Design, Control, and Evaluation of a Robotic Ankle-Foot Prosthesis Emulator

People with transtibial limb loss experience reduced mobility. Intelligent ankle-foot prostheses have the potential to improve quality of life in people with limb loss, but there are scientific, clinical, and commercial barriers that prevent widespread impact. Further research tools and experiments are needed to expand our understanding of how to design and control intelligent prosthetic limbs. We designed and built a robotic ankle-foot prosthesis with off-board actuation and control to serve as a platform for biomechanical lower limb loss research. Our prosthesis fits inside of a shoe during walking and attaches to standard clinical prosthesis componentry, including carbon fiber prosthetic footplates and pyramid adapters. Our novel mechanical architecture implements a custom torsion spring in parallel with the ankle joint to allow for dorsiflexion and plantarflexion torque control with a single off-board actuator. Benchtop tests show that our prosthesis has peak plantarflexion torques greater than 175 Nm and a torque control bandwidth of 6.1 Hz. Walking experiments with two participants with lower limb loss indicate that the prosthesis can achieve low torque tracking errors and push-off power greater than the biological ankle during walking. This device will enable future experiments on amputee gait biomechanics, human-robot interaction, and prosthesis control.

osteoarthritis [4], [5]. Further, balance during walking and standing is compromised, increasing the risk of falls and fear of falling [6], [7]. The prevalence of lower limb amputation in the United States is on the rise, primarily due to vascular diseases like diabetes and peripheral artery disease [8], [9]. One promising way to improve mobility in people with lower limb loss is through improvements to ankle-foot prosthesis technology.
Three primary classes of ankle-foot prostheses exist for people with limb loss: passive, semi-active, and powered robotic prostheses. Passive ankle-foot prostheses, the most common class, are typically made from composite materials and have fixed joint alignment and stiffness properties. Semi-active prostheses offer a wider range of behaviors by employing adjustable mechanisms and low-power electronics to alter mechanical properties or joint alignment depending on walking conditions [10], [11], [12], [13]. However, only powered robotic ankle-foot prostheses have the potential to fully replicate the capabilities of a biological limb by using batteries and powerful actuators to replace muscle work in each stride [14], [15], [16], [17]. Despite their promise, additional research is necessary to maximize the impact and adoption of robotic prostheses.
Open questions and barriers exist across all three classes of ankle-foot prostheses that limit their clinical impact. Optimal prescription for passive prostheses remains uncertain due to a plethora of subcategories with different features and insufficient clinical evidence to guide prescription [18]. Semi-active prostheses require further investigation into which adjustable mechanical properties are most consequential for end-users. Powered robotic ankles face challenges in mechanical design and control strategy [19], [20]. Few experiments have compared outcomes between control strategies, and no powered ankle-foot prosthesis has consistently and reliably improved mobility outcomes compared to conventional prostheses [21], [22], [23], [24]. A common challenge across all classes is predicting patient walking outcomes in response to changes in prosthesis properties or controller behaviors. This challenge is largely due to the inherent variability in human behavior and the difficulty in predicting gait adaptation in the presence of a new prosthesis or control strategy. In light of these challenges, there is a need for the development of tools that can facilitate the rapid exploration of various prosthesis behaviors and the associated human locomotor responses.
Robotic testbeds, or emulators, are powerful tools for rapid hypothesis testing in wearable assistive device behavior. These devices utilize off-board servomotors to transmit forces to 2576-3202 c 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information. wearable assistive devices through flexible actuation tethers. Emulators minimize body-worn mass by relying on external power sources and employ high-speed measurement and control hardware to achieve impressive mechatronic performance. They can be programmed to mimic the behavior of various hypothetical candidate devices, providing a versatile platform for testing different design and control strategies.
Although the use of off-board hardware limits experiments to treadmill walking, emulators offer significant advantages in research. They accelerate the exploration of prosthesis behaviors and associated human responses by eliminating the time-consuming process of building untethered devices. As a result, they facilitate research progress across all types of prostheses -through the evaluation of passive prosthesis prescription, emulation of mechanical behavior for proposed semi-active devices, and assessment of novel control strategies for robotic ankles [25], [26], [27]. Three noteworthy ankle-foot prosthesis emulator designs have been described in the literature. Caputo and Collins developed a universal prosthesis emulator capable of providing a wide range of ankle behaviors during walking [28]. Their wearable end-effector consisted of a sagittal plane joint with built-in series elasticity and plantarflexion torque control. Kim et al. expanded on this work by developing and testing an end-effector with two degrees of freedom for plantarflexion and inversion-eversion torque control [29]. Both devices used passive leaf springs in parallel with the prosthetic pylon to produce dorsiflexion moments in early stance, meaning that the off-board actuator cannot affect the net joint torque until the toe digits make contact with the ground. Chiu et al. developed an emulator with three off-board actuators for two torque-controlled forefoot digits and one torquecontrolled heel digit [30]. All three systems can produce high torques and joint powers, have lightweight end-effectors, and achieve low torque tracking errors during walking. While they are impressive engineering achievements, each of these end-effectors include design choices that depart from how prostheses are used in daily life -including a lack of footwear or adjustable-length footplates. The exploration of prosthesis end-effectors that more closely resemble real-world prosthesis use may lead to more ecologically valid findings.
This paper presents the design and preliminary evaluation of a novel robotic ankle-foot prosthesis with off-board actuation and control. The prosthesis, the COBRA Ankle, attaches to clinically-standard carbon fiber prosthetic footplates and fits within a cosmesis and shoe during walking. Our prosthesis design also features a novel mechanical joint architecture that includes a custom parallel torsion spring, enabling bidirectional control of sagittal plane ankle torque using a single off-board motor and Bowden cable tether. The incorporation of footwear, a prosthetic footplate, and bidirectional torque control using a single off-board actuator distinguishes our design from existing ankle-foot prosthesis end-effectors. The main contributions of this work include a detailed description of the mechatronic design and control of the prosthesis, as well as preliminary data from benchtop tests and walking experiments with two transtibial amputees. We conducted benchtop tests to assess the electromechanical performance of the prosthesis and evaluated its effectiveness during walking tests with two transtibial amputees. The COBRA Ankle and off-board actuation system provide a valuable platform for future research and development in the field of prosthesis design and control, enabling more targeted investigations into biomechanics, human-robot interaction, and overall impact on mobility and well-being for people with limb loss.

