Hip Trajectory Error: A Framework for Designing and Evaluating Passive Prosthetic Feet for People With an Above-Knee Amputation Open Access

A well-fitting, high-performance prosthesis for people with a lower limb amputation can greatly improve mobility and quality of life, but many amputees lack access to high-performance prosthetic components [1–5]. According to the global burden of diseases study from 2019, approximately 31.3 million people are living with a lower limb amputation (⁠$≈$25.6 × 10⁶ unilateral and 6 × 10⁶ bilateral amputees) [6]. Estimating how many of this population are people with above-knee amputation is nontrivial, as, to the knowledge of the authors, there are no global rates data for amputation levels. Data for high-income countries such as the U.S., Canada, and Germany predict that about 17–26% of lower limb amputees are people with an above-knee amputation (including through-knee and hip disarticulation) [7–9]. Data for lower and middle-income countries (North and South India, Iran, Trinidad, and Tobago) show even higher estimates; people with above-knee amputation make up 23–59% of all lower limb amputees [10–14]. Using these data, we can estimate that there are about 5–17 million people with an above-knee amputation. This demonstrates the large need for prosthetic leg component design, specifically for people with an above-knee amputation.

Despite the large demand, currently there are no commercially available prosthetic feet designed specifically for people with an above-knee amputation. A review of commercially available prosthetic feet from four prosthetic foot providers indicates that although there are feet marketed specifically for people with a below-knee amputation, there are no such equivalent designed specifically for people with an above-knee amputation [15–18]. The prosthetic feet that are marketed for people with an above-knee amputation are also marketed for people with a below-knee amputation, indicating that the two populations have similar user needs and can wear a universal foot design. However, people with above-knee amputation have distinctly different gait from people with a below-knee amputation, which suggests that this user population requires distinct ambulatory functionality from their prosthetic devices.

One of the main gait differences common to people with an above-knee amputation is the lack of early and midstance knee flexion, which occurs in up to 98% of above-knee prosthesis users [1,19,20]. The main function of early and midstance knee flexion, which is about 10–15 deg for able-bodied people and people with a below-knee amputation [21], is to provide shock absorption, stability, and continuous progression through the step [21–23]. Maintaining functions such as sufficient shock absorption and smooth progression of the leg throughout the step are important for preventing secondary health issues and supporting able-bodied walking speeds [1,3,22,24]. Prosthetic feet for people with a below-knee amputation are designed assuming the presence of early-stance knee flexion and therefore may not function as intended when used by the majority of people with an above-knee amputation. For people with an above-knee amputation who lack early and midstance knee flexion, using prosthetic feet designed for people with a below-knee amputation can result in an improper fit and poor interaction with the rest of the prosthetic leg such as too early initiation of flexion, potentially leading to falls and secondary health issues [1,3]. Therefore, a prosthetic leg designed for people with an above-knee amputation should incorporate features that provide shock absorption and progression throughout early and midstance in spite of early and midstance knee flexion absence.

The current solutions that account for the lack of early and midstance flexion in prosthetic leg designs do not sufficiently promote able-bodied gait or improve user mobility, especially in the case of passive prosthetic knees. For example, a common design idea is to control early stance knee flexion using a linear spring, due to the linear relationship of knee moment to knee angle at comfortable walking speeds [25–28]. However, Shamaei and Dollar [28] have analyzed how the knee moment-angle relationship changes with changes in walking speed, demonstrating that the required knee spring stiffness for appropriate knee flexion and extension changes significantly with walking speed. It is common for people with an above-knee amputation to choose a slower comfortable walking speed compared to able-bodied people [22,24], which would require the extension spring stiffness to be 30–60% higher than flexion stiffness. This result suggests that a passive prosthetic knee would require a complex spring mechanism for even a single selected speed, making the design costly and difficult to accommodate a range of people with an above-knee amputation. Another solution for providing the required early and midstance knee flexion functionality is the use of shock-absorbing pylons, however, experimental studies have not shown any noticeable effects or advantages [24,29]. Furthermore, neither of these solutions target predictable progression of the leg throughout stance, which may hinder the timely transition to late-stance and swing flexion. The remaining component of the prosthetic leg that could potentially provide early and midstance flexion functionality is the prosthetic foot, but solutions in this space have not yet been explored. Thus, a design objective needs to be defined to create and evaluate prosthetic feet for people with an above-knee amputation that can provide sufficient shock absorption and smooth progression, as well as overcome the disadvantages of currently available solutions.

