A Mode-Unified, Continuous, and Biomimetic Control Strategy of a Robotic Lower Limb Prosthesis
Existing biomimetic lower-limb prosthesis controllers are commonly designed for discrete locomotion modes, requiring separate parameterizations for level walking, ramps, and stairs. We present a mode-unified prosthesis control framework that represents knee and ankle biomechanics continuously as functions of terrain slope and gait phase across those locomotion conditions.
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A Mode-Unified, Continuous, and Biomimetic Control Strategy of a Robotic Lower Limb Prosthesis
Veröffentlicht unter CC BY 4.0. Wiedergabe mit Namensnennung, Original unter arXiv:2610.19183 (PDF).
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A Mode-Unified, Continuous, and Biomimetic
Control Strategy of a Robotic Lower Limb Prosthesis Hanjun Kim1, Sixu Zhou1,2, John Shim1,2, Xinyuan Guo1, Kinsey R. Herrin1,2, and Aaron J. Young1,2, Member, IEEE
Abstract—Existing biomimetic lower-limb prosthesis controllers are commonly designed for discrete locomotion modes, requiring separate parameterizations for level walking, ramps, and stairs. We present a mode-unified prosthesis control framework that represents knee and ankle biomechanics continuously as functions of terrain slope and gait phase across those locomotion conditions. Locally optimized ascent and descent polynomial models were unified using radial basis function (RBF) interpolation, preserving the accuracy of the partitioned models while outperforming a global polynomial representation. The resulting biomechanics were transformed into continuous and slope-dependent impedance parameters and implemented within a single finite-state-machine (FSM) controller using bilateral sensing for gait-state estimation. Experiments with 10 individuals with transfemoral amputation demonstrated terrain-adaptive locomotion from level walking through ramps and stairs up to ± 35 °, including 13 ° ramp and 15° stair conditions absent from the underlying able-bodied (AB) dataset. Experimental results reproduced the principal slope-dependent biomechanical trends predicted by the unified model, with high agreement for kneerelated measures (𝒓𝒓𝟐𝟐 = 0.82 - 0.99) and ankle kinematics (𝒓𝒓𝟐𝟐 = 0.78 - 0.96). These findings demonstrated that unified biomechanical parameterization enables a prosthesis control architecture across diverse terrain conditions while preserving key features of AB biomechanics without requiring independently tuned impedance parameter sets for each locomotion mode. Index Terms— Gaussian RBF, locomotion dynamics, lower limb prosthesis, unified biomechanics modeling, unified control.
I. INTRODUCTION
HE growing availability of open-source human locomotion datasets [1], [2], [3], [4], [5] spanning diverse conditions or environments has provided unprecedented insight into how lower-limb biomechanics adapt to changing terrain and task demands. This work was supported by the NIH Health Director New Innovator Award #DP2-HD111709, and the DoD CDMRP Award #W81XWH-22-11091. (Corresponding author: Hanjun Kim) This work involved human subjects or animals in its research. Approval of all ethical and experimental procedures and protocols was granted by the Georgia Tech’s Institutional Review Board under Protocol H21117. 1 Hanjun Kim and Xinyuan Guo are with the Woodruff School of Mechanical Engineering, Georgia Tech, Atlanta, GA 30332-0405 USA (email: [email protected]; [email protected]). 2 Sixu Zhou, John Shim, Kinsey R. Herrin and Aaron J. Young are with the Woodruff School of Mechanical Engineering, Georgia Tech, Atlanta, GA 30332-0405 and with the Institute for Robotics and Intelligent Machines, Georgia Tech, Atlanta, GA 30332-0405 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).
These biomechanical data have been widely utilized in developing lower-limb wearable robots because reproducing human dynamics offers nominal control behaviors without costly user-specific tuning [6], [7], [8], [9]. Although available datasets encompass level walking, ramps, and stairs, they remain discrete and finite, so prosthesis controllers inherit that same limitation. Accordingly, extending them to new environments needs extra modeling and may produce substantial kinematic and kinetic errors in unseen conditions [10], [11]. This is especially critical for prostheses because users cannot compensate with missing joints. Prior work interpolated biomechanics across adjacent ramp inclines, [12], [13], stair heights [14], or walking speeds [15] improving scalability within a domain. But discontinuities remain between domains such as overground walking and stairs, which are still commonly modeled as separate tasks. Task-agnostic or unified prosthesis controllers have also been proposed to enable multiple ambulation activities without explicit terrain classification [16], [17]. These approaches improve versatility across locomotion tasks but generally do not prescribe healthy individuals’ joint biomechanics as continuous functions of terrain geometry. Existing methods therefore address either biomechanical fidelity within separate domains or unified control across tasks, leaving the simultaneous achievement of biomimicry and task-unified control largely unaddressed. Gait phase estimation is another fundamental component of lower-limb prosthesis control because user-environment interaction dynamics vary continuously throughout the gait cycle. Conventional finite-state-machine (FSM) controllers partition the gait cycle into discrete states and assign corresponding control parameters to each state [18], [19], [20], [21]. Among various strategies, impedance control remains widely adopted because of its simplicity, compliance, and ability to reproduce human-like joint mechanics [22], [23]. To replicate human joint behavior, impedance parameters are typically modulated according to the current gait phase [24], [25]. However, these FSM-based control frameworks are commonly designed in a mode-specific manner, where locomotion tasks such as level walking, ramps, and stairs are separately modeled using different gait-state transition conditions. For example, transitions from early to late stance are often triggered when the ankle reaches a predefined dorsiflexion angle, requiring user-specific tuning of the transition threshold. If the angle threshold is not reached during a stride, the controller may fail to progress to the subsequent state as intended. Such threshold-dependent transitions increase tuning complexity and can limit generalizability across environments and prosthesis users. Recent studies proposed continuous gait phase estimation
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Fig. 1. Overview of the proposed mode-unified prosthesis control framework. (A) Discrete human locomotion data were transformed into continuous biomechanical models parameterized by terrain slope and gait phase. (B) These unified biomechanics were converted into continuous impedance parameters for prosthesis control within a unified FSM architecture. (C) Experimental evaluation was performed across level walking, ramps (0 ° - 13 °), and stairs (15 ° - 35 °). The available healthy human dataset did not contain the intermediate 13 ° ramp and 15 ° stair conditions, allowing assessment of the proposed framework in terrain regions not represented in reference biomechanics.
