Improving Sit/Stand Loading Symmetry and Timing Through Unified Variable Impedance Control of a Powered Knee-Ankle Prosthesis

Cara Gonzalez Welker; T Kevin Best; Robert D Gregg

doi:10.1109/TNSRE.2023.3320692

. Author manuscript; available in PMC: 2023 Dec 18.

Published in final edited form as: IEEE Trans Neural Syst Rehabil Eng. 2023 Oct 26;31:4146–4155. doi: 10.1109/TNSRE.2023.3320692

Improving Sit/Stand Loading Symmetry and Timing Through Unified Variable Impedance Control of a Powered Knee-Ankle Prosthesis

Cara Gonzalez Welker ¹, T Kevin Best ², Robert D Gregg ²

PMCID: PMC10726997 NIHMSID: NIHMS1938500 PMID: 37773917

Abstract

Individuals using passive prostheses typically rely heavily on their biological limb to complete sitting and standing tasks, leading to slower completion times and increased rates of osteoarthritis and lower back pain. Powered prostheses can address these challenges, but have control methods that divide sit-stand transitions into discrete phases, limiting user synchronization across the motion and requiring long manual tuning times. This paper extends our preliminary work using a thigh-based phase variable to parameterize optimized data-driven impedance parameter trajectories for sitting, standing, and walking, with only two classification modes. We decouple the stand-to-sit and sit-to-stand equilibrium angles through a knee velocity-dependent scaling term, reducing the model fitting error by approximately half compared to our previous results. We then experimentally validate the controller with three individuals with above-knee amputation performing sitting and standing transitions to/from three different chair heights. We show that our controller implemented on a powered knee-ankle prosthesis produced biomimetic joint mechanics, resulting in significantly reduced sit/stand loading asymmetry and time to complete a 5x sit-to-stand task compared to participants’ passive prostheses. Integration with a previously developed walking controller also allowed sit/walk transitions between different chair heights. The controller’s biomimetic assistance may reduce the overreliance on the biological limb caused by inadequate passive prostheses, helping improve mobility for people with above-knee amputations.

I. Introduction

The incidence of lower-limb amputation is increasing due to rising rates of vascular disease [1]. Because conventional passive and semi-active prostheses cannot supply net positive energy like biological joints, users compensate with their intact limb, leading to kinematic and kinetic asymmetries during walking [2], [3] and transitions between sitting and standing [4]–[6]. These compensations can cause secondary complications such as osteoarthritis and lower back pain [7].

Powered prostheses have the ability to supply net positive energy, and thus can reduce secondary complications by producing more normative mechanics [8]. However, the design of effective prosthetic control strategies, particularly for non-rhythmic tasks, remains a challenge. Almost half of movement bouts last less than 12 steps each [9], and a healthy adult transitions from sit to stand more than 60 times each day on average [10]. Therefore, in order to develop clinically viable powered prostheses, it is necessary to develop control strategies for transitional activities such as sitting and standing.

Although relatively sparse in the literature, there are examples of prior work investigating control strategies that allow a powered above-knee prosthesis user to transition between sitting and standing [11]–[17]. One commonly used approach is impedance control [11], [12], [15], which dictates the joint torque as a function of the joint’s angular position $θ$ and velocity $\dot{θ}$ , parameterized by a stiffness $K$ , damping $B$ , and equilibrium angle $θ_{eq}$ :

τ = K (θ_{eq} - θ) - B \dot{θ} .

(1)

In previous work, piecewise-constant values of $K$ , $θ_{eq}$ , and $B$ are individually tuned or configured throughout separate sitto-stand and stand-to-sit controllers [11], [12]. This manual tuning control approach has been successful in reducing asymmetry at peak vertical acceleration during sit-to-stand and stand-to-sit compared to a passive prosthesis [12].

Other prosthesis controllers for sit/stand do not utilize impedance control but still require manual tuning. For example, [17] reduced loading asymmetry in above-knee amputees with a powered prosthesis controller heuristically designed to approximate able-bodied knee kinetics, but the peak magnitude required tuning and was lower than able-bodied kinetics for every participant [17]. In addition, [16] investigated increasing knee torque magnitude during sit-to-stand for a single chair height, finding that higher magnitudes reduced completion time and asymmetry in the middle of the motion but increased asymmetry at the end of the motion. Similarly, studies on prosthetic ankle control during walking [18] have found that identifying the appropriate timing and magnitude of assistive torque is important for success, and tuning the controller for each individual can improve performance [19]. While these studies agree that tuning can improve outcome metrics, the tuning process can be time-consuming, with one study reporting tuning times lasting on the order of five hours for multi-activity controllers that include sit/stand transitions and walking at different speeds and inclines [20].

In addition to the practical challenges of long tuning times, risk of misclassification increases with additional distinct controllers for different activity modes. This misclassification can cause unwanted prosthesis behavior ranging from mildly uncomfortable to highly dangerous and likely to result in a fall, depending on the type and timing of the misclassification [21]. In the combined sit, stand, and walk framework developed by Varol et al., they were able to correctly identify true transitions, but had a 7% false positive rate [22]. Hunt et al. removed the problem of classification entirely by using measured electromyography (EMG) from the intact biceps femoris as an input to a continuous controller that allows users to transition between different activities such as sit, stand, walk, and lunge with a single controller [13]. They demonstrated the efficacy of this approach to reduce muscle activation and improve loading symmetry during sit-to-stand in a single session. However, using electromyography as a real-time input signal has the limitations of requiring detailed calibration for a specific surface placement and losing signal quality over time due to challenges with maintaining contact with that given surface [23].

Aside from EMG control, another approach that allows for fewer distinct states and less switching is phase-based control. In this approach, the prosthetic joints are controlled by a phase variable that tracks the progression of the user’s motion. Because the phase variable is a biomechanical signal controlled by the user, this allows for some amount of volitional control not only for rhythmic tasks, but also for non-rhythmic start and stop motions [24] and gait perturbations [25]. Previous work has shown the thigh angle to be an appropriate phase variable to control lower-limb powered prostheses during gait [24], [26]–[30]. Thigh angle has also been proposed as a phase variable in a position controller for sit/stand [14], but this controller was not experimentally validated. The sit/stand controller in [12] used axial load as a form of a phase variable, but it is unclear if this would be an appropriate choice for walking.

