Efficient and Scalable Tuning of Continuous Impedance Control for Powered Knee Prostheses

Abstract

Powered knee prostheses based on continuous impedance control enable smooth and human-like joint behavior; however, practical tuning remains challenging due to the high dimensionality of impedance functions and subject-dependent variability. In this work, we propose a structured low-dimensional tuning framework in which continuous stiffness, damping, and equilibrium-angle functions are first identified offline from able-bodied walking data using kinematics-informed convex optimization and then parameterized through principal component analysis (PCA). The resulting PC weights serve as interpretable tuning variables, enabling systematic and scalable adjustment of prosthetic knee behavior. Sensitivity analysis revealed that stiffness-related parameters exert the strongest and most consistent influence on knee kinematics, motivating their prioritization during tuning. The framework was experimentally validated through treadmill walking trials involving able-bodied individuals and an amputee participant by adjusting stiffness-related PC weights to achieve multiple predefined target knee profiles. Results demonstrate that diverse target behaviors can be reliably achieved using structured low-dimensional tuning. This study establishes PCA-based parameterization as an effective strategy for simplifying continuous impedance tuning and provides a principled foundation for personalized prosthetic control, in which target behaviors may be defined through user feedback, clinician input, or adaptive learning.

I. Introduction

Lower-limb amputation is a life-altering disability that significantly impacts individuals both physically and psychologically [1]. The incidence of major lower-limb amputations continues to rise globally, with projections indicating a substantial increase in the coming decades [2], [3]. The resulting mobility loss increases the risk of falls and secondary injuries among amputees [4]–[6]. To address this, lower-limb prostheses have evolved significantly and are widely used to restore mobility, stability, and participation in daily activities [4], [7]. Powered prostheses have specifically been studied for decades as a potential solution to achieving more natural gait patterns across various walking conditions [8]–[10]. These advanced devices also offer clinical benefits such as improved gait symmetry [6], [8], [11], reduced energy expenditure [12]–[14], and increased mobility [15], [16].

Humans perform various lower-limb tasks by dynamically modulating joint impedance, which allows for adaptable and stable interactions with the environment, an essential factor for safe and efficient locomotion [17]–[19]. For human-robot symbiosis in powered prostheses, robotic joint behavior should exhibit biomechanically consistent characteristics that synchronize with human movement across ambulation activities [20]–[22]. Impedance control is one approach for mimicking human ambulation, as it adjusts lower-limb joint impedance in response to the user’s gait progression, facilitating compliant interaction with the ground [23], [24]. Traditional impedance controllers rely on finite-state machine architectures that divide the gait cycle into discrete states (e.g., 4–5 states), each governed by its own impedance parameters (i.e., stiffness, damping, and equilibrium angle) [23], [24]. These parameters can be fine-tuned to provide tailored control based on user feedback, clinical observation, or specific ambulation tasks (e.g., ramp and stair walking) [25]–[27]. However, as the number of gait states and ambulation modes increases, the tunable parameter space grows rapidly, making subject-specific tuning time-consuming and clinically challenging. Moreover, this control approach employs discrete, piecewise impedance segments, which may limit smooth transitions across gait phases and introduce abrupt changes (i.e., jerkiness) in prosthesis mechanics [23], [28].

Continuously varying impedance control has emerged as a promising strategy for powered prostheses [8], [9], [29], [30]. One key challenge with continuous joint impedance, however, is estimating human joint impedance in a continuous manner. Early studies estimated human joint impedance empirically using perturbation systems, revealing piecewise trends over the gait cycle [31], [32]. Subsequent studies introduced continuously varying impedance profiles using optimization-based approaches to enable seamless, human-like prosthesis control across walking speeds and slopes [29], [30]. These frameworks have since been extended to tasks such as sit-to-stand transitions [8] and stair ambulation [9].