methods using global thigh orientation directly captured from inertial measurement units (IMUs) [6], [7], [26], [27] or model-based approaches informed by multiple sensors [28], [29], which replace discrete state transitions with continuously evolving phase variables. Nevertheless, these approaches continue to rely on predefined ambulation-mode classifications, particularly for stair locomotion, where distinct control structures or transition conditions remain necessary. Although gait phase evolves continuously, the overall controller remains partitioned across locomotion modes. To address these limitations, we propose a mode-unified prosthesis control framework that represents lower-limb biomechanics as continuous functions of terrain slope and gait phase. The proposed framework follows the hierarchical high-, mid-, and low-level control architecture commonly adopted in lower-limb prosthesis control [22], [23], [30]. First, knee and ankle biomechanics spanning level walking, ramps, and stairs were modeled within a continuous terrain slope and gait phase domain (Fig. 1A and Section II). Locally optimized models were developed separately for ascending and descending locomotion and subsequently unified while preserving their local biomechanical characteristics. Stance- and swing-phase representations were also modeled separately to retain phasedependent locomotion dynamics. These unified biomechanical representations were then transformed into continuously slope-dependent impedance parameters for prosthesis control (Section III). Finally, the proposed framework was experimentally validated in Section IV. The resulting
controller employs a bilateral sensing strategy to detect gait events from both-limb forward kinematics while maintaining a single FSM structure across all locomotion conditions. This architecture enables continuous modulation of prosthesis impedance with terrain slope without requiring modedependent gait-state definitions or independently tuned parameter sets. The proposed framework was experimentally evaluated across level walking, ramps, and stairs, including intermediate terrain conditions that were not explicitly represented in the underlying able-bodied (AB) biomechanical dataset (Fig. 1C). We tested two main hypotheses: (H1) the unified biomechanics model preserves the fidelity of the local ascent and descent fits and (H2) the proposed unified controller framework preserves the principal slope-dependent trends in healthy individuals’ biomechanics and interpolates these trends to intermediate conditions absent from the reference data.
II. MODE-UNIFIED LOCOMOTION BIOMECHANICS
This section develops continuous lower-limb biomechanical models parameterized by terrain slope and gait phase. First, partitioned polynomial models are constructed separately for the ascending and descending terrain domains from opensource locomotion datasets [1], [8]. These locally accurate representations are subsequently unified using radial basis function (RBF) interpolation, a well-established approach for constructing continuous multivariate functions from discrete observations [31], [32]. The resulting biomechanics surfaces
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Fig. 2. Determining the polynomial fitting domain. (A) Leave-all-stairs-out fitting fails to reconstruct stair biomechanics, whereas leave-all-ramps-out fitting exhibits moderate degradation, with lower accuracy than full ascent / descent fitting (i and ii). As a result, configurations (i) and (ii) were selected as final candidates. (B) Representative continuous biomechanics surfaces obtained after RBF unification of the partitioned polynomial models. The red dotted lines represent the locomotion data from healthy individuals, and the surfaces represent models across various fitting configurations.
serve as the foundation for the impedance parameterization framework in Section III.
A. Locomotion Biomechanics Dataset
We utilized knee and ankle kinematics (joint angles) and kinetics (joint moments) collected from healthy individuals [1], [8]. As illustrated in Fig. 1A, locomotion conditions consisted of level walking (LW, 0 °), ramp ascent/descent (RA / RD, ± 5 °, 7 °, 9 °, and 11 °), and stair ascent/descent (SA / SD, ± 20 °, 24 °, 30 °, 35 °). Four ramp and four stair conditions were selected to balance data distribution across locomotion modes. All kinematic and kinetic trajectories were segmented into stance and swing phases and phase-normalized to 101 samples. The LW trials were selected from the slow-speed condition, up to 0.6 m/s.