II. MECHATRONIC SYSTEM DESIGN
We designed and built a custom ankle-foot prosthesis and off-board actuation and control unit for use in walking experiments with transtibial amputees. Our intent for this work was to 1) develop a universal off-board actuation and control unit capable of providing robotic assistance to a variety of future wearable devices, and 2) develop a lightweight prosthesis end-effector capable of producing high ankle torques and power. The off-board actuation and control unit consists of a motor, controller, and experimenter interface that rest on a cart (Fig. 1). The prosthesis generates plantarflexion moments via a Bowden cable actuation tether, and dorsiflexion moments with a pair of torsion springs in parallel with the Bowden cable. These dorsiflexion moments can be further modulated by the applied torque from the off-board actuation unit. Sensor measurements from the prosthesis and instrumented treadmill are measured by the controller and are used for closed-loop torque control. This section describes the details of the design of both the Configurable Off-Board Robotic Actuator (COBRA) and the prosthesis end-effector (COBRA Ankle).

A. Off-Board Actuation and Control Hardware
The primary actuator for the COBRA system is a brushless servomotor (AKM74L, Kollmorgen, Radford, VA). The actuator is powered by a dedicated three-phase 208 VAC circuit that rests on the bottom of the cart. The actuator is commutated by a servo drive (AKD-P02406) that can operate in closed-loop position, velocity, or current control mode. The servo drive can measure and report motor position, velocity, and q-axis current. The torque and power output of the system are limited by the servomotor, which is rated for a continuous torque of 49.7 Nm, a peak torque of 143 Nm, and a continuous power output of 5.47 kW. The servomotor was selected using an optimization procedure and dynamical simulations of human walking with robotic assistance at the knee and ankle [31]. Our algorithm iterated over motors from the Kollmorgen catalog and varied mechanical design parameters while minimizing reflected inertia at the prosthesis. Low reflected inertia is beneficial for human-robot interaction and is usually lowest in electromechanical systems that use high torque motors and low gear ratios, as reflected inertia increases with the square of the gear ratio. Our system does not use an explicit gearbox, but some degree of mechanical advantage is achieved through differential Bowden cable leverage at the actuator and end-effector.
We designed and built a custom transmission that delivers motor power and torque to a Bowden cable actuation tether (Fig. 2). The frame of the transmission was machined from 6061 aluminum (Sound Machine Products, Kent, WA). The stator of the motor bolts directly to the frame. The motor's output shaft is coupled to a carbon steel drive shaft through a flexible coupler. The drive shaft is supported by a pair of ball bearings and was sized to resist large radial loads from the Bowden cable. The Bowden cable consists of a hollow sheath made of helically coiled steel (Lexco Cable, Norridge, IL) and a sliding inner synthetic rope cable (West Marine, Fort Lauderdale, FL). The outer sheath clamps to the transmission frame, while the inner rope wraps around a 2.51 cm radius pulley fixed to the drive shaft. Given the peak motor torque and pulley radius, peak cable forces at the motor frame could reach 5.7 kN, but peak forces at the end-effector are reduced due to frictional losses between the outer sheath and inner rope.
The measurement and control unit for the off-board system is a PXIe-8880 (NI, Austin, TX). A desktop PC functions as an experimenter interface and communicates with the PXIe-8880 (controller) through a local area network. Both the desktop PC and controller run custom LabVIEW (NI, Austin, TX)  2018 programs. During experiments, the controller streams lossy data from the prosthesis and torque control system to the desktop PC for data visualization. Lossless data is logged to a dedicated hard drive on the controller. High-level prosthesis behavior is governed by parameters that are periodically sent from the PC to the controller, while real-time control algorithms are implemented on the controller. The controller communicates with the servo drive through an ethernet cable and etherCAT protocol. The controller has multiple configurable high-speed data acquisition cards for various analog and digital wearable sensors. Both the PC and controller are mounted on a server rack on the bottom of the cart.