The design of high-performance prosthetic feet for people with an above-knee amputation must not only incorporate shock absorption and progression but also follow clinical guidelines, which provide suggestions on how to find a proper fit for each patient based on their ambulation level and needs. Current guidelines suggest that people with an above-knee amputation use either single-axis feet to provide greater stability or energy storage and return feet for better symmetry during walking [30–32]. This recommendation is similar to guidelines for people with a below-knee amputation but is less detailed, requiring prosthetists to determine a proper fit based on many fittings and adjustments. A more comprehensive model, involving clinically suggested energy storage and return foot designs, that accounts for the deflection in the prosthetic foot is required for a more accurate prediction of the resultant gait and knee–foot interaction.

The development of a comprehensive gait model for people with an above-knee amputation is especially important, as the prosthetic foot has a direct effect on the performance of the entire prosthetic leg, specifically on the operation of the knee mechanism throughout the stance phase. It is common for passive prosthetic knee designs to rely on accurate replication of able-bodied center of pressure (CoP) progression and orientation of ground reaction forces (GRFs) to maintain stability during early and midstance while allowing for timely knee flexion during late stance. Mechanical designs of such knees can include locks and latches attached to linkages that move depending on the orientation and location of applied forces and moments [33–35]. If an above-knee amputee is using a prosthetic foot that is designed for a below-knee amputee to replicate able-bodied motion and loading, the lack of early and midstance knee flexion could result in deviation from either able-bodied target motion or able-bodied loading. Improper loading at the foot can cause the knee to unlock too early or too late, which can result in a fall or stumble [3,36]. Therefore, a high-performance prosthetic foot for people with an above-knee amputation should accurately replicate able-bodied loading (CoP and GRF orientation and progression) to enable predictable knee mechanism operation. A framework is needed to design prosthetic feet based on above-knee gait to provide proper interaction with the prosthetic knee and promote overall mobility.

An example of a comprehensive model for prosthetic foot design is the lower leg trajectory error (LLTE) framework [37–41], which optimizes prosthetic foot behavior to accommodate the needs of people with a below-knee amputation. LLTE-designed feet have been shown to enable walking performance similar to commercial carbon fiber feet after short accommodation times for people with below knee amputation, demonstrating the utility of a predictive design framework. However, the LLTE framework optimizes the feet to replicate the kinematics of the lower leg, which would deviate in the gait of people with an above-knee amputation, and hence may not be appropriate for this population.

The purpose of this paper is to present a novel prosthetic foot design and evaluation framework, the hip trajectory error (HTE), to improve prosthetic foot design for people with above-knee amputation. Through simulation studies and the resulting design insights, we demonstrate the biomechanical advantages of HTE feet over products designed for below-knee amputees. The HTE framework can be used to quantitatively and predictively design prosthetic feet for people with above-knee amputations while accounting for the lack of early stance knee flexion. The shock absorption and progression through stance that was previously provided by knee flexion is instead accommodated using deflection within the prosthetic foot. This method is built on the success of the LLTE framework, which has been used to design high biomechanical performance prosthetic feet for people with a below-knee amputation [37–41]. Using the HTE framework, an architecture of a prosthetic foot is realized and the design solution space is explored for different ankle heights. Lastly, a simulation comparing the predicted hip motion for an above knee amputee using LLTE and HTE feet demonstrates potential advantages of prosthetic feet designed specifically for people with an above-knee amputation who lack early-stance knee flexion during walking, motivating the need to experimentally validate HTE framework.

where $xnknee$⁠, $ynknee$⁠, and $θnlower leg$ variables describe the predicted motion of the lower leg; $x¯nknee$⁠, $y¯nknee$⁠, and $θ¯nlower leg$ variables describe the able-bodied reference of lower leg motion; $Llower leg$ is the leg length from floor to knee center; $Lfoot$ is the length of the foot; and N is the number of stance frames used for the optimization [37,44]. Although past LLTE work has used able-bodied motion as the reference trajectory, the framework can be used for any desired motion of the lower leg.