B. Determining the Polynomial Fitting Domain
Several polynomial fitting configurations were investigated to determine an appropriate polynomial fitting domain prior to model unification (Fig. 2A). These included partitioned polynomial models for full ascent and descent, leave-ramp-out and leave-stair-out partitioning strategies, and a single global polynomial model spanning the entire terrain domain as below. (i) Full ascent partition: LW, RA, and SA (ii) Full descent partition: LW, RD, and SD (iii) Leave-all-ramps-out partition: LW (0 °) and either SA or SD (± 20 °, 24 °, 30 °, 35 °) (iv) Leave-all-stairs-out partition: LW (0 °) and either RA or RD (± 5 °, 7 °, 9 °, 11 °) (v) Global Polynomial: LW, RA, RD, SA, and SD Joint biomechanics 𝐵𝐵�𝐽𝐽,𝐷𝐷 (𝑠𝑠, 𝑝𝑝) were modeled as two-dimensional fourth-order polynomial functions of terrain slope and gait phase
4−𝑙𝑙
𝐵𝐵�𝐽𝐽,𝐷𝐷 (𝑠𝑠, 𝑝𝑝) = �� � 𝑐𝑐𝑙𝑙,𝑚𝑚 𝑠𝑠 𝑙𝑙 𝑝𝑝𝑚𝑚 � 𝑙𝑙=0 𝑚𝑚=0
(1)
𝐽𝐽,𝐷𝐷
𝑠𝑠 ∈ 𝑆𝑆𝑠𝑠 ⊆ 𝑆𝑆𝑈𝑈 = {−35, −34, … − 1, 0, 1, …, 34, 35} (2) 𝑝𝑝 ∈ 𝑃𝑃 = {𝑝𝑝1, 𝑝𝑝2, …, 𝑝𝑝101 } = {0, 0.01,0.02, …,0.99,1} (3) where 𝐽𝐽 and 𝐷𝐷 denote the target joint (knee or ankle) and the type of dynamics (angle or moment), respectively. The slope subset 𝑆𝑆𝑠𝑠 was defined according to configurations (i)-(v); for example, {0, 5, 7, 9, 11} for (iii) leave-all-ramps-out partition (ascending) or {35, -30, -24, -20, 0} for (iv) leave-all-stairs-out partition (descending). The coefficients 𝑐𝑐𝑙𝑙,𝑚𝑚 were estimated for each configuration using unconstrained linear least-squares (LSQ) regression, using the MATLAB R2024b Curve Fitting Toolbox. Based on the reconstruction performance summarized in Section II-D, the configurations (i) and (ii), the full ascent and full descent partitions, were selected as the local polynomial models for subsequent RBF unification. Configurations (iii) and (iv) were retained to evaluate cross-domain extrapolation, whereas configuration (v) served as a global polynomial benchmark.
C. RBF-Interpolation for Mode Unification
The selected models from configurations (i) and (ii) were unified using Gaussian RBF interpolation [31], leveraging the 𝐶𝐶 ∞ -continuity of the Gaussian kernel to provide a smooth transition between the two terrain partitions. This formulation reconciles trajectory mismatches at the shared level-walking condition while preserving the local biomechanical characteristics of the ascent and descent partitions. Let 𝑓𝑓𝑎𝑎 (𝑠𝑠𝑎𝑎, 𝑝𝑝), 𝑓𝑓𝑑𝑑 (𝑠𝑠𝑑𝑑, 𝑝𝑝) ∈ ℝ36×101 denote the local ascent and descent models obtained from Section II-B. Samples generated from both models over the full slope-phase domain were
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Fig. 3. Validation of the unified biomechanics. Proposed RBF-unified models were compared with the partitioned polynomial and global polynomials for (A) kinematics and (B) kinetics. The RBF-unified representation maintained the reconstruction accuracy of the partitioned polynomial models while consistently outperforming the global polynomial representation. Bars represent the mean ± 1 SD of the MAE across all evaluated terrain slope conditions. Statistical significance was evaluated using paired t-test (*𝑝𝑝 < 0.05, **𝑝𝑝 < 0.01, ***𝑝𝑝 < 0.001).
treated as observations in a two-dimensional space defined by terrain slopes 𝑠𝑠𝑎𝑎 ∈ {35, 34, …, 0} and 𝑠𝑠𝑑𝑑 ∈ {0, −1, …, −35}, and gait phase 𝑝𝑝. Then, the stacked representation of the local model can be defined as 𝑓𝑓 𝑀𝑀 = � 𝑎𝑎 � ∈ ℝ72×101 (4) 𝑓𝑓𝑑𝑑 The stacked representation contains duplicated coordinates at 𝑠𝑠 = 0 °, corresponding to the shared level-walking condition from both the ascent and descent models. The objective of RBF interpolation is to derive a unified representation of biomechanics 𝑓𝑓𝑈𝑈 (𝑠𝑠𝑈𝑈, 𝑝𝑝) ∈ ℝ71×101 for each joint in the slopephase domain, where 𝑠𝑠𝑈𝑈 ∈ 𝑆𝑆𝑈𝑈 . Let 𝑀𝑀𝑣𝑣 ∈ ℝ(72×101)×1 denote the vectorized form of 𝑀𝑀. A Euclidean coordinate matrix 𝑋𝑋 ∈ ℝ(72×101)×2 containing the slope-phase pair is defined below, over the same domain as the stacked sample matrix 𝑀𝑀𝑣𝑣 . 𝑠𝑠𝑖𝑖 𝑥𝑥𝑖𝑖 = �𝑝𝑝 � ∈ ℝ2 𝑖𝑖 +35 0 ⎡ +35 0.01⎤ 𝑥𝑥 𝑇𝑇 ⎥ ⎡ 1𝑇𝑇 ⎤ ⎢ ⋮ ⋮ ⎥ (5) ⎢ 𝑥𝑥 𝑋𝑋 = ⎢ 2 ⎥ = ⎢ +35 1 ⎥ ∈ ℝ7272×2 ⎢ ⋮ ⎥ ⎢ +34 0 ⎥ 𝑇𝑇 ⎣𝑥𝑥7272 ⎦ ⎢ ⋮ ⎥ ⋮ ⎣ −35 1 ⎦ Then the Gaussian kernel matrix 𝜓𝜓 ∈ ℝ7272×7272 is defined as 2 𝜓𝜓𝑖𝑖,𝑗𝑗 = exp �−�𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑗𝑗 �2 � (6) Using (6), the unified biomechanics model is represented as a weighted sum of Gaussian basis functions centered at each slope-phase coordinate. 7272
𝑓𝑓(𝑥𝑥) = � 𝑎𝑎𝑖𝑖 ∙ exp(−|𝑥𝑥 − 𝑥𝑥𝑖𝑖 |22 ) 𝑖𝑖=1
(7)
Let 𝐴𝐴 = [𝑎𝑎1 𝑎𝑎2 … 𝑎𝑎7272 ]𝑇𝑇 ∈ ℝ7272×1 denote the RBF coefficient vector. The components of 𝐴𝐴 were determined by reconstructing the stacked biomechanics samples 𝑀𝑀𝑣𝑣 from the kernel representation. Evaluating the above expression in (6) and (7) at all sample coordinates yields the kernel system.