Despite their versatility, generalized models used to characterize joint impedance often lack the flexibility required for personalization, as their parameters are typically predetermined through offline optimization and do not account for inter-individual variability in physiological traits and preferences. Given that nearly 68% of major lower-limb amputees wear their prosthesis for over seven hours daily [7], reliance on inadequately personalized assistance may compound adverse effects over time [6], highlighting the need for tailored support in daily activities. Furthermore, these models are typically derived from data collected from able-bodied (AB) individuals, which may not fully capture the physiological differences between amputee users and the healthy population. These considerations highlight the importance of preserving room for systematic tuning, enabling impedance behavior to be adapted to the needs of individual amputee users rather than enforcing a single generalized solution [14], [25]. While recent studies have explored user-specific adaptation by optimizing stance-phase control behavior based on performance metrics from individual users [33], such approaches differ from the present work in that they do not explicitly address systematic tuning of continuous impedance functions defined over the full gait cycle.

A key remaining challenge lies in effectively characterizing continuous impedance curves and determining how to adjust them in a structured and interpretable manner for individual users. However, achieving such structured tuning is nontrivial, as continuous impedance functions are typically high-dimensional. High-dimensional tuning spaces hinder both manual adjustment and even automated optimization strategies, such as human-in-the-loop optimization [34], [35] or reinforcement learning [36], [37]. As the number of joints or ambulation modes increases, the parameter space grows rapidly, making efficient exploration and adaptation increasingly challenging.

Thus, we propose parameterizing continuous impedance functions using Principal Component Analysis (PCA) to enable structured and low-dimensional personalization of continuous impedance control for powered prostheses. By capturing principal components (PCs) of variation across impedance curves derived from human walking data, PCA provides an interpretable representation of stiffness, damping, and equilibrium functions while preserving shared physiological characteristics [38], [39]. The objective of this study is to demonstrate the feasibility of this PCA-based parameterization for interactive tuning of continuous impedance control. The main contributions of this work are: 1) the development of a human-inspired continuous impedance model using kinematics-informed convex optimization, 2) the introduction of a PCA-based low-dimensional tuning scheme, and 3) experimental validation of the proposed framework during treadmill walking. The remainder of this paper is organized as follows. Section II presents the continuous impedance modeling and PCA-based parameterization (Fig. 1a–c). Section III describes the prosthesis system and experimental protocol (Fig. 1d–e). Section IV reports the results, which are discussed in Section V. Section VI concludes the paper with a summary of key findings.

Fig. 1.

Overview of the tuning scheme. (a-b) Individual impedance functions (i.e., stiffness, damping, and equilibrium) are generated via convex optimization using knee kinematics from 20 AB participants walking at 0.5–0.9 m/s. (c) Two PCs capturing more than 99% of the variance are extracted from each impedance function and used as tuning parameters. (d-e) Controller tuning is conducted by adjusting the given parameters as the user walks with the device.

II. Methodology

A. Human-Inspired Continuous Impedance Model

1). Data Inclusion:

To construct a continuous impedance model (i.e., stiffness, damping, and equilibrium angle) for the knee joint, we utilized an open-source dataset [40], which contains multimodal walking data from 22 AB participants at various walking speeds. For this study, we specifically used treadmill walking data at speeds of 0.5 to 0.9 m/s from 20 AB participants, shown in Fig. 1a. Data from two participants were excluded based on the predefined criterion that any joint position, velocity, or torque trajectory deviating by more than ±2 standard deviations (SD) from the mean at any point in the gait cycle was considered an outlier.

2). Kinematic-Informed Convex Optimization:

The human-inspired continuous impedance model was developed based on the impedance control framework, formulated as follows:

$$\hat{\tau }=K(\varphi )\cdot \left(\theta -{\theta }_{eq}(\varphi )\right)+D(\varphi )\cdot \dot{\theta }$$ (1)

, where $K(\varphi )$, $D(\varphi )$, and ${\theta }_{eq}(\varphi )$ represent the joint stiffness ($\mathrm{N}\cdot \mathrm{m}\cdot {\mathrm{k}\mathrm{g}}^{-1}\cdot {\mathrm{d}\mathrm{e}\mathrm{g}}^{-1}$), damping ($\mathrm{N}\cdot \mathrm{m}\cdot \mathrm{s}\cdot {\mathrm{k}\mathrm{g}}^{-1}\cdot {\mathrm{d}\mathrm{e}\mathrm{g}}^{-1}$), and equilibrium angle (deg) functions, respectively. The phase variable $\phi$ is defined as a time-based, normalized gait phase that progresses linearly from 0 to 1 over each gait cycle between consecutive heel-strikes. This phase variable represents the user’s gait progression and allows the impedance functions to vary continuously throughout the gait cycle. $\theta$ and $\dot{\theta }$ refer to joint angle and velocity of the prosthesis.