B. Ankle-foot Prosthesis Design and Characterization
Our design intent for the COBRA Ankle was to develop a prosthesis capable of replicating the sagittal plane biomechanics of the biological ankle for a 115 kg person walking at roughly 1.25 m/s. We used biomechanical ankle trajectories from [32] to set design criteria (Table I) and determine mechanical design parameters for the prosthesis. We aimed to minimize prosthesis mass where possible.
The COBRA Ankle consists of a custom rotary joint that bolts to a carbon fiber prosthetic footplate (LP Vari-flex, Össur, Reykjavík, Iceland). The ankle joint is a two-part structure that pivots about a carbon steel shaft on two flanged ball bearings. The joint housing structure was machined from high-strength 7075 aluminum alloy. The Bowden cable outer sheath attaches to a lever arm on the top side of the joint, and the inner rope attaches to a lever arm on the bottom side of the joint. When the servomotor rotates, it pulls the inner rope through the outer sheath, which pulls the lever arms towards each other and generates a plantarflexion moment about the joint. Equal and opposite forces in the outer sheath of the Bowden cable oppose the forces from the inner rope, so no net force is applied to the prosthesis, only a joint moment [28]. Dorsiflexion moments are provided by a pair of custom parallel torsion springs within the ankle joint. A clinically-standard pyramid adapter bolts to the top of the joint for attachment to the user's prosthetic pylon. Fig. 3 shows the design of the ankle-foot prosthesis.
The LP Vari-flex footplate was selected due to its low stack height and split toe design, which provides compliance between the user and the ground in the frontal plane. Our custom joint attaches to multiple different footplate sizes, from as low as 23 cm to 30 cm in length. During walking, the prosthesis fits inside of a cosmetic foot shell and shoe, which provide additional compliance in the frontal and transverse planes. Compliance in the frontal and transverse planes can improve balance and shear loading at the interface between the prosthetic socket and residual limb during walking [33], [34].
Mechanical hard stops limit the sagittal plane range of motion between 16 degrees of plantarflexion and 30 degrees of dorsiflexion. The stack height, as measured from the bottom of the prosthetic footplate to the top of the pyramid adapter, is 13.5 cm. With a size 27 cm footplate, the total ankle-foot prosthesis mass is 1.19 kg, 18% of which is due to the prosthetic footplate. The cosmesis and shoe add an additional 0.47 kg, for a total worn mass of 1.67 kg. Worn mass varies with footplate and shoe size, and the prosthesis can be used without the shoe and cosmesis if a lower mass is desired.
The purpose of the parallel torsion springs is to allow for the prosthesis to achieve both net plantarflexion and dorsiflexion moments with a single servomotor. The Bowden cable can only apply plantarflexion moments, while the parallel springs bias the ankle consistently into dorsiflexion. With no cable force, the springs drive the ankle into the dorsiflexion hard stop. The springs and Bowden cable are in an agonist-antagonist configuration, so the net ankle torque is the sum of the torques provided by both elements. Therefore, the servomotor can control both plantarflexor and dorsiflexor moments by modulating cable tension across the gait cycle. We chose the torsion spring stiffness and rest angle based on ankle biomechanics data, simple estimates of actuator work across a stride, and manufacturing considerations, with a goal of achieving suitably high dorsiflexion moments while minimizing actuator work. Detailed spring design and manufacturing were carried out by Precision Coil Spring in El Monte, CA.
There are three sources of significant series elasticity between the servomotor and the load. The first source of series elasticity is the Bowden cable [34], the second is the prosthetic footplate, which is designed to flex significantly during walking, and the third is the soft-tissue interface between the residual limb and prosthetic socket. The amount of series elasticity provided by the prosthetic footplate is variable across the stride and depends on the anterior-posterior distance between the ground reaction force vector and the ankle joint center. As there are mechanical elements that provide joint torques to the prosthesis both in series and parallel with the servomotor, the actuator is a series-parallel elastic actuator [35]. The equation describing the net torque applied to the ankle is: where τ a is the net applied prosthesis torque, F c is the applied Bowden cable force, r is the lever arm distance between the Bowden cable line of action and ankle joint center, τ s is the torque provided by the torsion springs, and θ a is ankle angle. Both spring torque and lever arm vary with ankle angle.
To control net ankle torque during operation, all terms in (1) must be measured or estimated in real-time. Force in the inner rope of the Bowden cable is measured with an inline load cell (FSH03887, Futek, Irvine, CA). Ankle angle is measured with a magnetic encoder (RM08, Renishaw, West Dundee, IL) and magnet recessed within the ankle joint shaft. We measured the line of action between the Bowden cable and ankle joint center, i.e., lever arm, at 11 equally-spaced points across the range of motion within the COBRA Ankle CAD model. We fit a 4th order polynomial to the data that maps ankle angle as measured from the dorsiflexion hard stop to the instantaneous lever arm of the prosthesis (Fig. 4).
We estimated the torques provided by the torsion springs through simple experiments. The prosthesis was mounted sideways in a rigid frame so that the footplate mass did not cause a gravitational torque about the joint. The prosthetic footplate was not in contact with anything, so the only torques about the joint were those applied by the Bowden cable and the torsion springs. During the experiment, the motor pulled the ankle slowly through the range of motion. The equation of motion for the joint during the experiment was: where I is the moment of inertia of the lower section of the ankle joint and footplate about the rotation axis. We assumed negligible inertial torques due to the low acceleration and mass of the footplate, reducing (2) to: We used (3), measurements of cable force and ankle angle, and our lever arm model to compute spring torque across the range of motion. A second-order polynomial was fit to the angletorque profile for use in the real-time control system (Fig. 4). The resulting polynomial model had an R 2 of 0.94. When the model was used to predict the measured spring torque trajectories, there was a root-mean-squared prediction error of 3.06 Nm ± 0.07 Nm.