𝑀𝑀𝑣𝑣 = 𝜓𝜓𝜓𝜓 (8) Note that the Gaussian kernel matrix 𝜓𝜓 contains duplicate level-walking coordinate pairs from both the ascent and descent models. Consequently, 𝜓𝜓 contains duplicated rows and columns, resulting in a singular or numerically illconditioned system. Hence, a diagonal regularization term with a small number 𝜀𝜀 was introduced to ensure a stable and unique solution, yielding 𝐴𝐴 = (𝜓𝜓 + 𝜀𝜀𝜀𝜀)−1 𝑀𝑀𝑣𝑣 (9) To obtain a mode-unified biomechanics surface, the RBF representation was subsequently evaluated on a unified slopephase domain defined by equations (2) and (3), without duplication. Similar to (5), a Euclidean coordinate matrix 𝑋𝑋𝑈𝑈 ∈ ℝ7171×2 corresponding to the unified slope-phase domain consisting of 𝑆𝑆𝑈𝑈 and 𝑃𝑃 can be defined. 𝑠𝑠𝑈𝑈 𝑥𝑥𝑈𝑈 𝑖𝑖 = � 𝑝𝑝 𝑖𝑖 � ∈ ℝ2 𝑖𝑖 𝑥𝑥𝑈𝑈 1𝑇𝑇 (10) 𝑋𝑋𝑈𝑈 = � ⋮ � 𝑥𝑥𝑈𝑈 𝑇𝑇7171 Then an interpolation kernel matrix 𝜓𝜓𝑈𝑈 was computed between the unified and the original sample coordinates 𝑋𝑋𝑈𝑈 and 𝑋𝑋 as 2 𝜓𝜓𝑈𝑈 𝑖𝑖,𝑗𝑗 = exp �−�𝑥𝑥𝑈𝑈 𝑖𝑖 − 𝑥𝑥𝑗𝑗 � � (11) 2 where 𝑥𝑥𝑈𝑈 𝑖𝑖 and 𝑥𝑥𝑗𝑗 denote the i-th and j-th coordinate of 𝑋𝑋𝑈𝑈 and 𝑋𝑋, respectively. Since the unified slope-phase domain contains 7171 query points and the original domain contains 7272 samples, each row of 𝜓𝜓𝑈𝑈 corresponds to a query coordinate in the unified slope-phase domain, while each column corresponds to an original sample coordinate used in equation (8). Therefore, 𝜓𝜓𝑈𝑈 ∈ ℝ7171×7272. The samples in the slope-phase domain were then obtained by evaluating the RBF representation using the coefficient vector 𝐴𝐴 ∈ ℝ7272×1, 𝑀𝑀𝑈𝑈 = 𝜓𝜓𝑈𝑈 𝐴𝐴 (12) Finally, a unified model 𝑀𝑀𝑈𝑈 ∈ ℝ7171×1 was reshaped into the slope-phase representation 𝑓𝑓𝑈𝑈 (𝑠𝑠, 𝑝𝑝) ∈ ℝ71×101, yielding a
This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. continuous and mode-unified biomechanics surface over the full slope domain, with no duplicated coordinate pair.
D. Validation of the Unified Biomechanics Representation
As illustrated in Fig. 2A, the leave-ramp-out models exhibited moderate degradation but retained reasonable reconstruction of the omitted ramp conditions, whereas the leave-stair-out models failed to extrapolate to stair locomotion. These leave-one-domain analyses were conducted to characterize cross-domain extrapolation rather than to construct the final unified model. The full-ascent and fulldescent models, configurations (i) and (ii), achieved the best reconstruction performance across the evaluated configurations and were therefore selected for RBF unification, as described in Section II-C. The quantitative comparison focused on the partitioned polynomial, global polynomial, and proposed RBF-unified models using polynomials from (i) and (ii). As shown in Fig. 3, the RBF-unified models consistently achieved lower meanabsolute errors (MAEs) than the global polynomials, while preserving the accuracy of the partitioned polynomials. The only significant difference between the RBF-unified and partitioned models occurred for the ankle swing angle during descent locomotion, where the RBF-unified model achieved a lower MAE. The resulting RBF-unified models are shown in Fig. 2B, demonstrating continuous slope-phase trajectories that closely reproduce the underlying AB biomechanics. These unified biomechanics subsequently serve as the foundation for the impedance parameterization in Section III.