$$K(\varphi )=\sum _{i=0}^4{k}_i{\varphi }^i,D(\varphi )=\sum _{i=0}^4{d}_i{\varphi }^i,{\theta }_{eq}(\varphi )=\sum _{i=0}^6{e}_i{\varphi }^i.$$ (2)

We defined the stiffness and damping functions as fourth-order polynomial functions and the equilibrium angle function as a sixth-order polynomial in (2). The polynomial order was selected to balance model flexibility and smoothness while avoiding overfitting. The polynomial coefficients for stiffness, damping, and equilibrium angle, denoted as $k_i,d_i$, and $e_i$, respectively, were estimated using a least-squares method. This method fits the generated torque $\hat{\tau }$ from (1) to the body-mass-normalized human knee joint torque $\tau$ provided by Camargo et al. across $n$ data points [40]. As shown in (3), the optimization problem was formulated to identify subject-specific continuous functions of stiffness, damping, and equilibrium angle. For each AB individual, multiple walking trials (over 15 gaits) were used in the optimization to account for intra-subject gait variability, resulting in a single set of continuous impedance functions per individual.

$$\underset{k_i,d_i,e_i}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{1}{n}\Vert \hat{\tau }-\tau {\Vert }_2^2$$ (3a)

$$\text{subject}\text{to:}0.50\le K(\varphi )\le 5.0$$ (3b)

$$0.01\le D(\varphi )\le 1.0$$ (3c)

$${\theta }_m-2{\theta }_{SD}\le {\theta }_{eq}(\varphi )\le {\theta }_m+2{\theta }_{SD}$$ (3d)

, where (3b) and (3c) refer to the lower and upper boundary constraints of stiffness and damping, which are heuristically determined based on physiological feasibility, stability, and hardware constraints [29], [30]. We used the human knee kinematics envelope (i.e., ${\theta }_m\pm 2{\theta }_{SD}$) as a boundary constraint for constructing the equilibrium angle model in (3d), ensuring consideration of human physiological traits. Note that ${\theta }_m$ and ${\theta }_{SD}$ refer to the mean and SD of 20 AB individuals’ knee profiles, respectively [40]. The given least-squares fit problem makes this process non-convex due to the nonlinear term of $K(\varphi )\cdot {\theta }_{\text{eq}}(\varphi )$ in (1), so we reformulate this problem to convex by introducing a new polynomial that has higher order and treats it as an independent term, as follows [30], [41]:

$$K(\varphi )\cdot {\theta }_{eq}(\varphi )=\sum _{i=0}^{10}{q}_i{\varphi }^i=Q(\varphi ).$$ (4)

Then, (1) becomes a linear combination of the polynomial coefficients defining $K(\varphi )$, $D(\varphi )$, and $Q(\varphi )$, allowing (3) to be treated as a convex optimization problem and reformulated into a quadratic form as shown below:

$$x={\left[k_0,\dots ,k_4,d_0,\dots ,d_4,q_0,\dots ,q_{10}\right]}^T$$ (5)

$${\alpha }_i={[{\theta }_i{\varphi }_i^0,\dots ,{\theta }_i{\varphi }_i^4,{\dot{\theta }}_i{\varphi }_i^0,\dots ,{\dot{\theta }}_i{\varphi }_i^4,-{\varphi }_i^0,\dots -{\varphi }_i^{10}]}^T$$ (6)

, where $i\in [1,n]$, referring to a data point. The final objective function from (3) becomes

$$\frac{1}{n}\sum _{i=1}^n{\tau }_i^2-f^{T}x+\frac{1}{2}{x}^{T}Hx$$ (7)

, where $f=\frac{2}{n}\sum _{i=1}^n{\tau }_i{\alpha }_i$ and $H=\frac{2}{n}\sum _{i=1}^n{\alpha }_i{\alpha }_i^T$. To solve this problem, we used the quadprog function in MATLAB (R2023b, MathWorks, Natick, MA, USA). After solving the convex reformulation, ${\theta }_{eq}(\varphi )$ is represented as a sixth-order polynomial consistent with the identified impedance structure.