In an effort to avoid the problems with discretely defined impedance parameters and to eliminate tuning, we previously developed a data-driven optimization framework to generate continuous, phase-based impedance trajectories for walking, which was validated on a powered knee-ankle prosthesis used by multiple participants [27], [28]. We recently extended this approach to sitting and standing transitions, fitting a unified polynomial impedance model to able-bodied kinematic and kinetic data [15]. A thigh-based phase variable parameterized the motion, allowing a single impedance model to control both sit-to-stand and stand-to-sit. A pilot study with a single amputee participant showed promise in reducing loading asymmetry and motion duration [15].

However, the preliminary sit-stand controller in [15] had a number of limitations that we address in the current manuscript. First, we previously used the same impedance parameter trajectories for both stand-to-sit and sit-to-stand motions, and similarly to [17], needed to reduce the torque magnitude below biomimetic values because our pilot participant felt that full stand-to-sit assistance made it difficult for him to sit down. To address this issue, we modify the structure of the impedance model to include a knee velocity-dependent equilibrium angle, allowing the model to produce more biomimetic torque profiles for both sit-to-stand and stand-to-sit without requiring explicit classification. Second, a researcher needed to manually configure the phase variable in [15] for each chair height to avoid phase saturation, which was similarly caused by an unmet assumption that the thigh angle was perfectly vertical while standing. To resolve these issues, we detect quiet sitting and standing and autonomously update the phase variable to accommodate different chair heights and standing postures. Finally, because we only tested with one participant with amputation using one chair height in [15], the number of conclusions that we could make was limited. This paper extends this validation to three participants with three different chair heights.

The specific contributions of the manuscript are as follows: 1) We develop a unified data-driven impedance control model for sit/stand with a knee velocity-dependent equilibrium angle, allowing distinctive behavior during sit-to-stand and stand-to-sit without explicit classification. 2) We implement autonomous phase variable detection for sit/stand to allow controller adaptation to different chair heights and standing postures. 3) We test this updated control strategy on three participants with above-knee amputation performing sit/stand with multiple chair heights, demonstrating greater similarity to able-bodied mechanics compared to previous work, as well as improved clinical outcome metrics such as loading symmetry and time to complete a given task. 4) We demonstrate proof-of-concept tests in which participants completed sequences of walking between chairs of different heights, demonstrating the controller’s efficacy in more realistic scenarios. To our knowledge, this is the first sit/stand controller validated to improve symmetry and timing at multiple chair heights without the use of EMG.

II. Sit/Stand/Walk Controller

In this article, we extend our hybrid kinematic/impedance controller for variable-task walking [28] to also enable sitting and standing behaviors. As in our preliminary work in [15], we generate a novel, data-driven impedance parameter model based on able-bodied data that produces biomimetic joint torque during both sitting and standing motions. To synchronize the model with the user’s motion, we develop an adaptive sit/stand phase variable that automatically adjusts to varying chair heights and standing postures. We review the high-level classifier originally presented in [15] that automatically switches between sit/stand and walk ambulation modes, enabling seamless sit/stand/walk sequences. For brevity, we omit the details of the walking controller in this article and instead refer to [28] for a thorough explanation.

A. High-Level Classification

As in our preliminary work [15], we use a high-level Finite State Machine (FSM) with two modes (Sit/Stand and Walk) to determine the user’s intent based on prosthesis sensor readings (Figure 1). By limiting the FSM to two control modes, we reduce complexity and the chances of incorrect transition decisions. Transition rules are developed based on pilot testing data using sensor measurements available from the powered knee-ankle prosthesis later used in experimental testing [31]. These signals include the global thigh angle $θ_{t}$ and angular velocity $ω_{t}$ obtained from the inertial measurement unit (IMU, 3DM-CX5–25, LORD Microstrain), the knee angle $θ_{k}$ measured from an optical joint encoder (E5, 3600 cpr, US Digital), and foot contact $F_{GC}$ determined by the load cell (M3564F, Sunrise Instruments). Further details on how these sensors are mounted in the powered prosthesis can be found in [31].

Fig. 1. — A diagram of the high-level control FSM that dictates transitions between walking and Sit/Stand modes, determined by the thigh angle $θ_{t}$ and angular velocity $ω_{t}$ , knee angle $θ_{k}$ , and foot ground contact ( $F_{GC}$ ). We use these signals to detect prosthetic-side heelstrike or late stance to transition to walking. We transition to standing when the user is stationary with prosthesis ground contact, a vertical thigh, and an extended knee.

We include two stand-to-walk transition criteria, as the user can initiate movement by leading with either the prosthetic or the biological leg (Figure 1). Thus, the FSM transitions from the Sit/Stand mode to the Walk mode if either prosthesis-side heel strike (HS) or late stance is detected. We define a HS transition to walking by a rising edge in the foot contact signal $F_{GC}$ while $10 < θ_{t} < 40 deg$ (hip flexion) and $ω_{t} < - 23 deg/sec$ (hip extension). The upper bound on thigh angle prevents erroneous transitions to walk mode while the user is seated. The threshold used for making F_GC was 75 N, while the threshold for breaking F_GC was 25 N. We define prosthesis-side late stance transition to walking if $θ_{t} < - 15 deg$ and $ω_{t} < - 23 deg/sec$ (hip extension).

We use only one set of criteria to detect a transition from Walk to Sit/Stand mode. Specifically, the FSM transitions when $| θ_{t} | < 10 deg$ , $| ω_{t} | < 11 deg/sec$ , and the foot is in contact with the ground, conditions indicative of upright stance (Figure 1). To prevent rapid state machine switching that could occur if the user paused during toe-off, we also do not allow a transition to the sit/stand mode if $θ_{k} > 15 deg$ . During each mode transition, we calculate each joint’s torque command as a convex weighted sum of the control torque given by both the Walk and Sit/Stand control modes. The weights linearly vary over 200 ms to produce a smooth mode transition with minimal jerk.