III. MODE-UNIFIED IMPEDANCE PARAMETERS
The RBF-unified biomechanics models derived in Section II provide continuous representations of joint kinematics and kinetics over terrain slope and gait phase. This section transforms these biomechanical surfaces into a continuous impedance parameter space for mode-unified prosthesis control. Consistent with the proposed FSM framework, stance and swing phase parameters were identified separately. The following sections focus on the stance phase parameterization, which governs the primary load-bearing dynamics of locomotion. At each terrain slope, 𝐾𝐾, 𝐵𝐵, and 𝜃𝜃𝑒𝑒𝑒𝑒 were defined according to the impedance relationship between joint moment 𝜏𝜏, joint angle 𝜃𝜃, angular velocity 𝜃𝜃̇, 𝜏𝜏 = −𝐾𝐾�𝜃𝜃 − 𝜃𝜃𝑒𝑒𝑒𝑒 � − 𝐵𝐵𝜃𝜃̇ (13) where 𝐾𝐾, 𝐵𝐵, and 𝜃𝜃𝑒𝑒𝑒𝑒 denote stiffness, damping, and equilibrium angle, respectively. The unified biomechanics models provide the target kinematic and kinetic behaviors to be reproduced by the controller. However, the impedance parameters were not directly identified from the biomechanical models. Instead, additional design constraints were incorporated during the parameterization process to ensure compatibility with the unified FSM framework.
A. Biomechanically Inspired Equilibrium Angle Trajectory
Previous studies identified stiffness, damping, and equilibrium angle simultaneously through optimization frameworks [6], [7]. In contrast, the proposed controller prescribes biomechanically
Fig. 4. Selection of the support equilibrium angle and resulting slope-dependent trajectories. (A) The support angle 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 is selected as the joint angle corresponding to the mid-to-late stance minimum of the biological joint moment. For Case I where there is no LS moment peak, this minimum occurs after the stance phase peak. On the other hand, for Case II with LS moment peak, the minimum takes place before the LS moment peak. (B) Continuous 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 trajectories across terrain slope. The ankle support angle is limited to 10 ° dorsiflexion to accommodate prosthesis constraints.
inspired equilibrium angle trajectories and subsequently identifies stiffness and damping through constrained optimization. This design follows established bio-inspired prosthesis controllers [19], [22], [23], [25] while extending their parameterization to a continuous terrain slope domain. To construct the unified equilibrium angle trajectories, a slope-dependent support angle (𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 ) was extracted from the stance phase angle-moment relationship (Fig. 4A). The 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 is defined as the joint angle corresponding to the mid-to-late stance minimum of the biological joint moment profile. Two representative cases are observed. In Case I, the biological moment does not exhibit a late stance (LS) moment peak, and the mid-to-late stance minimum occurs after the primary stance peak. This behavior is typically observed in the knee during level walking and ascending locomotion. In Case II, the biological moment exhibits a LS moment peak, and the minimum occurs before the LS peak, separating the weightacceptance and support phase. This pattern is observed in the knee during descending locomotion and in the ankle across all evaluated environments. Based on these observations, the proposed controller uses 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 as a common biomechanical