III. CONTROL SYSTEM DESIGN
We developed a hierarchical torque control system that prescribes and tracks ankle torque signals at 1000 Hz. All measurements used in the control system are low-pass Butterworth filtered at 100 Hz. A real-time gait cycle segmentation algorithm uses signals from the instrumented treadmill to measure foot-ground contact and provide a gait phase estimate used in the tracking controller. The prosthesis can operate in two high-level modes -passive emulation mode and active push-off mode. In passive emulation mode, the COBRA Ankle emulates an idealized passive ankle-foot prosthesis. In active push-off mode, additional torque is added in late stance to provide push-off work. Both modes were developed to evaluate the mechatronic performance of the prosthesis and were not necessarily intended to be optimal for walking. Torque setpoint signals are sent to a closed-loop tracking controller that uses classical feedback control and iteratively learned feedforward control (Fig. 5). Participant safety is ensured using softwarelevel safety limits, an emergency stop button that cuts motor power, and a fall harness.

A. Real-Time Gait Cycle Segmentation
Heel-strikes are identified in real-time with a threshold on the rate of the ground reaction force. Vertical ground reaction force, F z , is measured with a Bertec split-belt instrumented treadmill on the prosthesis side. The signal is then saturated at 200 N. The rate of force development for the saturated signal is computed using a finite-difference derivative. Prosthesisground contact is detected on the first loop iteration during swing phase where the vertical loading rate is higher than a preset threshold. This threshold is tuned for each participant. The ground reaction force signal is saturated at 200 N to prevent loading rate fluctuations in midstance from triggering an additional heel-strike. We found this method of detecting heel-strike to be slightly more responsive than a threshold on the ground reaction force magnitude in pilot testing. Toe-off is detected when the ground reaction force magnitude drops below 20 N.
The control system uses heel-strike information to estimate the instantaneous stride percentage, φ. To compute φ, a timer is started at each heel-strike. At each loop iteration, the current time is divided by the average stride duration over the previous five strides. φ is reset to zero on loop iterations where a heelstrike is detected. If φ reaches one, it is clamped there until the next heel-strike.

B. Ankle Torque Setpoint Generation
Both the passive and active torque setpoint generators use finite state machines to govern prosthesis behavior during walking (Fig. 6). In passive mode, the ankle emulates a virtual torsion spring with two linear stiffness regions that meet at a virtual rest angle. One virtual stiffness region generates plantarflexion moments and the other generates dorsiflexion moments, which approximates the differences in stiffness between the heel and forefoot of a passive ankle-foot prosthesis. The virtual rest angle is set near the pylon vertical angle and is tuned during experiments to prevent toe drag during Fig. 5. Torque control system for the COBRA Ankle.Torque setpoints, τ r , are generated using ankle angle, θ a , and ground reaction forces, F z . Torque setpoints are transformed into cable force setpoints, F r , using models of the ankle's lever arm and spring torque. A cascaded control architecture is used to track cable force setpoints, where an outer force tracking loop generates motor velocity commands, θ r , for an inner motor velocity loop. The force tracking loop uses feedback on measured cable force error with a damping term on measured motor velocity,θ m . The learning controller generates a feedforward trajectory as a function of gait phase, φ. (b) Finite state machine for ankle torque setpoint generation when the ankle is in passive mode. The active torque setpoint generator is similar to the passive setpoint generator, provides push-off in late stance when the requested torque exceeds a tunable threshold. When the threshold is exceeded, an additional constant offset torque is added to the passive stiffness, which is equivalent to an instantaneous change in the rest angle of the virtual spring. swing phase. The prosthesis produces torques proportional to the angular deviation from the virtual rest angle, with a stiffness constant that switches depending on whether the angle is plantarflexed or dorsiflexed relative to the virtual rest angle: In (4), τ r is the torque setpoint, K i is a stiffness constant that switches depending on ankle angle, θ a , and θ o is the virtual rest angle. K i switches to K pf when the ankle is dorsiflexed and to K df when the ankle is plantarflexed. K pf , K df , and θ o can be changed through the experimenter interface during walking experiments. Active push-off mode adds an additional state to the passive emulation mode in late stance. When the torque setpoint from the passive emulation mode exceeds a threshold, τ w , the controller switches to the push-off state: where w is a constant torque offset. The sudden addition of w is equivalent to instantaneously changing the rest angle of the virtual spring, which adds energy to the virtual spring and causes the prosthesis to push-off in late stance. The prosthesis stays in the push-off state until toe-off is detected. Both τ w and w are tunable parameters. Changing τ w influences pushoff timing and changing w influences the magnitude of the mechanical ankle work done in push-off. The torque setpoints in (5) are not intended to be followed precisely by the tracking controller. The setpoint changes instantaneously, but the true rate of torque development is gradual and dependent on the tracking controller behavior. Both controller modes limit their dorsiflexion torque setpoints based on the maximum amount of torque the parallel spring can provide at a given angle. If the dorsiflexion torque setpoint exceeds the torque that can be provided by the springs, the setpoint is clamped to the maximum spring torque (i.e., very low cable force). Both the passive and active mode control laws are variants of methods described in [17] and [36].