Lower leg trajectory error prosthetic feet have been shown to enable the replication of able-bodied loading and motion comparable or significantly better than daily use and commonly distributed carbon fiber prosthetic feet in people with a below-knee amputation [37,38]. The LLTE feet also showed the highest energy storage and return, a function that is strongly preferred by below-knee prosthesis users [37,38]. An LLTE stiffness sensitivity study demonstrated that the optimal stiffness prosthetic foot produced by the LLTE framework results in better replication of able-bodied kinematics and kinetics compared to prosthetic feet with lower and higher stiffnesses [39]. This demonstrates that the optimal LLTE prosthetic feet result in the best possible biomechanical performance, suggesting the validity of the choice of able-bodied lower leg trajectory as the goal for below-knee amputee gait. Lastly, a field study with LLTE feet has shown improvements in walking after long-term use, and general robustness of the prosthetic feet designed using this framework [45]. The resulting optimal designs can be mass-manufactured from plastic and distributed for a lower cost compared to other high-performance prosthetic feet [44,46].

Though the LLTE work is promising for improving access to well-fitting, high-performance prosthetic feet for people with a below-knee amputation, it is unlikely that LLTE feet will enable high-performance for people with an above-knee amputation due to the vast differences in ambulatory needs between these two populations [22,30,31]. Using lower leg position and orientation as the walking motion goal will result in undesired prosthetic leg performance, as the majority of above-knee amputees do not display stance phase knee flexion [1,19,20]. The inputs to the LLTE framework assume that the loading at the foot replicates able-bodied kinetics, so the foot will propel the knee to an able-bodied position for that point in the stance phase. However, in a locked knee configuration for a person with an above-knee amputation, this would result in large deviations from the desired hip motion; instead, the knee location should be posterior to the able-bodied reference knee location for that point in the stance phase. Based on gait training literature and LLTE foot performance with people with a below-knee amputation [38,47], it is more likely that hip kinematics will be maintained and the error from able-bodied references will manifest in the loading of the foot (CoP, $GRFx$⁠, and $GRFy$⁠). This deviation in loading may cause potential health issues with the residual limb and distal joints [3]. Although the LLTE feet are predicted to provide poor biomechanical performance for people with above-knee amputation, the methodology of LLTE can be applied to create a framework specifically for the needs of people using above-knee prostheses. The existence of such framework will be beneficial due to the quantitative and predictive nature of the framework and could be successful at improving mobility for people with above-knee amputation based on results from LLTE in vivo experimental studies.

Hip trajectory error is a novel framework to design prosthetic feet specifically to meet the needs of an above-knee amputee that does not flex their knee during the stance phase. The design framework presented in this paper uses the LLTE framework as a basis, but changes the optimization objective so that the resulting prosthetic feet facilitate lower limb motion that is beneficial for people with above-knee amputation. Prosthetic feet designed using the HTE framework are custom to the user based on their body parameters, such as body mass, leg length, ankle height, and foot length.

The goal of feet designed with the HTE framework is to provide the missing early and midstance knee flexion functions. As established in the Introduction, up to 98% of prosthetic knee users do not exhibit early and midstance knee flexion either due to training, user preference, or operation of the knee mechanism [19].

where $xnhip$ and $ynhip$ are the predicted motion of the hip; $x̂nhip$ and $ŷnhip$ are the able-bodied reference hip motion; $Lleg$ is the leg length from floor to hip center; and N is the number of stance frames used for the optimization.

The realization of the HTE prosthetic foot shape is based on the process used to design LLTE prosthetic feet [38–41]. An overview of the setup and the differences between LLTE and HTE optimizations are outlined in this section. The HTE prosthetic foot is modeled in the sagittal plane as a two-dimensional compliant structure described by threes wide Bezier curves with 12 independent coefficients (⁠$C1d$⁠, $C2x$⁠, $C2y$⁠, $C2d$⁠, $C3x$⁠, $C3y$⁠, $C3d$⁠, $C4x$⁠, $C4d$⁠, $C5d$⁠, $C6d$⁠, $C7d$⁠) (Fig. 3). Both the heel and the keel are flexible; the heel being flexible is especially important in the case of HTE feet, as the majority of shock absorption is experienced during the heel strike [21].