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Fig. 5. Three-region impedance design strategy. (A) Schematic construction of the stance equilibrium angle trajectory. (B) Resulting continuous equilibrium angle surfaces across terrain slope and stance phase. (C) Flowchart of LSQ-optimization for K and B.
reference to initiate the support phase under both cases. In Case II specifically, rather than reproducing the initial weightacceptance peak, the controller is designed to reproduce the support-related moment behavior following 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 . This is because accurately reproducing both moment peaks would require additional modeling complexity [7], [23], [33]. Using 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 as the primary biomechanical reference, the stance equilibrium angle trajectory was organized into three functional regions (Fig. 5A). During the weight acceptance (R1), 𝜃𝜃𝑒𝑒𝑒𝑒 transitions from the heel-contact angle (𝜃𝜃𝐻𝐻𝐻𝐻 ) toward 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 near the 25% stance, which corresponds to the first peak of the vertical ground reaction force (vGRF). During the single support (R2), 𝜃𝜃𝑒𝑒𝑒𝑒 remains at 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 so that increasing joint flexion produces support through impedance. Finally, during the unloading (R3) around 75% stance, 𝜃𝜃𝑒𝑒𝑒𝑒 transitions from 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 toward the toe-off angle (𝜃𝜃𝑇𝑇𝑇𝑇 ) in preparation for swing. For the knee, ascending locomotion directly follows the three-region structure described above. During descending locomotion, however, the transition in R1 is realized using a P-value strategy similar to previous prosthesis controllers [25], 𝜃𝜃𝑒𝑒𝑒𝑒 = 𝜃𝜃𝜃𝜃(𝑠𝑠) 0.29 (14) 𝑠𝑠 ≤ 0 𝑃𝑃(𝑠𝑠) = �0.70 − 35 𝑠𝑠, 0, 𝑠𝑠 > 0 As the descent becomes steeper, increasing 𝑃𝑃(𝑠𝑠) causes 𝜃𝜃𝑒𝑒𝑒𝑒 to follow the measured knee angle more closely during R1, permitting greater controlled knee flexion before converging to 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 . Once 𝜃𝜃𝑒𝑒𝑒𝑒 reaches 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠, it remains fixed throughout R2 so that additional knee flexion generates higher torque under the
impedance relationship. During R3, 𝜃𝜃𝑒𝑒𝑒𝑒 transitions toward 𝜃𝜃𝑇𝑇𝑇𝑇 for swing. The ankle follows the same three-region framework but incorporates two prosthesis-specific modifications. First, 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 was bounded to a maximum dorsiflexion angle of 10 ° (Fig. 4B) to accommodate the prosthetic ankle range of motion (± 30 °) and preserve sufficient plantarflexion torque. This limit also accounts for the reduced forward progression commonly observed by prosthesis users during stair ascent [24], [34]. Consequently, the bounded 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 preserves the slopedependent trend while remaining suitable for prosthesis control. During descending locomotion, the transition in R1 is similarly implemented using the P-value strategy, 𝜃𝜃𝑒𝑒𝑒𝑒 = �𝜃𝜃 − 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 �𝑃𝑃 + 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 (15) 𝑃𝑃 = 1.1 was selected, causing 𝜃𝜃𝑒𝑒𝑒𝑒 to follow the measured ankle angle while converging toward 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠, thereby providing controlled plantarflexion yielding during weight acceptance. Once 𝜃𝜃𝑒𝑒𝑒𝑒 reaches 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠, it