B. Sit/Stand Phase Variable

As in [15], we define a phase variable based on the user’s prosthetic global thigh angle $θ_{t}$ to parameterize the controller’s behavior during sit/stand motions. The global thigh angle is an appropriate choice for a phase variable, as it monotonically increases during sit-to-stand motions and monotonically decreases during stand-to-sit motions. During sit/stand, $s$ represents the user’s location between a sitting state ( $s = 0$ ) and a standing state ( $s = 1$ ). The phase variable can increase or decrease, making both sit-to-stand and stand-to-sit motions possible with one controller. Similar to previous work [15], [24], [27], [28], we define $s$ through an affine transformation of $θ_{t}$ :

s = (θ_{t}^{0} - θ_{t}) / (θ_{t}^{0} - θ_{t}^{f}),

(2)

where $θ_{t}^{0}$ is the value of $θ_{t}$ when the user sits in the chair and $θ_{t}^{f}$ is the value of $θ_{t}$ when the user stands comfortably. Typically, $θ_{t}^{f}$ is near 0 deg, but it can vary if a user prefers standing with one foot more anterior than the other.

The closed chain kinematics of sitting make $θ_{t}^{0}$ depend on both the user’s leg dimensions and the chair height. To allow for varying user geometries and chair heights, we calculate $θ_{t}^{0}$ in real-time with a moving average (0.5 second sliding window) while static sitting conditions are met, specifically $| ω_{t} | < 6 deg/s$ and $θ_{t} > 45 deg$ . Similarly, we calculate $θ_{t}^{f}$ with a moving average when static standing conditions are met: $| ω_{t} | < 6 deg/s$ , $θ_{t} < 15 deg$ , and $F_{GC} > 200 N$ .

C. Sit/Stand Impedance Model

1). Training Dataset Preparation:

We adapt a previously collected dataset of kinematics and kinetics from eight able-bodied participants transitioning from sit-to-stand and stand-to-sit 5 times each at a self-selected pace [32]. From these data, we use sagittal plane kinematics and kinetics for the knee and ankle, and calculate $θ_{t}$ in the sagittal plane using the hip and torso kinematic data. We normalize joint torques by subject mass and differentiate and filter joint angles with a fourth order Butterworth low-pass filter to obtain angular velocities. As 5–6 Hz is frequently used as a cutoff frequency during walking [33], we choose a cutoff frequency of 2 Hz due to the slower frequency of sit and stand transitions.

Next, we segment these data to determine the start and end of the sit-to-stand and stand-to-sit motions. We define the start and end of the motion as the points when $| ω_{t} |$ crosses a threshold of 5 deg/sec before and after each motion, respectively. In addition, as some subjects consistently reported a knee flexion angle much larger than zero (up to 10 degrees) during standing, we subtract out each subject’s average minimum knee angle over all five trials, as we suspect that this standing knee flexion is an artifact of motion capture rather than a true behavior. Finally, we calculate the phase variable trajectory for each trial using (2). The resulting dataset contains 80 trials of knee and ankle kinematic, kinetic, and phase trajectories, half sit-to-stand motions and half stand-to-sit motions.

2). Model Architecture:

Similar to [15], [28], we model the impedance parameters $K$ , $B$ , and $θ_{eq}$ as continuous fourth order polynomials in phase $s$ :

[\begin{matrix} K (s) \\ B (s) \\ θ_{eq} (s, \dot{θ}) \end{matrix}] = [\begin{matrix} k^{⊤} \\ b^{⊤} \\ e {(\dot{θ})}^{⊤} \end{matrix}] [\begin{matrix} s^{0} \\ ⋮ \\ s^{4} \end{matrix}],

(3)

where $k \in ℝ^{4 \times 1}$ and $b \in ℝ^{4 \times 1}$ are constant coefficient vectors defining the stiffness and damping polynomials, respectively. New in this work, we define the equilibrium angle parameter vector $e \in ℝ^{4 \times 1}$ as a function of the joint angular velocity $\dot{θ}$ . While our prior work [15] treated the equilibrium angle parameters as constants, experimental testing showed that the model provided excessive knee extension torque at the beginning of the stand-to-sit motion. This resulted in the participant struggling to initiate the sitting motion. To remedy this issue, we add a dependency on joint velocity to decouple the equilibrium angle trajectories for sit-to-stand and stand-to-sit motions:

e (\dot{θ}) = e_{1} + f_{j} (\dot{θ}) e_{2},

(4)

where $e_{1} \in ℝ^{4 \times 1}$ and $e_{2} \in ℝ^{4 \times 1}$ are constant vectors, and $f_{j} (\dot{θ}) \in [0, 1]$ is a saturating ReLU function that is active only for the knee joint, defined as

f_{knee} (\dot{θ}) = ReLU (\min (1, \dot{θ} / η)), f_{ankle} (\dot{θ}) = 0.

(5)

A fixed parameter $η$ controls the width of the linear window, which was set to 0.5 deg/s.

The knee kinematic trajectory is monotonic during both sitting and standing motions, meaning that $f_{knee} (\dot{θ}) = 0.0$ during sit-to-stand (negative knee velocity) and $f_{knee} (\dot{θ}) \approx 1.0$ during stand-to-sit (positive knee velocity). Therefore, the coefficients defining the equilibrium angle are $e (\dot{θ}) = e_{1}$ for sit-to-stand and $e (\dot{θ}) = e_{1} + e_{2}$ for stand-to-sit. During transitions between either sitting or standing motions (i.e., $0 < \dot{θ} < 0.5 deg/s$ ), the parameters continuously interpolate between the steady-state cases. In practice, this novel model definition allows for decoupled equilibrium angle trajectories for sitting and for standing, without necessitating discrete classification between cases.

3). Model Training:

The model is fully defined through the set of coefficients $κ = {k, b, e_{1}, e_{2}}$ . To train the model on the dataset, we construct an optimization problem to select the optimal $κ$ such that the impedance control equation (1) best reproduces the normalized torque profiles $τ$ given the joint kinematics $θ$ , angular velocities $\dot{θ}$ , and phase trajectories $s$ across all trials:

κ^{*} = \arg \min ‖ τ - \hat{τ} ‖_{2}^{2}, where \hat{τ} = K (s) (θ_{eq} (s, \dot{θ}) - θ) - B (s) \dot{θ} .

(6)

As in [15], we include the dataset phase trajectories $s$ in the optimization, allowing it to internally account for the expected shape of the $θ_{t}$ trajectory and the resulting nonlinearity in the phase variable. This improves upon our previous work which assumed a perfectly linear phase trajectory and necessitated an additional online linearization step [27], [28].

