 11). However, the fitting percentage shows that the nonlinearity in the system (possibility due to nonlinear friction in Bowden cable) is not completely explained by these models (Table 3).

The dynamic range of LC-SEA was found to be about three times higher than HT-SEA (Table 1). This is because of the higher effective stiffness of HT-SEA as compared to LC-SEA and more uniform scaling of torque in LC-SEA.

LC-SEA can be controlled to achieve good tracking performance (RMSE <12% and fidelity >97%) for torques of peak magnitudes between 0.15 and 0.3 N·m (Table 4). In addition, the identified stiffness was found to be fairly constant at all peak torque magnitudes. Gradually increasing the desired peak torque for the HT-SEA showed that the achievable peak torque was limited to 0.3 N·m with off-the-shelf torsion springs (Fig. 12). In addition, the torque was found to scale more nonlinearly for HT-SEA as compared to LC-SEA. Thus, LC-SEA can be used as a good torque source for the hand exoskeleton.

The results show that the tracking performance did not deteriorate significantly even when severe disturbance was applied at the output end for the LC-SEA (Fig. 13). However, the torque output showed increasing error as the severity of the disturbance was increased. We anticipate a mild to medium disturbance during the operation of our device. Thus, LC-SEA ensures good torque tracking performance during the operation of the device. Since the disturbance mainly affects the friction between the sheath and Bowden cable, which is similar for both types of SEAs, we anticipate that a similar deviation in torque tracking would be observed for HT-SEA from the baseline tracking performance.

Results show that with low stiffness the joint torques do not increase significantly when the exoskeleton motion is impeded (Fig. 14(b)). However, the exoskeleton is not able to track the reference trajectories at low stiffness (Fig. 14(a)). With high stiffness, on the other hand, the exoskeleton is able to track the reference trajectories within its range of motion closely (Fig. 14(a)). However, considerable increase in joint torques is observed when the motion is impeded (Fig. 15(b)). This is because the applied torque is proportional to the error in tracking the reference trajectory for this controller.

We present two designs for actuators for small-scale robotic devices. Both designs are Bowden-cable-based SEAs that allow for accurate torque control. One of the designs uses linear springs (LC-SEA) and is previously introduced, and the second design, introduced in this paper, uses helical torsion springs (HT-SEA). We have carried out thorough analysis and comparison in performance of the two designs. Experiments using an SEA test rig showed that with a feed-forward PID controller the desired torque was tracked with sufficient accuracy (RMSE <12%) and fidelity (>97%) for both the actuators. This was achieved without an explicit sensor to measure the deflection of the elastic elements. While LC-SEA is longer (6 mm) than HT-SEA, it is overall more compact. In addition, HT-SEA has more components than LC-SEA and is more complex in design. However, the springs are contained within the assembly in HT-SEA, but are installed externally in LC-SEA. Though the two SEAs are equivalent mathematically, the differences in physical design lead to significantly distinct performance characteristics. The torque bandwidth for HT-SEA was found to be higher (∼4.5 Hz) than LC-SEA (∼2.5 Hz), even when a higher catalog stiffness was used for LC-SEA. On the contrary, the dynamic range (7.5–27.5) and the achievable peak torque (±0.3 N·m) was found to be limited for the HT-SEA with off-the-shelf springs having maximum stiffness (0.86 N·m/rad) in the limited space. Also, in HT-SEA there is a limit to the angular deflection of the springs beyond which the applied torque does not hold a linear relationship with the angular deflection. LC-SEA, on the other hand, was found to be performing well even at different peak torque magnitudes and under disturbance. In addition, the torque was found to scale more linearly with the increase in the peak torque magnitude for LC-SEA as compared to HT-SEA. The experiments also showed that the exoskeleton was able to successfully render both low and high impedances.

HT-SEA would be preferred over LC-SEA where high frequency torques are needed as the bandwidth of HT-SEA is higher, since there is less damping present in the HT-SEA design. Furthermore, HT-SEA design is cleaner from an end user's perspective as the springs are enclosed in the casing. Also, since the effective stiffness is higher for HT-SEA as compared to LC-SEA, HT-SEA is more efficient in terms of the energy requirement to generate the same amount of torque. Applications that would require the SEA to run off a battery pack would benefit from HT-SEA, since it has lower power requirement. In addition, since the effective stiffness is higher for HT-SEA the tensions induced in the cable at the motor end are lower as compared to LC-SEA. So, a smaller size motor with a smaller diameter sheath and cable combination could generate the same amount of torque as that generated by an equivalent LC-SEA.