B. PCA-Based Impedance Function Tuning

As illustrated in Fig. 1b, a set of continuous impedance functions (i.e., $K(\varphi )$, $D(\varphi )$, and ${\theta }_{\mathit{eq}}(\varphi )$) can be generated through offline convex optimization for each individual. These subject-specific impedance functions are obtained under the physical constraints in (3), ensuring that stiffness and damping remain within physiologically and hardware-feasible bounds. Since these functions were derived from AB individuals, additional tuning may be required to accommodate amputees’ gait characteristics and preferences [25]. To address this, we aim to develop a system that enables personalized control through fine-tuning. While each individual exhibits different gait patterns, there are shared characteristics due to common human physiological traits [38], [39]. To capture these shared features, we perform PCA separately for each impedance function (i.e., stiffness, damping, and equilibrium angle) during the offline process [41]. The first two PCs from each function, which capture the majority of variance (see Fig. 1c), are retained as low-dimensional basis functions. During online operation, their corresponding weights ($W_{\mathrm{1,2}}^{k,d,eq}$) are adjusted to reconstruct the impedance functions for personalized assistance, as shown in Fig. 1d and (8).

$$K(\varphi )=W_1^k\cdot P{C}_1^k(\varphi )+W_2^k\cdot P{C}_2^k(\varphi )$$ (8a)

$$D(\varphi )=W_1^d\cdot P{C}_1^d(\varphi )+W_2^d\cdot P{C}_2^d(\varphi )$$ (8b)

$${\theta }_{eq}(\varphi )=W_1^{eq}\cdot P{C}_1^{eq}(\varphi )+W_2^{eq}\cdot P{C}_2^{eq}(\varphi )$$ (8c)

III. Prosthetic Walking Experiments

A. Powered Knee Prosthesis and Control System

A custom-built powered knee prosthesis (Fig. 2a) was employed to mimic human knee motion during walking [42]. This device features a passive ankle joint and an active knee joint driven by a DC motor (RE 40, Maxon Motor, Sachseln, Switzerland) through a slider-crank mechanism, enabling human-like knee movement. A potentiometer (RDC503013A, ALPS, Japan) and a 6-axis load cell (Mini58, ATI, NC, USA) were integrated into the prosthesis to measure knee angle, velocity, and dynamic loading. The control system was implemented on the TwinCAT software platform (Beckhoff Automation, Verl, Germany), offering a real-time operation at 1 kHz. Data acquisition was conducted using the C6 Router (KEB Automation, Barntrup, Germany), and the controller was built in Simulink (MathWorks, Natick, MA, USA). The prosthesis operated under a continuous impedance control framework using a time-based phase variable ($\varphi \in [0,1]$), which progressed linearly from 0% to 100% of the gait cycle between consecutive heel-strike events detected in real time. To improve robustness, each participant’s stride time was measured and used to determine the slope of the phase variable. This strategy enabled stable and repeatable gait phase estimation during treadmill walking. Based on six given weights ($W_{\mathrm{1,2}}^{k,d,eq}$), phase-dependent functions for stiffness, damping, and equilibrium angle were generated. These functions were used by the impedance controller (1) to compute joint-level torque commands using real-time feedback of knee position and velocity, ensuring responsive and adaptive assistance throughout the gait cycle.

Fig. 2.

Powered knee prosthetic system. (a) A custom-built powered knee prosthesis. Three participants were recruited: (b) one transfemoral amputee (TF01) and two AB individuals – (c) AB01 and (d) AB02.

B. Experimental Protocol

The experimental protocol consisted of three phases. The first phase was a training session for participant familiarization and prosthesis setup. The second phase involved sensitivity analysis conducted over multiple sessions to evaluate stiffness and damping modulation with equilibrium parameters held constant. The third phase focused on tuning validation, including baseline walking trials with fixed stiffness parameters ($W_1^k=0.30,W_2^k=-0.10$) followed by tuning to achieve the specified target knee trajectories.