The input reference data were collected for ten able-bodied subjects walking at a self-selected comfortable walking speed (Froude number 0.2, speed 1.4 m/s) on a level ground [37]. The average loading data (⁠$GRFx$⁠, $GRFy$⁠, and CoP), normalized by body size according to Ref. [48], were used to calculate the deflection of the foot. At multiple instances throughout the stance phase, the deflection of the prosthetic foot is calculated by applying the instantaneous expected loading (GRFs and CoP) to the constitutive model of the prosthetic foot. This deflection was then propagated to predict the hip center location at that instance, assuming that the pylon, knee, and socket are rigidly attached to each other. At the end, the prosthetic foot performance was evaluated by comparing the predicted hip location to reference target data. The able-bodied hip center motion data (⁠$xhip$ and $yhip$⁠) were used as the target reference motion (Eq. (2)). The HTE framework requires body mass, leg length, ankle height, and foot length to be defined for each person to customize the foot to their body. Nylon 6/6 was selected as the material for HTE feet for the same reasons it was chosen for LLTE feet; namely, low-cost, high strain-energy density, ease of manufacturing, and consistency for comparison with previous LLTE work [37–39]. The optimization uses the following Nylon 6/6 material characteristics as optimization parameter inputs: tensile Young's modulus $E=3.15 GPa$⁠, tensile yield stress $σy=92.7 MPa$⁠, Poisson ratio $ν=0.4$⁠, and density $ρ=1150 kg/m3$⁠. Based on the normal walking loads experienced by able-bodied subjects [21,49], a safety factor on maximum stress of 1.75 was chosen. This is the same safety factor used for LLTE feet in past studies [37–39], selected for consistency to allow direct comparison.

The reference kinetic and kinematic data, as well as the chosen material properties, are entered into a custom structural analysis algorithm in MATLAB (Mathworks, Natick, MA), where the deflection of the two-dimensional prosthetic foot structure is calculated using a finite element analysis [37,38]. Using the built-in genetic algorithm, a population of solutions is evaluated using the HTE framework objective (Eq. (2)). The Bezier coefficients that define a foot shape (Fig. 3) that result in the lowest HTE score while satisfying the constraints, such as maximum stress and a shape within a standard foot outline, are chosen as the optimal solution.

The deflection of the foot and the resulting location of the hip center is calculated for each evaluated frame, or a point in time throughout the stance phase. This process is computationally expensive, therefore, it is beneficial to minimize the number of frames analyzed per foot. The number of evaluated frames was chosen based on the Nyquist sampling theorem, where it was assumed that walking has a frequency of 2 Hz [50]. The analysis showed that a minimum of six frames are required throughout the stance phase (heel strike to toe off) to accurately replicate the walking signal with a safety factor of two. In the HTE optimization, additional stance frames were included for a more detailed performance prediction, resulting in the final frames being: 8%, 15%, 22%, 29%, 36%, 43%, 50%, 57%, 64%, and 68% of the stance phase while keeping the optimization time under 1.5 h.

The design of the prosthetic foot must integrate with the prosthetic knee used, requiring the foot to fit within the available space below the length of the prosthetic knee. One of the foot variables that can be set for each subject is the ankle height or the height of the prosthetic foot $hankle$ (Fig. 3). A design space exploration was performed to understand how changing the ankle height affects the resulting optimal HTE score. A taller ankle height could allow for larger displacements within the foot, hence providing more shock absorption. However, the ankle height setting needs to account for the available height based on the length of the knee mechanism distal to the knee rotation axis, which can vary across knee designs. For example, the Ottobock Genium X3 and C-leg have lengths of 310–514 mm and 289–534 mm, respectively. This is substantially longer than mechanical polycentric knees from Ottobock, which are 62 mm long [51]. Anthropometric data show that the length of a human shank ranges from 345 to 488 mm [52], which suggests that the ankle height might be restricted by the length of the chosen prosthetic knee. Therefore, the first goal of the design variables analysis was to determine how the HTE score is affected by ankle height. Additionally, the HTE framework uses a genetic algorithm, which is a stochastic process. Therefore, the optimization output is not necessarily the global minimum. Multiple optimizations are required to increase the chance that the resulting prosthetic foot shape has an HTE score close to global optimal. Thus, the second goal of this analysis was to identify how many times the optimization needs to be performed to achieve consistent HTE scores.

The design variables analysis was performed for each combination of heights 1.6 m, 1.7 m, and 1.8 m and body masses 60 kg, 70 kg, and 80 kg. These ranges were chosen based on the height and body mass distribution of adult individuals around the world [53]. The results are presented for an adult subject of 70 kg body mass and 1.7 m height; the effects of ankle height for larger and lower body mass and heights are presented in  Appendix. To investigate the number of optimization runs required based on the second goal of the study, the optimization was run five times for each subject and each ankle height from 0.08 m to 0.15 m with 0.005 m intervals.