remains fixed throughout R2, allowing the ankle to provide stable support during single support. During R3, 𝜃𝜃𝑒𝑒𝑒𝑒 transitions from 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 toward 𝜃𝜃𝑇𝑇𝑇𝑇 to facilitate prosthetic push-off. Applying the above design principles across the continuous terrain slope domain yields the unified equilibrium angle trajectories for the stance phase (𝜃𝜃𝑒𝑒𝑒𝑒 (𝑠𝑠, 𝑝𝑝)) shown in Fig. 5B. For the swing phase, equilibrium angles were defined from two reference points of the RBF-unified kinematic models. The swing flexion state used the maximum flexion angle to promote sufficient limb shortening and ground clearance, whereas the swing extension state used the heel-contact angle (𝜃𝜃𝐻𝐻𝐻𝐻 ) to
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Fig. 6. Continuous impedance parameterization. (A) Identified stiffness and (B) damping surfaces. Green dashed curves denote slopedependent upper bounds 𝐾𝐾𝑚𝑚𝑚𝑚𝑚𝑚 (𝑠𝑠) and 𝐵𝐵𝑚𝑚𝑚𝑚𝑚𝑚 (𝑠𝑠). (C) Moments reconstructed from the impedance model closely match the AB reference (red dotted lines).
prepare the limb for stable ground contact. Extending these references across the continuous terrain slope domain yielded unified swing phase equilibrium trajectories consistent with the stance phase design philosophy.
B. Stance Phase Parameterization
The stance-phase stiffness and damping trajectories were parameterized using the same three-region structure introduced for the equilibrium angle (Fig. 5A). Both parameters increase during weight acceptance (R1), remain approximately constant throughout single support (R2), and remain constant or decrease during unloading (R3). This structure translates the biomechanical design principles into continuous impedance trajectories compatible with the proposed unified FSM framework. To obtain compact impedance representations suitable for optimization and real-time implementation, both stiffness (K) and damping (B) were represented as fourth-order polynomials of normalized stance phase for each terrain slope (s), 4
𝐾𝐾(𝑠𝑠, 𝑝𝑝) = � 𝑘𝑘𝑠𝑠 𝑖𝑖 𝑝𝑝𝑖𝑖, 𝑖𝑖=0
𝐵𝐵(𝑠𝑠, 𝑝𝑝) = � 𝑏𝑏𝑠𝑠 𝑖𝑖 𝑝𝑝𝑖𝑖
(16)
𝑖𝑖=0
Substituting (16) into the impedance relationship (13) yields a linear least-squares problem, min|𝜏𝜏 − 𝑪𝑪𝐱𝐱|22 (17) 𝐱𝐱
where 𝐱𝐱 = �𝑘𝑘𝑠𝑠 𝑜𝑜 … 𝑘𝑘𝑠𝑠 4 𝑏𝑏𝑠𝑠 𝑜𝑜 … 𝑏𝑏𝑠𝑠 4 � ∈ ℝ10×1, 𝑪𝑪 ∈ ℝ101×10 is constructed from the RBF-unified joint kinematics and equilibrium angle trajectories, and 𝜏𝜏 is the RBF-unified moment. The j-th row 𝑪𝑪𝑗𝑗,: ∈ ℝ1×10 corresponds to the impedance equation evaluated at a stance phase sample 𝑝𝑝𝑗𝑗, 𝑪𝑪𝑗𝑗,: = �∆𝜃𝜃𝑗𝑗 𝑝𝑝𝑗𝑗0 … ∆𝜃𝜃𝑗𝑗 𝑝𝑝𝑗𝑗4 𝜃𝜃̇ �𝑠𝑠, 𝑝𝑝𝑗𝑗 �𝑝𝑝𝑗𝑗0 … 𝜃𝜃̇�𝑠𝑠, 𝑝𝑝𝑗𝑗 �𝑝𝑝𝑗𝑗4 � (18) 𝑗𝑗 ∈ {0, 1, …, 100}
where ∆𝜃𝜃𝑗𝑗 = 𝜃𝜃�𝑠𝑠, 𝑝𝑝𝑗𝑗 � − 𝜃𝜃𝑒𝑒𝑒𝑒 �𝑠𝑠, 𝑝𝑝𝑗𝑗 � and 𝜃𝜃̇ �𝑠𝑠, 𝑝𝑝𝑗𝑗 � is RBF-unified kinematics derived in Section II. The overall optimization procedure is summarized in Fig. 5C. First, the unified biomechanics �𝜏𝜏, 𝜃𝜃, 𝜃𝜃̇ � together with the equilibrium angle trajectories are separated into early-stance (ES) and late-stance (LS) regions. ES governs the loading response from heel-contact to p = 0.49, whereas LS is initialized from the terminal ES solution and governs the subsequent support and unloading phases. The corresponding parameters are then identified sequentially through constrained least-squares optimization before being combined into continuous stiffness and damping surfaces over the terrain slope domain. Increasing terrain steepness requires greater joint support during ascent and greater energy dissipation during