C. Torque Tracking Controller
The purpose of the tracking controller is to drive the ankle torque to the commands sent by the torque setpoint generators in real-time. The tracking controller first transforms the desired torque setpoint to a desired cable force setpoint using the polynomial prosthesis models developed in Section II. The tracking controller uses a cascaded architecture with an outer Bowden cable force control loop and an inner motor velocity control loop. The cable force control loop is implemented on the NI PXIe-8880 and uses a feedback controller with an iteratively learned feedforward term that corrects for tracking errors that happen at repeatable points in the gait cycle. The motor velocity controller is implemented on the Kollmorgen AKD-P02406 and uses a proportional-integral controller with feedforward friction compensation. Controller gains for the motor velocity loop were set using an auto-tuning procedure when the Bowden cable was detached from the motor and transmission. The controller sampling frequency is 1,000 Hz for the outer force control loop and 1.5 MHz for the inner velocity control loop.
Torque setpoint signals are transformed into cable force setpoints using measurements from the ankle encoder and the polynomial models that describe spring torque and lever arm. Eq. (1) is solved for the cable force required to achieve the desired ankle torque: where F r is the reference cable force that is required to drive the prosthesis to the desired net ankle torque. Requested cable force is saturated within the control system to be between 5 N and 2224 N. The lower limit is to prevent cable slack, and the upper limit is the capacity of the load cell that measures cable tension. The force tracking loop uses parallel feedback and feedforward control algorithms to send a command to the inner motor velocity loop. The feedback controller uses proportional control on the cable force error and a damping term on the motor velocity measurement. The feedforward controller is an iterative learning controller (ILC) that leverages the cyclic nature of walking to reduce systematic tracking errors at consistent points in the gait cycle. The command sent to the motor velocity controller at every loop iteration is: In (7), K p is and K d are positive scalar feedback gains, e F is the difference between desired and measured cable force, andθ m is the motor velocity measurement.θ L is the iteratively learned feedforward trajectory and is a function of gait phase, φ, and the current stride number, n.θ L is initialized with an empty trajectory and is updated with the following equation at every heel-strike: In (8), K L is a learning gain that scales the cable force error trajectory from the previous stride, β is a forgetting gain (0.95 ≤ β ≤ 0.99) that prevents instability, and D is a phase delay term. In effect, the feedforward motor velocity trajectory for a given stride is a weighted average of the cable force tracking error and the feedforward trajectory from the previous stride. The delay, D, shifts the error trajectory from the previous stride in phase so that the learned motor velocity signal preempts systematic errors. ILC is a common solution for challenges in force control of Bowden cables due to their highly nonlinear friction characteristics [37], [38], and our algorithm is an independent implementation of the "PD-LRN" controller described in [37]. The ILC can be activated and deactivated through the experimenter interface, so the system can operate with feedback control only when desired. When the prosthesis is in active push-off mode, the ILC is 'clamped' to the stiffnessbased feedforward trajectory, because the sudden discontinuity in ankle torque setpoint at push-off caused high-frequency ripples to form in the learned signal.

IV. EXPERIMENTAL METHODS
We conducted experiments with the COBRA Ankle that evaluated 1) the electromechanical performance of the system in isolation and 2) the biomechatronic performance of the prosthesis during walking with two transtibial amputee participants. We evaluated the peak torque and controller bandwidth in benchtop experiments. During walking experiments, we evaluated the tracking controller performance and the ability of the prosthesis to produce push-off power and work during walking.