The resulting HTE score distribution for each ankle height is presented in Fig. 4. The data suggest that independent of ankle height, the lowest optimal HTE score achievable was in the range of 0.01–0.015. Examples of designs corresponding to different ankle heights are presented with Fig. 4 (A–D), demonstrating four resulting foot shapes in the sagittal plane. Some characteristics of the foot shapes are consistent between all designs, such as the heel portion that is longer that the toe and a generally straight and vertical keel. However, some shape changes can result in a different biomechanical performance, which is captured in the HTE score. This is demonstrated with optimized shapes A and B (Fig. 4), where the same ankle height of 0.08 m can have significantly different HTE scores (0.033 and 0.012 for A and B, respectively). In the comparison of A and B cases, the longer heel and a more horizontal keel portion in design A resulted in a worse predicted performance. For both designs C and D, the optimized shape resulted in a thicker toe, which suggests a less compliant behavior in late stance.

The variation across HTE scores from different optimization runs decreases as ankle height increases (Fig. 4). The results of this analysis suggest that below an ankle height of 0.1 m at least five optimizations are required to find an optimal foot shape closer to global minima, whereas for ankle heights above 0.1 m 2–3 optimization runs are sufficient. Similar trends were observed for other heights and body masses ( Appendix).

It is hypothesized that HTE prosthetic feet will result in a better biomechanical performance in people with an above-knee amputation than a prosthetic foot for a person with a below-knee amputation. In this study, better biomechanical performance would be evidenced by closer replication of able-bodied hip motion and smooth progression of the leg through the stance phase, replacing the functionality of early stance knee flexion. To elucidate the potential advantages of prosthetic feet designed specifically for people with an above-knee amputation, a simulation study compared the performances of feet designed using the LLTE framework and the HTE framework. LLTE framework was chosen to represent the performance of a prosthetic foot designed for a person with a below-knee amputation due to the predictive and quantitative characteristics of the framework. The goal of the simulation case study was to evaluate how LLTE and HTE feet would affect the operation of the prosthetic knee and the gait of a person with an above-knee amputation.

Lower leg trajectory error and HTE prosthetic feet were optimized for a subject of height 1.70 m and body mass 70 kg (Fig. 5(c)), which corresponds to the average subject evaluated in Sec. 3.3 design space exploration. The subject's leg length, lower leg length, and foot lengths were calculated to be 0.9 m, 0.485 m, and 0.26 m, respectively, by scaling anthropometric data on the lengths of body segments for a standard human [49]. Five optimizations were run for both the LLTE and HTE objectives, and the resultant foot design with the lowest corresponding objective scores was chosen for each. The resulting optimal shapes of the LLTE and HTE feet have three major shape differences: the HTE foot resulted in a longer heel, thicker toe portion, and thinner keel compared to the LLTE foot. The longer heel adds compliance during heel strike and shock absorption that would otherwise be provided by early-stance knee flexion in below-knee amputee and able-bodied gait. In the case of the LLTE prosthetic foot, the hip center locations were calculated as the extension of the lower leg, assuming that the knee remains locked during early and midstance for above-knee amputee users (Fig. 5(b)). The hip trajectory evaluation was only done for the duration of early and midstance knee flexion, evaluating time points within heel strike to initiation of late stance and swing knee flexion, or 0–68% of stance phase. This time segment was chosen based on the ideal operation of knee mechanisms [33,54]; after approximately 68% stance phase the prosthetic knee becomes unlocked to initiate flexion for the swing phase, and the foot and knee start to interact with each other. To evaluate the performance of each foot for enabling replication of the desired hip motion, an HTE score was calculated for both of the prosthetic feet for target able-bodied hip center motion shown in Fig. 5(a).