1). Participants:

Three participants participated in this study: two AB individuals (AB01 and AB02) and one transfemoral amputee (TF01); all male, 30.3±2.9 years old, 172.2±3.5 cm, 73.9±5.7 kg. The AB participants used an L-shaped adapter to walk with the knee prosthesis, while the amputee participant connected the device using his daily-use socket. All participants completed training sessions to acclimate to the powered knee prosthesis until they could walk comfortably and confidently without handrails. Across three training visits, we identified each participant’s preferred treadmill walking speed with the prosthesis and measured stride duration to configure the time-based gait phase estimator. All experiments were conducted on a treadmill at each participant’s preferred walking speed (0.6 m/s for AB01 and AB02; 0.7 m/s for TF01), with safety ensured by handrails on both sides of the treadmill and a ceiling-mounted harness. Written consent was obtained from all participants prior to the experiments, and the study was reviewed and approved by the Institutional Review Board of North Carolina State University.

2). Sensitivity to Stiffness and Damping:

One of the primary objectives of this study was to examine how the proposed tuning parameters, specifically stiffness and damping, affect knee kinematics. Since the equilibrium parameters ($W_1^{\text{eq}}$ and $W_2^{eq}$) are directly related to kinematics, they are generally expected to have stronger effects than stiffness or damping. To isolate the contribution of stiffness and damping, we fixed the equilibrium angle parameters at $W_1^{eq}=0.278$ and $W_2^{eq}=-0.231$, determined through optimization to minimize the root-mean-square error (RMSE) between the equilibrium angle trajectory and normative knee kinematics [36].

Stiffness parameter validation:

To evaluate the influence of stiffness, we conducted two modulation sessions:

• $W_1^k$ varied from 0.1 to 1.0 in increments of 0.1, with other parameters fixed: $W_2^k=0.0,W_1^d=0.6,W_2^d=0.0$.

• $W_2^k$ varied from −1.0 to 1.0 in increments of 0.2, with other parameters fixed:$W_1^k=0.5,W_1^d=0.6,W_2^d=0.0$.

Parameters were updated every minute in randomized order, with at least 25 gait cycles per condition to account for user adaptation. A five-minute break was provided at the midpoint of each session to mitigate fatigue.

Damping parameter validation:

To evaluate damping, we followed a similar procedure:

• $W_1^d$ varied from 0.1 to 1.0 in increments of 0.1, with other parameters fixed: $W_1^k=0.5,W_2^k=0.0,W_2^d=0.0$.

• $W_2^d$ varied from −1.0 to 1.0 in increments of 0.2, with other parameters fixed:$W_1^k=0.5,W_2^k=0.0,W_1^d=0.6$.

The same timing, randomization, and rest procedures as in the stiffness validation were applied.

3). Parameter Tuning Validation:

Building on the sensitivity analysis, which demonstrated that stiffness parameters exert a dominant influence on knee kinematics relative to damping, tuning validation focused on stiffness modulation. To isolate the effects of stiffness tuning, damping parameters were fixed across all conditions. Equilibrium parameters were specified separately for each target trajectory prior to tuning, as they define the baseline knee kinematics for that reference.

Basic reference:

Stiffness parameters were first tuned to reproduce a normative knee profile (NP), suggested by the physical therapist [36]. The equilibrium parameters for NP were set to $\left(W_1^{eq},W_2^{eq}\right)=(0.278,-0.231)$, determined via optimization to minimize the RMSE against the target.

Additional references:

To further examine generalizability, two additional reference profiles were introduced for TF01: the average knee kinematics of 20 AB individuals [40] (ABP), and TF01’s intact-side knee profile using his own prosthetics, obtained from motion capture data (IP). For each reference, equilibrium parameters were re-optimized to match the new target trajectory, resulting in $\left(W_1^{eq},W_2^{eq}\right)=(0.227,-0.726)$ for ABP and (0.186, –0.502) for IP.