We employ the approach presented in [28] to approximate the cost function in (6) with a convex equivalent. If we define $s = {[s^{0} \dots s^{d}]}^{⊤}$ and $δ_{i} (s) = (k^{⊤} s) (e_{i}^{⊤} s)$ , for $i \in {1, 2}$ , we can re-write the nonlinear term as $K (s) θ_{eq} (s, \dot{θ}) = δ_{1} (s) + δ_{2} (s)$ . By treating the coefficients in the $δ_{i}$ polynomials as independent decision variables, the cost function becomes linear in the decision variables and reduces to a convex quadratic program. As in [28], the original coefficients $e_{1}$ and $e_{2}$ can be recovered from the solution by simply approximating the rational function $δ_{i} (s) / K (s)$ with a fourth order polynomial (see [28] for further detail).

We add constraints to the optimization problem based on desired controller behavior and pilot testing to ensure reasonable impedance trajectories for both sitting and standing motions. Namely, we constrain stiffness to be greater than 0.0087 Nm/(deg·kg) at the knee and 0.0175 Nm/(deg·kg) at the ankle. We constrain damping to be between 1.75 · 10⁻⁴ and 2.6 · 10⁻³ Nm·sec/(deg·kg) for both joints. Finally, we add soft constraints to ensure that (1) produces minimal torque at either joint when $s = 0$ (sitting) and minimal torque at the knee when $s = 1$ (standing). Given the positive stiffness and damping constraints, these sitting and standing torque constraints indirectly constrain $θ_{eq} (0)$ for both joints and $θ_{eq} (1)$ for the knee to be equal to the mean sitting and standing joint angles, respectively.

4). Model Results:

Figure 2 shows the optimized impedance parameter functions for both joints. The equilibrium angle for the knee is shown for both the positive velocity (sit-to-stand) and negative velocity (stand-to-sit) conditions. The optimal parameter trajectories for the ankle resemble those of our preliminary work [15], but the knee parameters show significant differences. As in prior work, the optimal ankle stiffness maintains a static value at the minimum constraint, which we hypothesize is due to the fact that the ankle torque in the dataset was fairly small in magnitude with large variance [32]. While this suggests that lowering the minimum stiffness could have increased dataset fit, a controller with very low stiffness would be unable to reject disturbances or robustly handle inter-subject variation.

Fig. 2. — Plots of the optimal impedance parameter trajectories for both joints. Positive equilibrium angles correspond to knee flexion and ankle dorsiflexion. Phase progresses from 0 to 1 during sit-to-stand and from 1 to 0 during stand-to-sit. Two curves are shown for the knee equilibrium angle, one for sit-to-stand and one for stand-to-sit. The controller moves smoothly between each curve based on (4)-(5), providing appropriate assistance levels during both standing and sitting without explicit classification.

To determine goodness of fit of our optimization results, we calculated the model joint torque using (1), (3), and $κ^{*}$ at each point in the dataset. We compared the root mean squared error (RMSE) of our model torque to typical human variation by normalizing it by the standard deviation of the joint torque seen in the dataset. Compared to prior work [15], the new model’s normalized RMSE in the ankle decreased by 9.7% (1.21 compared to 1.34), while the normalized RMSE of the knee is decreased by 51.1% (0.68 compared to 1.39).

III. Experimental Validation

A. Overview

To experimentally determine the effectiveness of our novel controller, we implemented it on a powered knee-ankle prosthesis previously validated for open-loop impedance control [31]. We conducted an experiment with three participants with above-knee amputation, comparing outcomes while using their passive prostheses versus the powered prosthesis with our novel controller. The demographics of each participant are described in Table I. Participant 1 had participated in a prior walking study [28] and a sit/stand study with a preliminary version of this controller [15]. Participant 2 had completed only one prior walking study [28]. Participant 3 had no prior experience with the powered prosthesis.

TABLE I.

Participant Demographics

Participant	Sex	Age	Body Mass (kg)	Height (cm)	Side of Amputation	Residual Limb Length (cm)	Time since Amputation (years)	Prescribed Prosthesis (knee/ankle)	Etiology	Device Experience

1	M	26	108.8	192	L	26.8	26	C Leg 4/Triase	Congenital	Moderate
2	M	40	79.2	180	L	38	23	Rheo Knee/Freedom Maverick Xtreme AT	Cancer	Low
3	M	18	69.3	183	L	15.9	18	Genium X3/Pro-Flex LP	Congenital	None

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The experimental protocol was approved by the Institutional Review Board of the University of Michigan (HUM00166976), and participants wore a ceiling-mounted safety harness for the duration of the experiment. A certified prosthetist assisted in fitting the powered prosthesis, changing out the two prostheses, and ensuring the safety of the participants throughout the experiments.

B. Protocol

The experimental protocol investigated sit-to-stand, stand-to-sit, walking, and turning motions while participants wore either their standard prosthesis or the powered prosthesis. Specifically, we investigated similarity in kinematics and kinetics with our controller compared to able-bodied, as well as how the powered prosthesis compared to the participant’s standard prosthesis in functional metrics. Because passive and semi-active above-knee prostheses are associated with increased asymmetry and completion time in sit/stand transitions compared to able-bodied movement [4]–[6], we investigated both of these metrics. Using these metrics, we tested both the generalization and functionality of our controller by investigating different sitting and standing speeds and chair heights. Finally, we demonstrated the controller’s ability to perform in more “real-world” conditions through a test involving sitting, standing, turning, and walking between chairs of different heights.

The detailed protocol included both training and experimental validation components for both the powered and passive prostheses, and the order in which these two conditions were tested was randomized for the three participants. The powered prosthesis condition included an initial fitting with the help of a certified prosthetist, which included walking between parallel bars to adjust alignment of the prosthetic leg. The sitting and standing conditions used a height-adjustable four-legged stool. Two participants completed the protocol within a one-day session, while Participant 3 became fatigued after completing the protocol with one prosthesis condition, so the training and testing for the second prosthesis condition was performed on a second day.

Conditions for both prostheses included approximately 15 minutes of symmetry training where the participants practiced sitting and standing with one leg each on two in-ground force plates, from which they received visual feedback regarding the loading symmetry in the vertical direction between the two legs, as described in [15]. Of these 15 minutes of training, the first five minutes included instruction on the visual feedback, and they were instructed to try different strategies to see how they affected symmetry. In the next five minutes, they were instructed to purposefully sit and stand asymmetrically on one side or another. In the last five minutes, they were instructed to try to sit and stand as symmetrically as possible. After the training was complete, the visual feedback was removed.