The simulation case study demonstrates that HTE prosthetic feet will result in better replication of able-bodied hip motion than LLTE prosthetic feet (Fig. 6). The results in Fig. 5(b) show the hip center location predicted by the finite element analysis based on the compliant behavior of the foot. Based on the predicted hip motion, the calculated HTE score (Eq. (2)) for the LLTE-objective prosthetic foot is approximately six times higher than the HTE score for the HTE-objective prosthetic foot (0.064 compared to 0.011 for LLTE and HTE feet, respectively). HTE feet showed a more uniform distribution of the hip location throughout the early and midstance phase (Fig. 5(b)), suggesting a smooth progression and transition into the swing phase. The effect of the framework objective function choice on predicted gait is especially visible at the point where able-bodied peak knee flexion during early and midstance would typically occur (Fig. 5(c)). LLTE prosthetic feet are designed with the goal of replicating able-bodied knee location and lower leg orientation. This results in the predicted hip center for above-knee amputees, who typically do not display early stance flexion, being substantially anterior compared to the predicted hip motion when using an HTE prosthetic foot. Overall, the results suggest that HTE prosthetic feet designed specifically for above-knee amputee gait would result in significantly better biomechanical performance for people with an above-knee amputation.

This paper introduces the HTE framework for quantitative and predictive design of prosthetic feet for people with an above-knee amputation. The HTE framework is an adaptation of the LLTE framework, which has successfully designed prosthetic feet that enable accurate biomechanical performance for people who have a below-knee amputation [37–41,44]. Unlike the LLTE framework, which promotes able-bodied knee location and lower leg orientation throughout stance, the HTE framework accounts for the fact that the majority of people with an above-knee amputation do not flex their knee during early and midstance. The lack of early and midstance knee flexion can result in insufficient shock absorption unless accounted for in other components of the prosthetic limb. The HTE framework provides a design of a compliant foot that enables close replication of able-bodied hip motion, which is correlated with sufficient shock absorption. Providing sufficient shock absorption may facilitate walking speeds closer to able-bodied gait [24] in above-knee amputees and could prevent secondary health issues associated with using above-knee prostheses [3]. Similar to LLTE feet, HTE feet can also be qualified as energy storage and return prosthetic feet, which in previous studies have been shown to promote walking symmetry for people with an above-knee amputation [31,32].

To the knowledge of the authors, this is the first prosthetic foot design metric that is specific to the needs of people with above-knee amputations. Based on the simulated performance demonstrated in the case study presented, HTE feet are likely to provide better biomechanical performance than prosthetic feet designed for people with a below-knee amputation, such as LLTE feet. Specifically, the resulting simulated hip motion associated with using HTE feet replicated the able-bodied target motion six times better than LLTE feet based on the HTE score. This suggests that HTE prosthetic feet are predicted to provide a smoother progression of the hip center through the stance phase and into the swing phase, motivating the need for experimental investigation. Overall, this paper not only demonstrates that existing prosthetic design methods and goals will likely result in undesirable prosthetic leg performance for above-knee amputees but also proposes a novel performance evaluation metric that is based on the needs of people with an above-knee amputation.

Lastly, the HTE framework can be extended to design rehabilitation walking devices where the knee does not flex during stance. For example, the HTE framework could be used to design a compliant foot for hands-free knee braces, as knee braces do not flex during stance and conventionally have a noncompliant foot attachment [55]. Therefore, the HTE framework has broader implications beyond prostheses.

This paper describes a method to design and optimize a prosthetic foot architecture based on the HTE framework objective that might improve the mobility of people with an above-knee amputation. The model incorporates the characterization of the compliant behavior of the foot to predict the motion of the prosthetic leg. This is an enhancement on the limitations of previously published models that used a simple inverted pendulum model for the prosthetic foot performance evaluation during the stance phase [56]. The HTE framework enables prosthesis designers to optimize a prosthetic foot for a target gait pattern for a patient based on their body parameters, such as body mass, leg length, and foot length. This paper also demonstrates how the resulting predicted performance is affected by changes in ankle height and number of optimizations conducted. These results can be used by prosthetic foot designers not only to customize the foot design based on users' body parameters but also to ensure that the foot design fits well with the chosen prosthetic knee mechanism.