A. Benchtop Experiments
We built a rigid test frame for benchtop experiments with the prosthesis. The top of the prosthesis joint was bolted to the upper section of the test frame and the footplate pressed into a reaction bar bolted to the bottom section of the frame. The prosthesis ankle joint was not explicitly locked during the benchtop tests and could pivot slightly when the carbon fiber footplate deflected. All tests were conducted with a 27 cm footplate inside of a cosmetic foot shell and shoe. The reaction bar was placed at the midpoint between the ankle joint center and the toe of the footplate.
Torque step tests were conducted to evaluate the maximum plantarflexion torque for the prosthesis.
Step input tests were conducted with peak torques at 75 Nm, 125 Nm, and 175 N, with a 25 Nm preload. The highest torque value was limited by the capacity of the prosthesis load cell. For each test, the system was held at the preload for one second, commanded to the intermediate torque value for one second, and then returned to the preload for one second. Three tests were conducted for each magnitude.
We conducted bandwidth tests to evaluate the responsiveness of the torque controller with sinusoidal torque setpoint signals. The torque setpoint signals had a constant offset of 100 Nm and oscillated between 50 Nm and 150 Nm at fixed frequencies. We tested eight discrete oscillation frequencies ranging from 0.4 Hz to 6 Hz. Experiments were stopped at 6 Hz due to excessive vibrations at the off-board motor and cart when higher frequencies were tested.

B. Walking Experiments
We recruited two participants with unilateral transtibial limb loss to walk on the COBRA Ankle for an IRB approved research study at the Center for Limb Loss and Mobility (CLiMB) within the Veterans Affairs Puget Sound Health Care System in Seattle, WA. The purpose of the experiment was to evaluate the electromechanical performance of the prosthesis during walking and was not to measure or improve the participants' walking patterns in response to the prosthesis.
Before data collections, the prosthesis was configured to match the side of the participant's amputation and was fit with a footplate that matched the size of their prescribed prosthesis. Upon arrival, the participants were weighed, and their leg lengths were measured as the distance from the head of the greater trochanter to the floor on the prosthetic-side limb. A certified prosthetist then fit and aligned the experimental prosthesis to the participant. For all walking experiments, both participants walked at a non-dimensionalized Froude walking speed of Fr = 0.16, where Fr = v 2 /gl, v is walking speed, g is the gravitational constant, and l is leg length. We elected to have participants walk at this Froude speed because it is near self-selected walking speeds for people without limb loss, and it allows for comparisons to other studies of powered prosthetic ankles [22], [32]. Table II shows the participant  demographics and walking speeds. For experiments with passive emulation mode, we collected data at low, medium, and high virtual forefoot stiffness values (K pf ), corresponding to values of 0.093, 0.116, and 0.139 Nm/deg-kg. The virtual stiffness values are near the quasistiffness of the biological ankle during walking and were based on [36]. For all walking trials, the virtual heel stiffness (K df ) was held at 0.038 Nm/deg-kg. The virtual rest angle of the prosthesis, θ o , was set so that the prosthesis was comfortable during standing and the toe had significant ground clearance during swing phase. We collected data in an additional condition with the medium forefoot stiffness and the iterative learning controller disabled to evaluate the effect of the ILC on the tracking error. Table III shows the settings of the prosthesis across passive emulation mode conditions.
Initially, both participants were given between five and ten minutes to adapt to the experimental prosthesis on the medium stiffness condition. The participants then walked on the low, medium, and high stiffness settings for several minutes each. Two ten-second trials were logged at each condition.
During experiments with active push-off mode, the participants walked with the medium virtual forefoot stiffness. The push-off torque threshold, τ w , was initially set so that pushoff would be triggered slightly before the peak ankle torque in passive emulation mode, and then conversationally tuned with feedback from the participant. The additional push-off torque, w, was initially zero and increased over several minutes, until the experimenter observed peak ankle power values near values reported in the literature. Ten-second trials were logged at several increasing values of w.

C. Data Analysis
For the step response benchtop experiments, we averaged the measured torque trajectories at each magnitude and computed the 90% rise and fall times and percent overshoot. For the frequency response experiments, we computed the gain and phase lag of the closed-loop control system at each frequency and fit a transfer function model to the frequency response data for future analyses of the controller performance. We defined the closed-loop control bandwidth as the frequency where the magnitude response crossed −3 dB.
For the walking experiments, we evaluated the torque tracking performance of the prosthesis when it was in passive emulation mode, and the average peak sagittal plane ankle torque and power when the prosthesis was in active push-off mode. All results were computed using the instrumented treadmill and sensors within the COBRA Ankle. Ankle angles and moments were filtered with a fourth-order zerolag Butterworth filter at 12 Hz and 20 Hz, respectively. We computed the ankle angular velocity using finite-difference derivatives, and computed ankle power by multiplying ankle angular velocity with ankle torque. We segmented each stride using an offline implementation of the real-time heel-strike algorithm described in Section III-A. Segmented strides were then normalized in time between 0% and 100% of the gait cycle using linear interpolation. Stride-averaged signals were computed for all signals across each condition.
In passive emulation mode, we evaluated both total and systematic RMS tracking errors during stance phase. Total RMS error computes the error between the torque setpoint and measurement across the concatenation of all trials for a given condition prior to segmentation or time normalization. Systematic RMS error was computed using the torque setpoint and measurement for the average stride for a condition [29]. We did not evaluate torque tracking performance in active push-off mode, because tracking error is not a meaningful metric for our push-off controller. A large, instantaneous torque tracking error is imposed by our setpoint generator, and the response of the tracking controller response is what causes push-off.