As the baseline, participants walked for two minutes with an initial physiologically feasible parameter set $W_1^k\left.=0.30,W_2^k=-0.10\right)$, selected from the population-level PC weight distribution derived from PCA of continuous impedance functions identified from 20 AB individuals [40]. In the subsequent sessions, stiffness tuning was initiated from this baseline to match the given reference, guided by real-time comparison between measured knee kinematics and the target profile. Tuning was performed sequentially and informed by the sensitivity analysis, with parameters exhibiting stronger sensitivity adjusted first using relatively coarse step sizes, followed by finer adjustments as kinematic error decreased. Tuning continued until the stopping criterion was satisfied, defined as achieving average knee angles at three key gait events—stance flexion (STF), stance extension (STE), and swing flexion (SWF)—were within 5° of the reference trajectory. Control parameters were updated every 20 gait cycles to allow for user adaptation. Each tuning session was limited to five minutes to prevent fatigue, with five-minute rest breaks provided between sessions.

IV. Results

A. Effects of Stiffness and Damping Variations

Figure 3 summarizes the effects of varying stiffness and damping on knee kinematics during walking trials. As shown in Fig. 3a, $W_1$ primarily scaled the magnitude of stiffness or damping, while $W_2$ modulated the phase-dependent profile across the gait cycle. To quantify sensitivity, we modeled the relationship between parameter variations and peak joint angles at STF, STE, and SWF using linear mixed-effects fits. Fig. 3b presents only the parameters with sensitivities exceeding $3^{\circ }/\mathrm{\Delta }W$ (see Fig. 4). As seen, $W_1^k$ showed the strongest influence, with SWF increasing by approximately ${34}^{\circ }/\mathrm{\Delta }{W}_1^k$ and additional increases in STF (${6.16}^{\circ }/\mathrm{\Delta }{W}_1^k$) and STE (${22.62}^{\circ }/\mathrm{\Delta }{W}_1^k$). $W_2^k$ also influenced kinematics but with smaller slopes (SWF: ${6.81}^{\circ }/\mathrm{\Delta }{W}_2^k$, STE: ${3.91}^{\circ }/\mathrm{\Delta }{W}_2^k$) In contrast, damping parameters produced relatively minor, negative effects, slightly reducing flexion and extension peaks (e.g., $W_1^d$: SWF $-{3.17}^{\circ }/\mathrm{\Delta }{W}_1^d$, STE $-{3.54}^{\circ }/\mathrm{\Delta }{W}_1^d$). $W_2^d$ had the weakest influence, with sensitivities below $3^{\circ }/\mathrm{\Delta }{W}_2^d$ (see Fig. 4). These findings demonstrate that stiffness dominates the modulation of knee kinematics, while damping plays a secondary role. This motivated subsequent tuning experiments to focus primarily on stiffness while holding damping constant.

Fig. 3.

Effects of stiffness and damping parameter variations on knee kinematics. (a) Knee stiffness and damping profiles across the gait cycle with variations in $W_1^k$ (0.1 to 1.0), $W_2^k$ (-1.0 to 1.0), $W_1^d$ (0.1 to 1.0), and $W_2^d$ (1.0 to 1.0). (b) Sensitivity of knee kinematics at key gait events: stance flexion (STF), stance extension (STE), and swing flexion (SWF). Subject means (TF01: red; AB01: blue; AB02: yellow) are shown with ±SD, and linear mixed-effects fits (black) indicate slopes $\left(°/\mathrm{\Delta }\mathrm{W}\right)$ quantifying tuning sensitivity.

Fig. 4.

Summary of parameter sensitivity slopes for knee kinematics. Sensitivity was quantified as the change in joint angle per unit parameter variation $\left(°/\mathrm{\Delta }\mathit{W}\right)$ for SWF, STF, and STE. Bars represent mean slopes across participants with error bars denoting SD.

B. Parameter Tuning for Personalization

Figure 5 presents the knee stiffness derived from the initial stiffness PC coefficients ($W_1^k=0.30,W_2^k=-0.10$ for all participants) and the tuned values for each participant: TF01 ($W_1^k=0.55,W_2^k=-0.20$), AB01 ($W_1^k=0.38,W_2^k=-0.70$), and AB02 ($W_1^k=0.57,W_2^k=-0.10$). After tuning, each participant exhibited a distinct set of coefficients, resulting in individualized stiffness functions. We evaluated the tuned parameters by comparing the resulting gait patterns with the normative profile (NP; black dashed line in Fig. 5). As summarized in Table I, tuning performance was quantified using the RMSE and cross-correlation (Corr), computed for each gait cycle and averaged across 20 gait cycles. These metrics quantify the similarity between the participants’ knee kinematics and the reference trajectory.