Prosthetic feet designed using the HTE framework incorporate functions that are important for the performance and simplification of the entire prosthetic leg. The design of a prosthetic foot directly impacts the performance of the entire prosthetic leg, especially in how the foot interacts with the knee mechanism. The majority of prosthetic knees remain locked during early and midstance, with flexion starting in late stance into swing phase to promote toe clearance. The HTE framework allows for the prediction of the location and orientation of the knee mechanism throughout early and midstance, which can be used to design the knee mechanism to ensure knee stability before the initiation of late stance knee flexion. Furthermore, accurate replication of CoP progression and GRFs orientation is critical to facilitate a smooth and timely transition into swing knee flexion, especially for passive prosthetic leg designs [33–35,57]. It is hypothesized that HTE prosthetic feet will not only promote able-bodied hip motion but will also replicate able-bodied loading at the foot based on past LLTE studies [37–39]. Providing accurate replication of able-bodied loading could prevent untimely unlocking and initiation of flexion, which can cause injuries from falls and stumbles [3,36]. Furthermore, by providing shock absorption through the foot design, the design of the rest of prosthetic leg components can be simplified, eliminating the need for complex early stance springs or shock-absorbing pylons. The reduction of components and their complexity could improve the prosthesis fitting process, resulting in more time for acclimation and training.

The prosthetic feet predictively designed with the HTE framework could be especially beneficial in low and middle-income countries. Based on literature data, the rates of above-knee amputation are higher in low-resource settings, where lengthy fitting and alignment iterations may be perceived as cumbersome or be impractical based on prosthetist availability and distance to clinics for amputees [57]. The developments in LLTE work, such as design for low-cost mass-manufacturing for low and middle-income countries [45,46], can be applied to the HTE framework. Applying these insights would allow HTE feet to be distributed to users at a low cost, therefore providing the large population of people with an above-knee amputation with better mobility and quality of life.

The HTE framework has multiple limitations due to assumptions made. First, the HTE framework assumes that closely replicating able-bodied hip motion at a self-selected comfortable walking speed will ensure that sufficient shock absorption is achieved. This assumption is based on experimental results in existing literature, where it is suggested that deviation in pelvic obliquity is correlated to the lack of shock absorption during the loading of the prosthetic foot [21,24]. In these studies, it was noted that the lack of shock absorption could have resulted in slower walking speeds. While we acknowledge that people with above-knee amputation tend to select slower walking speeds, we felt it pragmatic to use able-bodied comfortable walking speed as a reference for this study since an “ideal” reference biomechanical dataset is unknown. It was assumed that promoting able-bodied hip center progression would result in sufficient shock absorption, which in turn could result in faster walking speed [21,24]. Experimental studies and user feedback will be required to confirm whether using able-bodied walking speed in the HTE framework is appropriate or whether a different target walking speed would be perceived as more comfortable by people with an above-knee amputation. A benefit of the HTE framework is that it can be used to design a foot to replicate any desired reference walking speed and motion of the hip. If future experimental studies suggest the need, a different reference dataset can be used with the HTE framework, including able-bodied hip motion at slower speeds or hip motion from a dataset for below-knee amputees.

The HTE framework presented in this paper aims to improve biomechanical walking performance for people with an above-knee amputation by providing smooth hip progression throughout the stance and shock absorption in early stance. However, there are other aspects of gait biomechanics not explored in this study that could also improve mobility, such as adding coronal compliance or minimizing circumduction and vaulting. Exploring other design goals and adding other functions, such as coronal compliance, to the prosthetic foot may also be beneficial for meeting the needs of users with above-knee amputations. Future work can update the objective function of the framework accordingly and still leverage the compliant behavior analysis outlined in this paper.

The other important assumption is the input of able-bodied kinetics (⁠$GRFx$⁠, $GRFy$⁠, and CoP) to optimize the prosthetic foot shape. The rationale for this assumption stems from the results of the LLTE experimental studies, which have shown that feet produced using a similar predictive design framework enable close replication of able-bodied kinetics in the gait of people with a below-knee amputation [38]. This could be a result of the optimized constitutive behavior of the prosthetic foot, which connects the motion of the foot to the loading applied. Since the HTE framework uses the same approach, it is hypothesized that a participant will similarly accurately replicate the kinetics as in LLTE studies, though this assumption needs to be validated in a experimental setting.

The presented design optimization uses the same number and definition of Bezier curve coefficients and shape bounds that were previously tested for LLTE feet [37–39]. Changes in the shape definition could result in better predicted performance of HTE feet by expanding the solution space. However, the purpose of the simulation case study was to specifically test the efficacy of the objective functions (Eqs. (1) and (2)) for designing prosthetic feet for people with an above-knee amputation. Therefore, it was decided to keep the shape input parameters consistent between the two design frameworks to not introduce additional confounding variables when analyzing differences in the predicted walking performance. Future work can expand the shape definition variable bounds to explore if better performance can be achieved.