Following training, sitting and standing was performed in two different speed conditions (self-paced and rapid) and three different stool height conditions (tall, medium, and short, corresponding to 53.5 cm, 51.0 cm, and 48.5 cm, respectively), asking participants to keep their hands crossed on their chest. In the self-paced speed conditions, the participants were cued to sit and stand five times each at a comfortable pace, with the goal of maximizing the symmetry that had been practiced with the visual feedback. In the rapid speed conditions (5x sit/stand), participants were asked to sit and stand fives times each as fast as comfortably possible. Both conditions with five sit/stand cycles each were repeated three times with a one minute break in between, for a total of fifteen sit/stand cycles per condition.

Finally, the participants completed a multi-activity task comprising a sit/stand/walk sequence between different chair heights. In this test, the participants started seated on the stool at the tallest setting, stood up, walked 12 meters to a chair of height 44.5 cm, sat down, stood back up, walked back to the stool where they had started, and sat down. They were asked to complete this task as fast as comfortably possible and repeated it five times with each prosthesis. Note that due to a mechanical failure in the robotic hardware, participant 1 was only able to complete two trials. We timed each sit/stand/walk sequence using a stopwatch.

C. Data Analysis

1). Comparison to Able-Bodied Mechanics:

We compared the observed phase, joint angle, and joint torque trajectories in the self-paced trials to the able-bodied dataset. Phase was determined using the thigh IMU and Eq. (2), joint angles were measured by the joint encoders, and joint torque was calculated using motor current and a validated motor model [31]. In order to ensure uniformity in the segmentation, data from the powered prosthesis was segmented in the same manner as the able-bodied dataset (See Section II-C-1), and percent motion was calculated by normalizing time between the start and endpoints identified in this segmentation.

2). Functional Comparisons Between Prostheses:

Next, we calculated various clinical metrics and checked for significant differences due to the powered prosthesis. The degree of asymmetry in leg loading $D o A$ was calculated for both the self-paced and rapid trials as

D o A = (F_{z, bio} - F_{z, prosth}) / (F_{z, bio} + F_{z,prosth}),

(7)

where $F_{z,bio}$ and $F_{z,prosth}$ are the vertical forces applied by the biological and prosthetic foot, respectively. Note that datapoints where the combined vertical force from both legs $F_{z, feet} = F_{z, bio} + F_{z, prosth}$ was less than 10% of the participant’s bodyweight were excluded. These points were excluded because a low value of $F_{z, feet}$ causes the $D o A$ metric (7) to become sensitive to minor changes in loading and sensor noise, and clinically makes asymmetrical loading more inconsequential, as the main comorbidities (e.g., osteoarthritis and lower back pain) are caused by large joint loads. Positive $D o A$ value corresponds to increased loading of the biological limb and a negative value corresponds with increased loading of the prosthetic limb, while $D o A = 0$ indicates perfectly balanced loading of both limbs.

For the self-paced conditions, we segmented the data into separate sit-to-stand and stand-to-sit segments based on the derivative of $F_{z, feet}$ , denoted as ${\dot{F}}_{z,feet}$ . During sit-to-stand motions, ${\dot{F}}_{z,feet}$ starts at zero, increases to a peak and then returns to zero in an under-damped pattern. Therefore, the start of a sit-to-stand motion was defined as the first instance when ${\dot{F}}_{z,feet}$ was greater than 300 N/s prior to the peak, and the end of the motion was determined when $| {\dot{F}}_{z,feet} |$ remained below 300 N/s for 60 ms after the peak. Likewise, the start of a stand-to-sit motion was defined as the next point where $| {\dot{F}}_{z,feet} |$ returned to above 300 N/s, and the end of a stand-to-sit motion was defined as when $| {\dot{F}}_{z,feet} |$ returned to below 300 N/s for 60 ms after the negative peak in ${\dot{F}}_{z,feet}$ .

Using the segmented self-paced data, we calculated the average asymmetry during the motion $\bar{D o A}$ and RMS loading of the prosthetic leg normalized by bodyweight ${\bar{F}}_{z,prosth}$ , similar to [13]. Although other previous work has calculated asymmetry at the peak of the ground reaction force during sit-to-stand, we chose to calculate average asymmetry because this peak does not always exist during stand-to-sit motions. We selected these metrics because they quantify the user’s ability to symmetrically load both legs during discrete motions, which should prevent detrimental compensations.

For the rapid 5x sit/stand trials, our primary outcome metrics were the time required to complete the task $T_{5 x}$ and average loading asymmetry $\bar{D o A}$ throughout the entire trial. Thus, no segmenting was required for these trials. We selected these metrics to investigate if the powered prosthesis allowed the user to complete the sit/stand motions faster and if symmetry was maintained during fast motions.

3). Statistical Analysis:

We fit linear mixed effects models to each of the clinical sit/stand metrics. The fixed effects included the categorical prosthesis type (powered or passive) and the continuous chair height, and the participant was treated as a random effect by including a subject-specific intercept. Statistically significant influences of the fixed effects were determined with appropriate Bonferroni corrections. As two comparisons were made for each independent set of data, significance was set to p < 0.025. A student’s t-test was used to determine significance in the sit/stand/walk test, with p = 0.05 set as the level of significance.

IV. Experimental Results

A. Comparison to Able-Bodied Mechanics

Figure 4 shows the average (a) phase, (b) kinematic, and (c) kinetic trajectories observed across all self-paced trials with the tall stool height and the powered prosthesis.

Fig. 4. — Inter-subject average behavior of the powered prosthesis during self-paced sit/stand motions at the tall stool height, plotted with respect to the percentage of motion completion. a) Trajectory of the phase estimate, determined by normalized thigh angle. b) Kinematic data at the knee and ankle joints. c) Kinetic data at the knee and ankle joints. These data are compared to the mean able-bodied data used in the impedance model development [32], which was collected in similar chair height conditions. Shaded regions represent ±1 standard deviation, with each trial treated as an independent sample.

For comparison, the figure also shows the equivalent mean trajectories from able-bodied data collected at a chair height approximately equal to participant knee height, similar to the tall stool height. The shaded regions represent ±1 standard deviation, where each sit-to-stand or stand-to-sit across all subjects was treated as an independent sample for both the able-bodied dataset and experimental results.