Fig. 5.

Stiffness tuning to match the target normative profile (NP). Knee stiffness profiles and the resulting knee kinematics using initial parameters (green) and tuned parameters for each participant (TF01: red; AB01: blue; AB02: yellow) compared with the target NP (black dashed line). Shaded regions indicate ±SD across gait cycles.

TABLE I.

Tuning evaluation for the normative reference.

Metric	Participant	Initial	Tuned
RMSE	TF01	6.20 ± 0.97	4.58 ± 0.31
	AB01	11.53 ± 0.45	5.92 ± 0.63
	AB02	8.52 ± 0.30	5.77 ± 0.43
Corr	TF01	0.951 ± 0.009	0.972 ± 0.004
	AB01	0.757 ± 0.021	0.940 ± 0.010
	AB02	0.906 ± 0.016	0.939 ± 0.012

To further validate the framework, TF01 participant was tested with two additional target profiles: ABP and IP. As shown in Fig. 6, the tuned prosthetic knee motion was closely aligned with both references. The corresponding optimized stiffness and equilibrium parameters are reported in Table II, while the damping parameters were fixed at $(W_1^d,W_2^d)=(0.1,0.1)$. For both references, tuning resulted in RMSE values below 5° and correlation above 0.97, demonstrating the robustness and adaptability of the proposed personalization framework.

Fig. 6.

Tuned knee angle trajectories for additional target profiles. (a) TF01 tuned to the averaged knee profile of 20 AB individuals (ABP). (b) TF01 tuned to his intact limb knee profile obtained from motion capture data (IP). Dashed black lines indicate the target profiles, while colored lines show the tuned prosthetic knee motion (ABP: red; IP: blue).

TABLE II.

Optimized parameters and performance metrics for ABP and IP.

	ABP	IP
$W_1^k$	0.55	0.50
$W_2^k$	0.30	0.70
$W_1^{\mathit{eq}}$	0.227	0.186
$W_2^{\mathit{eq}}$	−0.726	−0.502
RMSE	4.48 ± 1.49	4.77 ± 0.43
Corr	0.973 ± 0.017	0.974 ± 0.006

V. Discussion

To investigate the sensitivity of tuning parameters, we modeled their effects on knee kinematics using linear mixed-effects fits, which capture first-order sensitivity (slope) within the tested range (see Fig. 3b). While higher-order models could improve the fit in some cases (e.g., $W_1^k$ for SWF and STE), they did not alter the direction or relative ranking of effects. We therefore report linear slopes for clarity and interpretability, as the goal is to validate parameter influence rather than to develop a predictive model. As illustrated in Figs. 3b and 4, stiffness parameters had the greatest impact on knee kinematics, with $W_1^k$ showing the strongest effects, particularly for SWF (${34.07}^{\circ }/\mathrm{\Delta }{W}_1^k$) and STE (${22.62}^{\circ }/\mathrm{\Delta }{W}_1^k$) across participants ($R^2>0.7$). In contrast, STF showed high variability for $W_1^k(R^2=0.125$). The second stiffness component also contributed positively to both SWF and STE, although greater inter-participant variability was observed in STE ($R^2=0.473$). By comparison, damping parameters had much weaker influences: $W_1^d$ reduced SWF ($-{3.17}^{\circ }/\mathrm{\Delta }{W}_1^d$) and STE ($-{3.54}^{\circ }/\mathrm{\Delta }{W}_1^d$), while $W_2^d$ produced negligible changes ($<3^{\circ }/\mathrm{\Delta }W$). These results indicate that stiffness parameters, particularly $W_1^k$, dominate knee kinematic modulation and therefore serve as the primary focus during tuning. Moreover, individual parameters influence multiple kinematic features (e.g., STF, STE, and SWF), highlighting the need for coordinated multi-parameter adjustment.