A. Benchtop Results
The peak ankle plantarflexion torque achieved in the step response tests was 178.3 Nm (Fig. 7). For the 175 Nm test, the 90% rise and fall times were 146 ms and 139 ms, respectively. The percent overshoot was 5.3%. A time delay in torque development of approximately 48 ms was noted for all three step test conditions, which was not present in the motor current or velocity signals. The highest frequency successfully evaluated in the bandwidth tests was 6 Hz, which had a magnitude response of −2.83 dB and phase delay of −249.3 degrees. The frequency response was well characterized by a second-order transfer function with a time delay: 1.001 0.001s 2 + 0.024s + 1.000 e −0.048s The time delay of 48 ms observed during the step responses was fixed during model fitting. The −3 dB crossing of the Step responses collected with the foot mounted in a test stand. Right: Frequency response collected with the foot mounted in a test stand. Discrete sine waves of torque setpoint were sent to the controller with an offset of 100 Nm and peak-to-peak amplitude of 100 Nm. The inset shows torque tracking data from one representative test at 2 Hz. A second-order transfer function model with a time delay was fit to all experimental gain and phase data.

B. Walking Results
During the walking tests, both participants walked comfortably on the COBRA Ankle (Fig. 8). For both participants, the total RMS error across all passive emulation conditions was 4.12 Nm, while the RMS torque tracking error for an average stride (systematic error) was 2.71 Nm, both of which are less than 5% of the peak torques during the walking experiments. Table IV shows tracking errors across all stiffness conditions for both participants. During tests of the medium stiffness condition where the ILC was disabled, the total and systematic tracking RMS errors were 8.49 Nm and 6.14 Nm, respectively, indicating that the ILC reduced tracking errors by roughly 50%. Fig. 9 shows ankle biomechanics for the three stiffness conditions.
During active push-off trials, the peak ankle powers achieved by the COBRA Ankle were 351.6 W ± 33.88 W and 368.03 W ± 12.67 W for participants one and two, respectively. Body mass normalized, these values are 3.08 W/kg ± 0.30 W/kg and 3.51 W/kg ± 0.12 W/kg. As a point of comparison, Russell Esposito et al. found peak ankle powers of 3.15 W/kg ± 0.73 W/kg for the BiOM and 2.29 W/kg ± 0.42 W/kg for intact ankles when walking at the same Froude speed [22]. Fig. 10 shows how ankle power in active push-off mode increased with the parameter w. Fig. 9. COBRA Ankle biomechanics across three stiffness conditions. The top row shows data from Participant I and the bottom from Participant II. Peak ankle torques remained relatively constant across conditions, while peak ankle dorsiflexion angles decreased as stiffness increased. Peak ankle power was lowest for the stiffest condition. Fig. 10. Ankle power during active push-off mode tests. As the push-off torque parameter w from (5) was increased, peak ankle power increased.