There is a close correlation between the averaged able-bodied and experimental phase during sit-to-stand (Figure 4a), although the standard deviation in the phase variable is larger experimentally. During stand-to-sit, the experimental phase variable closely tracks the model dataset for the beginning of the motion, but consistently overestimates during the end of the motion. The RMSE between the average phase of the dataset and the experimental results was 4.91% for sit-to-stand and 7.38% for stand-to-sit, similar to our previous work in walking with an RMSE of 6.25% [28].

We observe high similarity between the experimental and average able-bodied joint angles (Figure 4b), with the largest discrepancies occurring at the knee in the middle of the sit-to-stand motion, and discrepancies in the ankle occurring at standing. Figure 4c demonstrates strong similarities between the experimental and able-bodied joint torques, with the largest discrepancy occurring at the end of the stand-to-sit motion, likely due to an overestimate in the phase variable.

B. Functional Comparisons Between Prostheses

Differences in $\bar{D o A}$ were observed during the self-paced trials between prosthesis types (Figure 5a). During stand-to-sit, the mixed effects model determined a reduction in $\bar{D o A}$ with the powered prosthesis (0.09 ± 0.12 versus 0.45 ± 0.17), indicating that participants were significantly more symmetric compared to the passive prosthesis ( $p \approx 10^{- 56}$ ). Participant 2 showed the greatest symmetry improvement with the powered prosthesis. The symmetry improvements with the powered prosthesis for sit-to-stand were less pronounced (0.26 ± 0.10 versus 0.39 ± 0.17), but this change was still significant ( $p \approx 10^{- 22}$ ). In both sit-to-stand and stand-to-sit, chair height had no significant effect on $\bar{D o A}$ . Supplemental Figure 1 shows $\bar{D o A}$ plotted over the motion for sit-to-stand and stand-to-sit for each chair height, demonstrating that the largest powered prosthesis improvements occurred during the first half of the motion for sit-to-stand and the second half of the motion for stand-to-sit.

Fig. 5. — Functional comparisons for the self-paced trials (Color = participant, powered prosthesis = solid markers, passive prosthesis = hollow markers). a) Average vertical loading asymmetry $\bar{D o A}$ between both legs, with a positive value corresponding to increased biological leg loading. b) RMS prosthetic leg loading ${\bar{F}}_{z,prosth}$ normalized by bodyweight, with higher values indicating higher average force on the prosthetic foot. The powered prosthesis improvements in both $\bar{D o A}$ and ${\bar{F}}_{z,prosth}$ were statistically significant for both stand-to-sit and sit-to-stand.

Similarly, we observed significant improvements in ${\bar{F}}_{z,prosth}$ with the powered versus passive prosthesis for both sit-to-stand (0.38 ± 0.05 versus 0.32 ± 0.09, $p \approx 10^{- 17}$ ) and stand-to-sit (0.46 ± 0.06 versus 0.29 ± 0.09, $p \approx 10^{- 52}$ ) self-paced trials. Similar to $\bar{D o A}$ , this improvement was more pronounced in stand-to-sit motions. Again, the effect of chair height was not significant. Interestingly, Participant 2 showed little benefit during sit-to-stand, but the most benefit during stand-to-sit. Conversely, Participants 1 and 3 showed consistent benefits during both directions of motion.

With increased speed during the rapid trials, the participants still demonstrated reduced asymmetry $\bar{D o A}$ with the powered prosthesis (0.078 ± 0.077 versus 0.328 ± 0.043, Figure 6a, $p \approx 10^{- 23}$ ). Again, no significant effect of chair height was observed. The time to complete one 5x sit/stand trial $T_{5 x}$ was also significantly improved with the powered prosthesis (18.0 ± 2.6 versus 21.3 ± 1.9 sec, Figure 6b, $p \approx 10^{- 8}$ ) and Participant 2 showing the greatest improvement. As expected, the chair height significantly affected completion time ( $p \approx 10^{- 5}$ ), with the shorter chair heights resulting in higher completion time.

Fig. 6. — Functional comparisons for the rapid 5x sit/stand tests. Color represents participant, with powered prosthesis results in solid markers and passive prosthesis in hollow markers. a) Average vertical loading asymmetry between both legs $\bar{D o A}$ , with a positive value corresponding to increased biological leg loading. b) Time required to complete each 5x sit/stand trial, $T_{5 x}$ . The improvements elicited by the powered prosthesis in $\bar{D o A}$ and $T_{5 x}$ were found to be statistically significant, as well as a significant correlation between $T_{5 x}$ and chair height.

In the sit/stand/walk trials, participants were able to intuitively change prosthesis modes, allowing them to rise from one chair, walk across the room, and sit in the second chair without issue (see supplemental video). The controller smoothly switched between Walk and Sit/Stand modes at the appropriate moments, and there were no mode misclassifications in these trials, nor in any of the sit/stand trials. However, despite the powered assistance in the sit/stand/walk trials, the participants were, on average, faster with their passive prostheses than the powered prosthesis (20.6 ± 3.9 versus 24.3 ± 3.9 sec; p = 0.031). These differences were most pronounced in Participant 3, who had the least training with the powered device in previous studies (Figure 7, Table I). This suggests that additional training might be beneficial.

Fig. 7. — Lap times for each participant for the sit/stand/walk trials using each prosthesis. On average, participants were faster with the passive versus powered prosthesis, although Participants 1 and 2 reported similar times. Note Participant 1 only completed two laps due to a mechanical failure with the powered prosthesis.

V. Discussion

Comparisons between our novel controller implemented on a powered prosthesis and able-bodied sit/stand data demonstrate that the controller generally produces biomimetic kinematics and kinetics (Figure 4), with some notable deviations. The largest discrepancies in ankle mechanics occur during standing, though the magnitude of the discrepancy is quite small. The difference may be due to the participants electing to stand with their weight more towards the heel of the prosthesis, resulting in less dorsiflexion compared to the dataset. In addition, during sit-to-stand, the prosthetic knee trajectories lead those from the able-bodied dataset in the middle of the motion, due to the experimental phase estimate leading that calculated from the able-bodied dataset. In contrast, during stand-to-sit, the experimental phase lags that of the able-bodied dataset at the end of the motion, likely contributing to excessive applied knee torque. This excessive knee torque at the end of stand-to-sit is the largest deviation from able-bodied mechanics, but it is unclear if this had a negative effect in the measured clinical outcomes. It may in fact improve clinical outcomes, as previous studies have reported that decreasing knee torque correlated with increased symmetry at the end of the motion [17]. In contrast to our prior work [15] and others [17] where torque magnitude had to be reduced from biomimetic kinetics to allow the pilot participant to sit down, all participants were able to effectively use the biomimetic knee torque during stand-to-sit without any changes (Figure 4c). Qualitative feedback from the participants, detailed further below, echoed this sentiment.