The simulation case study presented in this paper demonstrates potential advantages of using the HTE framework over current methods, however, it is not a replacement for experimental studies with people with an above-knee amputation. We are using these simulation results to establish a new methodology and motivate future experimental studies, an approach commonly used in the field of assistive device technology [40,43,58–62]. In this paper, the simulation showed substantially improved hip motion replication when using HTE feet compared to LLTE feet, supporting the need for a design framework for people with an above-knee amputation. Future in vivo experiments with above-knee prosthesis users will validate the simulation results and quantitatively demonstrate the efficacy and practical utility of the HTE framework.

Additionally, the simulation relied on the assumption of able-bodied loading of the prosthetic foot (⁠$GRFx$⁠, $GRFy$⁠, and CoP); however, as stated previously in the LLTE background section, people tend to replicate able-bodied motion rather than able-bodied loading. As both LLTE and HTE frameworks connect the loading to the kinematic behavior of the prosthetic leg, it is likely that people with an above-knee amputation using LLTE prosthetic feet would lead to deviation in the loading rather than in hip motion. Erroneous loading can affect the function of the knee mechanism and would further demonstrate the need for prosthetic feet designed specifically for above-knee amputees. Future experiments should compare the contributing errors between kinetic (⁠$GRFx$⁠, $GRFy$⁠, and CoP) and kinematic (⁠$xhip$ and $yhip$⁠) variables to validate this hypothesis.

Lastly, there are some limitations of the HTE framework that could affect the user's satisfaction. Lack of knee flexion during stance might be noticeable during walking, which would go against the need of being perceived as able-bodied, common for users living in low and middle-income countries [4,5,63,64]. Though the majority of above-knee prosthesis users do not exhibit early and midstance knee flexion [19], a perception study is required to identify what gait deviations would be perceived as typical gait. Finally, the addition of the shock absorption within the foot might initially feel unstable to the user, though it will likely be less unstable than the knee flexion mechanisms. An experimental study with subjective feedback collection is required to assess the user's perception of the stability of HTE prosthetic feet.

This paper proposes a novel HTE framework for designing prosthetic feet specifically for people with an above-knee amputation. The HTE framework takes into account common gait deviations for users of above-knee prostheses, such as lack of knee flexion during early and midstance phase. This gait deviation results in the loss of shock absorption during the loading response, which can result in potential health issues for the user or further gait deviations from able-bodied. The HTE framework incorporates the shock absorption function within the prosthetic foot design by targeting able-bodied hip center motion, which is correlated with identifying whether sufficient shock absorption was achieved during the stance phase. The study presents an optimization and performance evaluation process resulting in a prosthetic foot structure that not only closely replicated able-bodied hip center motion but could be manufactured for a low-cost. An additional design space exploration provides insights on how the HTE score, and therefore the predicted biomechanical performance, is affected by ankle height and the number of optimizations performed. Lastly, a simulation case study shows the advantages of using an above-knee-specific objective, such as hip motion replication and predictable behavior for interaction with the prosthetic knee, to design prosthetic feet compared to an objective set based on the needs of people with a below-knee amputation. The HTE framework provides a method to design and evaluate prosthetic feet designs for the use of people with above-knee amputations, which could provide 5–17 million people around the world with high-performance, low-cost prosthetic feet that are designed to interact properly with prosthetic knees.

The author would like to thank GEAR Lab members for proofreading and helping with the writing process. Also the author would like to thank Victor Prost for his help in understanding the LLTE framework to adjust it for people with an above-knee amputation. Lastly, we would also like to thank the National Science Foundation (Award No. 1653758) and MathWorks for providing funding for the project.

• National Science Foundation (Award No. 1653758; Funder ID: 10.13039/100000001).

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

Figure 6 shows how ankle height changes affect the optimal HTE scores for each combination of heights 1.6 m, 1.7 m, and 1.8 m and body masses 60 kg, 70 kg, and 80 kg. These ranges were chosen based on the height and body mass distribution of adult individuals around the world [53]. The trends are the same as the represented in Fig. 4 for a single subject of the body mass 70 kg and height 1.7 m in Sec. 3.3. Additionally, the trend of more optimization runs required for lower ankle heights is consistent for the body mass and height ranges. Similar to the example presented in Fig. 4, at least five optimizations need to be run to find a prosthetic foot shape with an HTE score close to the general optimal minima of 0.01–0.015.