VI. DISCUSSION
We designed, built, and evaluated a robotic ankle-foot prosthesis and off-board actuation and control unit for walking experiments with transtibial amputees. The COBRA Ankle uses a novel mechanical layout that pairs an off-board motor and Bowden cable tether with a parallel spring to provide bi-directional torque control during walking. We performed benchtop tests that showed that the prosthesis can provide high plantarflexion torques and that the torque feedback controller is sufficiently responsive for walking. Walking experiments indicated that the system achieves low tracking error and can provide peak ankle push-off power greater than that of the biological ankle and similar to other robotic prostheses. Our experiments indicate that COBRA and the COBRA Ankle are highly performant experimental platforms that will enable a wide variety of future research in prosthesis design and control.
One of the biggest differences between the COBRA Ankle and other research prostheses with off-board actuation and control is that it can estimate and control the torque about the ankle joint throughout the entire gait cycle, from heel-strike to heel-strike. Other emulator designs provide dorsiflexion moments via passive elements in parallel with the prosthetic pylon [28], [29]. These dorsiflexion moments are not measured or estimated in real-time, so total control of the net ankle torque can only be achieved after the passive heel element lifts off the ground in mid-to-late stance. The key difference for our design is that our dorsiflexion spring is embedded within the mechanical joint and its deflection is measured with an encoder. These capabilities allow for different types of experiments to be carried out with the COBRA ankle, e.g., experiments that explore the effects of systematic variation of dorsiflexion moments in early stance.
Inclusion of the parallel torsion spring allowed for active modulation of dorsiflexion torque but required several design trade-offs. In terms of benefit, torsion springs are relatively easy to design and are readily available by a variety of manufacturers. Our design is relatively compact and simple compared to prosthesis designs that achieve parallel elasticity using compression springs [39], [40]. Our parallel spring is biased into dorsiflexion, such that the rest angle of the spring is outside of the range of motion of the ankle. In this configuration, the Bowden cable must overcome the spring torque before it can develop a net ankle plantarflexion torque. This configuration is energetically inefficient and would require more consideration for a battery-powered prosthesis, as energy is expended biasing the spring. However, the spring torque and energy storage are small compared to the capabilities of the off-board motor and power supply, so the trade-off is reasonable. Torsion springs can exhibit more hysteresis than leaf springs or compression springs due to inter-coil friction during loading and unloading. Hysteresis, as see in Fig. 4, makes it challenging to model the spring torque as a function of ankle angle and affects our ability to estimate total joint torque. Issues with our spring could be improved by winding the torsion spring with more spacing between adjacent coils to avoid friction losses or redesigning the joint to use a composite leaf spring instead. Alternatively, a more advanced real-time model could be developed that more accurately characterizes the dynamic torque output of the spring.
Inclusion of the carbon fiber prosthetic footplate also involved design trade-offs. The compliance in the footplate protects our mechanical joint from shock loading, e.g., at heel-strike. The engineering design process was also made substantially simpler by incorporating the widely available commercial part that is well-suited for human walking. We are also able to swap footplates to match the participant's shoe size, which allows us to match foot length between the amputated and intact sides. Foot length is known to affect the mechanics and energetics of walking [41]. The split-toe design of the LP Variflex also provides beneficial compliance in the frontal plane.
The footplate deflects due to loading during walking, which is not measured. Our encoder measures the angular displacement of the mechanical ankle joint, whereas many of ankle angle measures from the biomechanics literature are derived from ankle-foot models that combine angular displacements from the talocrural, subtalar, and forefoot joints [42]. Therefore, comparing the presented kinematics and power of the COBRA Ankle to values from the literature must be done with caution. For example, depending on whether the footplate is absorbing/returning energy during late stance, the total ankle-foot power would be lower/higher than the ankle power measured using onboard sensors. This phenomenon is minimized in emulator systems with very stiff forefoot designs, such as [28], [29], [30].
The shoe and cosmetic foot shell provided significant transverse plane compliance to our prosthesis. Shoes are known to change the stiffness and other relevant mechanical properties of prosthetic feet during walking [43], though this effect has not been quantified in the transverse plane to our knowledge. An appropriate amount of transverse compliance is important during walking, as can reduce painful shear forces between the residual limb and prosthetic socket [33]. Prosthetic foot emulators with low transverse plane stiffness could cause amputees to walk differently than they would in other scenarios to avoid uncomfortable loads on the residual limb. Emulators that use shoes may therefore have more external scientific validity.
Our torque controller bandwidth of 6.1 Hz is low relative to other ankle-foot prostheses with off-board actuation, with one system reporting bandwidths up to 33 Hz [30]. This is partly due to the experimental setup for our benchtop tests. It is common to lock the ankle joint during bandwidth tests [17], [25], [29], [30], but this method would likely overestimate our controller's ability to develop torque during walking because it does not account for series elasticity between the ankle, ground, and user. Both the footplate and the interface between the socket and residual limb contribute to additional series elasticity. Even with a relatively compliant test setup, our bandwidth is similar to other robotic ankles [15], [17] and is clearly high enough to systematically alter prosthesis behavior during walking (Fig. 9). Given our controller bandwidth, the COBRA Ankle currently cannot haptically emulate other devices with high fidelity. The controller is not fast enough to emulate passive devices when large external disturbances occur, e.g., at heel-strike. Future work to improve controller bandwidth will include 1) improving the interface between the motor and the cart to reduce vibration and 2) improving the signal-to-noise ratio in our ankle encoder measurements, which will allow us to use more aggressive feedback gains. However, not all research applications require high-fidelity emulation, and our systematic torque tracking errors were similar to other published work [28], [29], [30] due to the iterative learning controller.
It is important to note that our walking experiments were conducted with a limited sample size of two participants with limb loss. While these preliminary results are promising, a larger number of subjects would have provided a more comprehensive understanding of the system's performance and applicability. Moving forward, we plan to expand our research to include a more diverse participant pool.

VII. CONCLUSION
We designed and built the COBRA Ankle and off-board actuation and control system, offering bidirectional torque control for wearable robotic devices. Through successful benchtop tests and walking experiments, our system demonstrates its effectiveness as a powerful experimental platform for prosthesis research. By utilizing the COBRA Ankle in future studies, we aim to advance wearable robotic device design and control, ultimately enhancing mobility and quality of life for individuals with limb loss and other clinical populations.