In comparison to participants’ passive prostheses, the powered prosthesis significantly reduced average loading asymmetry, with increased prosthetic leg loading during sit-to-stand and stand-to-sit (Figures 5 and 6). Chair height did not have a significant effect, suggesting that the controller is equally effective across multiple chair heights. In general, these improvements were more significant during stand-to-sit compared to sit-to-stand, especially for Participant 2, who consistently loaded the powered prosthesis at least 50% more than his passive prosthesis and was able to achieve close to perfect symmetry during stand-to-sit with the powered prosthesis. Compared to prior work that used a powered prosthesis to significantly reduce stand-to-sit loading asymmetry by 42% to approximately 0.25 [17], our controller reduced loading asymmetry by 79% to 0.09, likely due to the larger magnitudes of torque provided. Compared to stand-to-sit, the relationship between applied torque and loading asymmetry during sit-to-stand seems more complicated, as prior work has shown increasing powered prosthesis knee torques to reduce asymmetry during the middle of the sit-to-stand motion, while increasing asymmetry at the end of the motion [16]. Prior work investigating a 5x sit-stand-sit test in healthy older adults reports an average loading asymmetry of 0.09 [34], suggesting that, at least for stand-to-sit, our controller performance is nearing that of able-bodied. RMS loading was also increased with our controller compared to previous work; while previous work by our group and others reported average RMS loading of 30–35% during sit-to-stand and stand-to-sit [13], [15], this study’s average sit-to-stand RMS loading was 38% and stand-to-sit RMS loading was 46% across participants and chair heights (Figure 5). In addition, while loading asymmetry has been shown to correlate with ostearthritis and lower back pain [7], it is unclear how much of an improvement is clinically meaningful.

In addition to improved loading symmetry, we also demonstrated that the novel controller significantly reduced the time required to complete a 5x sit/stand test to an average of 18 sec (Figure 6). Although 5x sit/stand is an imperfect metric, it is frequently used clinically and has been shown to correlate with balance [35], [36]. Despite the improvement when using the powered versus passive prosthesis, participant completion times were still above the standard deviation reported for healthy individuals (11.4 ± 3.4 sec) [36]. This timing was also the only measured metric where we determined a statistically significant effect of chair height, with a shorter chair height corresponding with longer 5x sit/stand times. This is expected when raising the participants’ center of mass a larger distance from the shorter chair height and was corroborated by the qualitative feedback from the participants regarding the difficulty of the task.

We used a combined sit/stand/walk test with chairs of different heights to demonstrate proof of concept of the versatility of the controller to adapt to different scenarios, but did not improve average speed in this combined test (Figure 7). Although Participant 3 was consistently faster with his passive prosthesis, Participants 1 and 2 had similar times with both prostheses, with a difference of approximately 10%, consistent with our previous findings using similar tests [15]. Interestingly, the powered prosthesis did improve average overall speed for Participant 1 by approximately 10%, due to the fact that during one of the passive prosthesis trials, he tried to stand up from the lowest height chair and fell back to the chair before starting again (see Supplemental Video). However, we can make limited conclusions from this participant because he only completed two trials prior to a hardware failure. We believe that both the unfamiliarity and added mass of the powered prosthesis contributed to the sit/stand/walk result. We suspect that, given the high variance in the powered prosthesis lap times, more acclimation and practice balancing, walking, and turning with the robotic device could improve this result.

It should be noted that our results are likely influenced by the devices used in this study. For example, Participant 2’s prescribed device was a Rheo Knee, which controls knee flexion resistance through the use of a magnetorheological damper. According to this participant, loading prevented the device from flexing the knee, forcing him to load most of his weight on the biological leg in order to sit down with his prescribed device. In addition, the powered device used for validation employed quasi-direct drive actuators, which allowed for open-loop impedance control. An additional low-level controller would likely be necessary if implementing this control strategy on powered prostheses with less transparent drivetrains.

Although we focused on collecting quantitative metrics during this experiment, we also noted qualitative feedback from participants of their impressions of the experiment and the novel controller. In general, all participants liked using the powered prosthesis and found it helpful compared to their passive devices. One noted that “standing up from the short chair was way harder” with his passive device compared to the powered prosthesis. Another noted that “it’s nice that I don’t have to predict where my rear is going to go” during stand-to-sit with the powered prosthesis, because his passive device “just releases, it’s just a guess”. And although we did not directly measure fatigue, one participant noted that “tired-wise, I think [the powered prosthesis] is helpful... but overall I’m fatiguing during the whole experiment”. Future work should investigate participant preferences further, as this is important for adoption [37].

This study was not without its limitations. First, it should be noted that the small sample size limits any generalized claims. In addition, our able-bodied dataset only included sit-to-stand and stand-to-sit experimental data with a single chair height. Although we demonstrated that our algorithm was able to adapt to multiple chair heights, it is possible that better performance could have been achieved by providing the optimization data of multiple chair heights, and possibly parameterizing the impedance parameters as functions of chair height. We also only provided 15 minutes of training with the novel prosthesis controller, and it is possible that increased training times could lead to improved performance in the clinical metrics tested. Further, although our rule-based high-level classification method produced no errors during this study, it is possible that our chosen thresholds could be inappropriate for individuals with strongly atypical gaits. Future work is needed to investigate this limitation and employ methods to automatically adapt the thresholds to a specific gait. Finally, although we demonstrated that we can improve clinical performance with a tuning-free “average” controller, future work should investigate tuning methods that might improve performance with minimal tuning.

VI. Conclusion

This work presented a novel Sit/Stand controller for a knee-ankle prosthesis comprising a thigh-based phase variable and a variable impedance controller. The novel sit/stand impedance model incorporated a dependency on knee velocity, allowing a unified model for both sit-to-stand and stand-to-sit motions with distinct behavior for each. We fit the model to an able-bodied dataset using convex optimization and experimentally demonstrated the controller’s ability to produce biomimetic joint kinematics and kinetics for three individuals with amputations during both sitting and standing. The improved biomimicry of our controller compared to previous work resulted in improved clinical metrics, including improved leg loading symmetry and increased speed during a timed 5x sit/stand test. We also demonstrated the controller’s ability to adjust to different chair heights in sit/stand trials, as well as sit/stand/walk trials. Our results suggest that the controller’s biomimetic assistance may reduce the overreliance on the biological limb, potentially improving mobility and quality of life for people living with above-knee amputations.