As shown in Fig. 5, participants exhibited distinct knee responses despite identical initial parameters, highlighting the necessity of participant-specific tuning. After tuning, each participant converged to a unique stiffness configuration that more closely matched the target profile. For TF01, tuning reduced variability in knee kinematics compared to the initial set, suggesting that the initial parameters were suboptimal and that personalized tuning improved gait stability. As summarized in Table I, all three participants demonstrated lower RMSE and higher correlation with the target after tuning. The supplemental video further illustrates TF01’s gait responses to different stiffness conditions. Notably, the shared characteristics identified in this study were derived from AB walking data and may not fully generalize to amputee locomotion, particularly due to long-term adaptation and compensatory strategies. Therefore, the AB-derived impedance functions are treated as adaptable baselines rather than fixed solutions, highlighting the importance of preserving room for tuning toward user-specific needs. We further validated the framework by introducing two additional target profiles for TF01: ABP and IP (see Fig. 6). Prior to tuning, equilibrium parameters were optimized offline to match each target profile, ensuring physiologically plausible baseline trajectories (see Table II). This is consistent with motor control theory, which suggests that the central nervous system generates time-varying (or virtual) equilibrium trajectories, with joint impedance modulated around them to stabilize motion and regulate interaction forces [43], [44]. By fixing equilibrium parameters, the observed effects could also be attributed to stiffness modulation rather than trajectory shifts.

As shown in Fig. 6, tuned stiffness parameters aligned prosthetic knee motion with both reference profiles, achieving RMSE < 5° and correlation > 0.97. Our use of normative and AB profiles was not intended to replicate clinical tuning but to validate the feasibility of the proposed framework. These profiles provide physiologically meaningful kinematic trajectories that serve as suitable first-step validation targets, and the focus on kinematics further reflects the clinical practice of tuning devices based on observed joint motion. To further address individual specificity, we additionally included the amputee participant’s intact-limb profile, demonstrating that the framework can accommodate user-specific targets. Since the proposed approach operates under impedance control, the objective is not strict trajectory tracking; RMSE and correlation were therefore used solely as performance metrics to confirm tunability with respect to a given reference. We note that the impedance functions were derived from nominal walking data, which limits strict parameter identifiability due to the underdetermined structure of the impedance model. Accordingly, the resulting impedance functions should be interpreted as regularized, human-informed baselines that define a structured tuning space rather than exact estimates of intrinsic joint impedance. Future work will focus on improving identifiability and extending the framework to patient-preferred trajectories and clinical contexts.

This study focused on stiffness tuning, motivated by sensitivity analyses showing a stronger kinematic influence than damping. However, damping and equilibrium parameters are expected to play more prominent roles in multimodal ambulation tasks such as walking at variable speeds [30], ramp ascent and descent [26], [29], and stair navigation [9], [10], [27]. Furthermore, a time-based gait phase estimator was employed to synchronize control with the participant’s gait. As this work represents an initial feasibility investigation, this design choice was intended to reduce variability arising from participant-dependent phase estimation and to isolate the effects of parameter tuning. By using a uniform phase timing strategy during treadmill walking, differences in gait patterns were primarily attributable to parameter adjustments rather than phase-estimation variability. Incorporating adaptive phase estimation will enable deployment in level-ground walking, multimodal ambulation, and unstructured environments [9], [29], [30], [45]. In this study, parameter tuning was manually performed. Although dimensionality reduction simplifies exploration, manual tuning remains burdensome and may overlook optimal solutions [24]. Integrating reinforcement learning within the proposed structured tuning space provides a promising pathway toward automated, personalized, and task-adaptive prosthetic control [36], [37], [46].

VI. Conclusion

This study presented and experimentally validated a structured, low-dimensional tuning framework for continuous impedance control of powered knee prostheses. By integrating kinematics-informed convex optimization with PCA-based dimensionality reduction, the proposed approach enables systematic and interpretable tuning of continuous impedance functions using a small set of parameters. Sensitivity analysis identified stiffness as the primary factor of knee kinematic modulation, informing the prioritization of parameters during experimental tuning. Experimental results across multiple target profiles demonstrated that distinct and repeatable knee behaviors can be reliably achieved through parameter adjustment, supporting the feasibility and robustness of the framework. These findings establish a principled foundation for personalized continuous impedance control and provide a scalable pathway toward user-specific objectives and automated tuning strategies.