The higher bandwidth of HT-SEA is because the effective stiffness for LC-SEA was found to be much smaller than the catalog stiffness as compared to HT-SEA (Table 2). This may be due to the difference in the way sheath is clamped in the two SEAs. In LC-SEA, the sheath interfaced with the linear compression spring directly, whereas in HT-SEA the sheath interfaced with the housing and the cable is attached to the pulley, which housed the springs. Also, the springs in LC-SEA show some sideways deflection, which might contribute to the reduction in effective stiffness. In addition, in LC-SEA there is some friction between the spring and the interface where the linear compression spring rests, which reduces the effective stiffness. No such friction is present in HT-SEA. This results in an observable resonance peak in the frequency of HT-SEA [55], which increases its bandwidth.

The output torque range of the SEAs can be adjusted as per the application by choosing the appropriate spring stiffness. However, the design with helical torsional springs is limited by the availability of off-the-shelf springs. Good performance of the SEA requires appropriate pretension of the Bowden cable to ensure that the cable is not slack throughout the output torque range. Both highly stiff and highly compliant springs deteriorate the performance of the actuator. A very stiff spring reduces the torque resolution due to the noise in the angle sensor measurements at the joint, whereas a very compliant spring limits the maximum achievable torque at the joint. Despite the nonlinear friction in the Bowden cables, the performance of the actuator was satisfactory with feed-forward PID control. However, significant reduction in effective stiffness was observed with the introduction of Bowden cable in the transmission. This reduction in stiffness was also found to be dependent on the physical design. Furthermore, the cables also introduced some friction in the system, which was also dependent on the configuration of the cable. Also, it was observed during the experiments that the major contribution of the elasticity was due to the compression of the sheath rather than the elongation of the Bowden cable. In the future, we plan to carry out a detailed analysis of the force transmission through the Bowden cable considering its flexibility. Furthermore, we plan to carry out experiments with metal sheath, which could significantly reduce the losses in the transmission and help in achieving higher torques and bandwidths for the same size of the SEAs. Our goal in the future is to develop a hand exoskeleton supporting all the hand digits using the SEAs presented in this work. The developed SEAs could also be used for several small-scale robotic applications including wearable haptic devices, surgical robots, and small-scale robotic manipulators.

This work was supported, in part, by the National Science Foundation (NSF) Grant No. CNS-1135949 and National Aeronautics and Space Administration (NASA) Grant No. NNX12AM03G.

b_j =

damping at the SEA joint

$Ci(s)$ =

controller transfer function

$Ci,fb(s)$ =

feedback component of controller transfer function

$Ci,ff(s)$ =

feed-forward component of controller transfer function

e =

error for PID control

f₀ =

frequency of the chirp signal at t = 0

f₁ =

rate of increase of frequency for the chirp signal

$Gi(s)$ =

subsystem transfer function

$GCL(s)$ =

closed-loop transfer function of the system

$GOL(s)$ =

open-loop transfer function of the system

I_j =

inertia of output link of SEA

k =

stiffness of linear compression spring

k_j =

stiffness of torsional spring

$kp,kd,ki$ =

proportional, derivative, and integral gains for the PID controller

r_j =

SEA joint pulley radius

r_m =

motor pulley radius

t =

time

T =

sampling time

T_i =

tension in Bowden cable

$Ti0$ =

initial tension in Bowden cable

u =

control input to the system

β =

Bowden-cable coefficient for the friction model

$Δli$ =

change in spring length l_i

$Δθ$ =

torsional spring deflection for HT-SEA

θ_j =

joint angle of output link of SEA

θ_m =

motor angle

θ_p =

joint pulley angle for HT-SEA

$θj0$ =

initial joint angle of output link of SEA

$θm,ff$ =

feed-forward motor angle

τ_A =

amplitude of the desired sinusoidal torque trajectory

τ_d =

desired torque

τ_j =

output torque at SEA joint

τ_r =

reference output torque at SEA joint

τ_jm =

measured output torque at SEA joint

$τ̂j$ =

estimated output torque at SEA joint

$σ(.)$ =

sign function