Supplementary Material

Supplementary figure

NIHMS1938500-supplement-Supplementary_figure.pdf^{(437.5KB, pdf)}

Supplemental video

Download video file^{(392.7MB, mp4)}

Fig. 3. — Photos of Participant 3 practicing sit/stand symmetry with the powered prosthesis during training. The bar on the screen depicts real-time loading symmetry between the left and right foot. When the loading symmetry was outside the recommended range (± 15%), the bar turned yellow and then gradually red as it moved from the center line.

Acknowledgement

The authors thank Dr. Friedl De Groote for providing the de-identified sit/stand data, Dr. Curt Laubscher and Hannah Frame for assistance in data collection, Leslie Wontorcik, CP, for clinical support, and our participants.

This work was supported by the National Institute of Child Health & Human Development of the NIH under Award Number R01HD094772 and by the National Science Foundation under Award Number 2024237 and Graduate Research Fellowship Grant No. DGE 1841052. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH and NSF.

References

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Supplementary figure

NIHMS1938500-supplement-Supplementary_figure.pdf^{(437.5KB, pdf)}

Supplemental video

Download video file^{(392.7MB, mp4)}

[R1] [1].Ziegler-Graham K, MacKenzie EJ, Ephraim PL, Travison TG, and Brookmeyer R, “Estimating the Prevalence of Limb Loss in the United States: 2005 to 2050,” Arch. Phys. Med. Rehab, vol. 89, no. 3, pp. 422–429, Mar. 2008. [DOI] [PubMed] [Google Scholar]

[R2] [2].Schaarschmidt M, Lipfert SW, Meier-Gratz C, Scholle HC, and Seyfarth A, “Functional gait asymmetry of unilateral transfemoral amputees,” Hum. Movement Sci, vol. 31, no. 4, pp. 907–17, Jan. 2012. [DOI] [PubMed] [Google Scholar]

[R3] [3].Kaufman KR, Frittoli S, and Frigo CA, “Gait asymmetry of transfemoral amputees using mechanical and microprocessor-controlled prosthetic knees,” Clin. Biomech, vol. 27, no. 5, pp. 460–5, Jun. 2012. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R4] [4].Burger H, Kuzeliˇ cki J, andˇ CR Marinek, “Transition from sittingˇ to standing after trans-femoral amputation,” Prosthet. Orthot. Int, vol. 29, no. 2, pp. 139–51, Aug. 2005. [DOI] [PubMed] [Google Scholar]

[R5] [5].Highsmith M et al. , “Kinetic asymmetry in transfemoral amputees while performing sit to stand and stand to sit movements,” Gait Posture, vol. 34, no. 1, pp. 86–91, Apr. 2011. [DOI] [PubMed] [Google Scholar]

[R6] [6].Wolf EJ, Everding VQ, Linberg AA, Czerniecki JM, and Gambel CJM, “Comparison of the Power Knee and C-Leg during step-up and sit-to-stand tasks,” Gait Posture, vol. 38, no. 3, pp. 397–402, Jul. 2013. [DOI] [PubMed] [Google Scholar]

[R7] [7].Gailey R, Allen K, Castles J, Kucharik J, and Roeder M, “Review of secondary physical conditions associated with lower-limb amputation and long-term prosthesis use,” J. Rehabil. Res. Dev, vol. 45, no. 1, pp. 15–29, Jun. 2007. [DOI] [PubMed] [Google Scholar]

[R8] [8].Torrealba RR and Fonseca-Rojas ED, “Toward the Development of Knee Prostheses: Review of Current Active Devices,” Appl. Mech. Rev, vol. 71, no. 3, pp. 1–22, May 2019. [Google Scholar]

[R9] [9].Orendurff MS, Schoen JA, Bernatz GC, Segal AD, and Klute GK, “How humans walk: Bout duration, steps per bout, and rest duration,” J. Rehabil. Res. Dev, vol. 45, no. 7, pp. 1077–1090, Feb. 2008. [DOI] [PubMed] [Google Scholar]

[R10] [10].Dall PM and Kerr A, “Frequency of the sit to stand task: An observational study of free-living adults,” Appl. Ergon, vol. 41, no. 1, pp. 58–61, Jan. 2010. [DOI] [PubMed] [Google Scholar]

[R11] [11].Varol HA, Sup F, and Goldfarb M, “Powered sit-to-stand and assistive stand-to-sit framework for a powered transfemoral prosthesis,” in IEEE ICORR, Kyoto, Japan, 2009, pp. 645–651. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R12] [12].Simon AM, Fey NP, Ingraham KA, Finucane SB, Halsne EG, and Hargrove LJ, “Improved Weight-Bearing Symmetry for Transfemoral Amputees during Standing Up and Sitting Down with a Powered Knee-Ankle Prosthesis,” Arch. Phys. Med. Rehabil, vol. 97, no. 7, pp. 1100–6, Jul. 2016. [DOI] [PubMed] [Google Scholar]

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PERMALINK

Improving Sit/Stand Loading Symmetry and Timing Through Unified Variable Impedance Control of a Powered Knee-Ankle Prosthesis

Cara Gonzalez Welker

T Kevin Best

Robert D Gregg

Abstract

I. Introduction

II. Sit/Stand/Walk Controller

A. High-Level Classification

Fig. 1.

B. Sit/Stand Phase Variable

C. Sit/Stand Impedance Model

1). Training Dataset Preparation:

2). Model Architecture:

3). Model Training:

4). Model Results:

Fig. 2.

III. Experimental Validation

A. Overview

TABLE I.

B. Protocol

C. Data Analysis

1). Comparison to Able-Bodied Mechanics:

2). Functional Comparisons Between Prostheses:

3). Statistical Analysis:

IV. Experimental Results

A. Comparison to Able-Bodied Mechanics

Fig. 4.

B. Functional Comparisons Between Prostheses

Fig. 5.

Fig. 6.

Fig. 7.

V. Discussion

VI. Conclusion

Supplementary Material

Fig. 3.

Acknowledgement

References

Associated Data

Supplementary Materials

ACTIONS

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