descent. Therefore, slope-dependent upper envelopes, 𝐾𝐾max (𝑠𝑠)and 𝐵𝐵max (𝑠𝑠), were defined using arctangent functions connecting experimentally validated impedance parameters from 35° stairascent and stair-descent controllers [22], [23]. Constant lower bounds, 𝐾𝐾min and 𝐵𝐵min, were selected as empirical safety limits, yielding 𝐾𝐾min ≤ 𝐾𝐾(𝑠𝑠, 𝑝𝑝) ≤ 𝐾𝐾max (𝑠𝑠) (19) 𝐵𝐵min ≤ 𝐵𝐵(𝑠𝑠, 𝑝𝑝) ≤ 𝐵𝐵max (𝑠𝑠) To facilitate implementation within the unified FSM, the optimization was solved separately for ES and LS. Equality constraints were set to anchor the impedance trajectories at key transition points, 𝑨𝑨eq 𝐱𝐱 = 𝜷𝜷eq (20) ES begins with the minimum impedance values at heel-contact, whereas LS is initialized using the terminal ES solution and constrained to preserve the support-level impedance until the onset of unloading. Therefore, the impedances at the heel-contact were constrained by
This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. TABLE I. 𝜙𝜙 𝑇𝑇 (0) 𝟎𝟎 𝐾𝐾min 𝑨𝑨ES �, 𝜷𝜷ES � (21) eq = � eq = �𝐵𝐵 2 𝟎𝟎 𝜙𝜙 𝑇𝑇 (0) min 𝑟𝑟 VALUES OF RECONSTRUCTED MOMENT MODELS where 𝜙𝜙(𝑝𝑝) = [1 𝑝𝑝 𝑝𝑝2 𝑝𝑝3 𝑝𝑝4 ]𝑇𝑇 . Subsequently, the LS optimization marked the terminal ES values on p = 0.50 and 0.75. 𝜙𝜙 𝑇𝑇 (0.50) 𝟎𝟎 ⎡ 𝑇𝑇 ⎤ 𝜙𝜙 (0.75) 𝟎𝟎 ⎢ ⎥ LS 𝑨𝑨eq = ⎢ , 𝟎𝟎 𝜙𝜙 𝑇𝑇 (0.50)⎥⎥ ⎢ ⎣ 𝟎𝟎 𝜙𝜙 𝑇𝑇 (0.75)⎦
𝐾𝐾ES (0.49) 𝐾𝐾 ES (0.49) � 𝜷𝜷LS eq = � 𝐵𝐵ES (0.49) 𝐵𝐵ES (0.49)
(22)
And the finalized equality constraints were obtained as 𝑨𝑨ES 𝜷𝜷ES eq eq 𝑨𝑨eq = � LS �, 𝜷𝜷eq = � LS � (23) 𝑨𝑨eq 𝜷𝜷eq The remaining portions of the optimization were formulated as inequality constraints with bound and shape components, 𝑨𝑨ineq 𝐱𝐱 ≤ 𝜷𝜷ineq (24) ES 𝑨𝑨bound 𝜷𝜷ES bound ⎡ ⎤ ⎡ ⎤ LS LS ⎢𝑨𝑨bound ⎥ ⎢𝜷𝜷bound ⎥ 𝑨𝑨ineq = ⎢ ES ⎥, 𝜷𝜷ineq = ⎢ ES ⎥ (25) 𝑨𝑨 𝜷𝜷shape ⎢ shape ⎥ ⎢ ⎥ LS LS ⎣ 𝑨𝑨shape ⎦ ⎣ 𝜷𝜷shape ⎦ The bound constraints enforce the admissible stiffness and damping envelopes in (19) throughout the corresponding optimization interval. Defining 𝒫𝒫𝐸𝐸𝐸𝐸 = [𝜙𝜙(0) … 𝜙𝜙(0.49)]𝑇𝑇, 𝒫𝒫𝐿𝐿𝐿𝐿 = [𝜙𝜙(0.50) … 𝜙𝜙(1)]𝑇𝑇, and 𝒍𝒍𝑗𝑗 = [1 … 1]𝑇𝑇 ∈ ℝ𝑗𝑗×1, the corresponding bound matrices are 𝐾𝐾 (𝑠𝑠)𝒍𝒍50 𝒫𝒫ES 𝟎𝟎 ⎡ max ⎤ −𝒫𝒫 𝟎𝟎 −𝐾𝐾min 𝒍𝒍50 ⎥ ES 𝑨𝑨ES �, 𝜷𝜷ES =⎢ (26) bound = � 𝟎𝟎 bound 𝒫𝒫ES ⎢𝐵𝐵max (𝑠𝑠)𝒍𝒍50 ⎥ 𝟎𝟎 −𝒫𝒫ES ⎣ −𝐵𝐵min 𝒍𝒍50 ⎦ 𝐾𝐾𝐸𝐸𝐸𝐸 (0.49)𝒍𝒍51 𝒫𝒫LS 𝟎𝟎 ⎡ ⎤ −𝒫𝒫LS 𝟎𝟎 −𝐾𝐾min 𝒍𝒍51 ⎥ 𝑨𝑨LS �, 𝜷𝜷LS =⎢ (27) bound = � 𝟎𝟎 bound 𝒫𝒫LS ⎢𝐵𝐵𝐸𝐸𝐸𝐸 (0.49)𝒍𝒍51 ⎥ 𝟎𝟎 −𝒫𝒫LS ⎣ −𝐵𝐵min 𝒍𝒍51 ⎦ These bound constraints prevent the LS solution from exceeding the support-level impedance established during ES and preserves the intended plateau-to-unloading structure. The shape constraints are then constructed from the derivative polynomial basis 𝜙𝜙̇(𝑝𝑝) = [0 1 2𝑝𝑝 3𝑝𝑝2 4𝑝𝑝3 ] to regulate the impedance progression within the envelopes. Since partial derivatives of 𝐾𝐾(𝑠𝑠, 𝑝𝑝) and 𝐵𝐵(𝑠𝑠, 𝑝𝑝) are represented as 4
∂𝐾𝐾(𝑠𝑠, 𝑝𝑝) = � 𝑖𝑖𝑘𝑘𝑠𝑠 𝑖𝑖 𝑝𝑝𝑖𝑖−1, ∂𝑝𝑝 𝑖𝑖=1
∂𝐵𝐵(𝑠𝑠, 𝑝𝑝) = � 𝑖𝑖𝑏𝑏𝑠𝑠 𝑖𝑖 𝑝𝑝𝑖𝑖−1 ∂𝑝𝑝
(28)
𝑖𝑖=1
𝑇𝑇 Defining derivative terms 𝒫𝒫̇𝐸𝐸𝑆𝑆1 = �𝜙𝜙̇(0) … 𝜙𝜙̇(0.24)�, 𝒫𝒫̇𝐸𝐸𝑆𝑆2 = 𝑇𝑇 𝑇𝑇 �𝜙𝜙̇(0.25) … 𝜙𝜙̇(0.49)�, 𝒫𝒫̇𝐿𝐿𝑆𝑆 = �𝜙𝜙̇(0.50) … 𝜙𝜙̇(0.74)�, and 1
𝑇𝑇 𝒫𝒫̇𝐿𝐿𝑆𝑆2 = �𝜙𝜙̇(0.75) … 𝜙𝜙̇(1)� yields −𝒫𝒫̇ 𝟎𝟎 𝜖𝜖𝐸𝐸𝑆𝑆,𝐾𝐾 𝒍𝒍25 ⎡ ES1 ⎤ ⎡ 1 ⎤ ̇ −𝒫𝒫ES1 ⎥ ⎢ 𝟎𝟎 ⎢𝜖𝜖𝐸𝐸𝑆𝑆1,𝐵𝐵 𝒍𝒍25 ⎥ ⎢ 𝒫𝒫̇ 𝟎𝟎 ⎥ ⎢𝜖𝜖𝐸𝐸𝑆𝑆,𝐾𝐾 𝒍𝒍25 ⎥ ES2 ⎢ ⎥, 𝑨𝑨ES = 𝜷𝜷ES =⎢ 2 shape shape 𝜖𝜖𝐸𝐸𝑆𝑆,𝐾𝐾 𝒍𝒍25 ⎥ 𝟎𝟎⎥ ⎢−𝒫𝒫̇ES2 ⎢ 2 ⎥ ⎢ 𝟎𝟎 𝒫𝒫̇ES2 ⎥ ⎢𝜖𝜖𝐸𝐸𝑆𝑆2,𝐵𝐵 𝒍𝒍25 ⎥ ⎢ ⎥ ⎣𝜖𝜖𝐸𝐸𝑆𝑆2,𝐵𝐵 𝒍𝒍25 ⎦ ⎣ 𝟎𝟎 −𝒫𝒫̇ES2 ⎦
(29)
𝒫𝒫̇ 𝟎𝟎 𝜖𝜖𝐿𝐿𝑆𝑆,𝐾𝐾 𝒍𝒍25 ⎡ LS1 ⎤ ⎡ 1 ⎤ ̇ −𝒫𝒫 𝟎𝟎 ⎢ ⎥ LS1 ⎢𝜖𝜖𝐿𝐿𝑆𝑆1,𝐾𝐾 𝒍𝒍25 ⎥ ⎢𝟎𝟎 𝒫𝒫̇LS1 ⎥ ⎢𝜖𝜖𝐿𝐿𝑆𝑆,𝐵𝐵 𝒍𝒍25 ⎥ ⎢ ⎥, 𝑨𝑨LS = 𝜷𝜷LS =⎢ 1 (30) shape shape 𝜖𝜖𝐿𝐿𝑆𝑆1,𝐵𝐵 𝒍𝒍25 ⎥ ⎢ 𝟎𝟎 −𝒫𝒫̇LS1 ⎥ ⎢ ⎥ ⎢ 𝒫𝒫̇ 𝟎𝟎 ⎥ ⎢𝜖𝜖𝐿𝐿𝑆𝑆2,𝐾𝐾 𝒍𝒍25 ⎥ ⎢ LS2 ⎥ ⎣𝜖𝜖𝐿𝐿𝑆𝑆2,𝐵𝐵 𝒍𝒍25 ⎦ 𝒫𝒫̇LS2 ⎦ ⎣𝟎𝟎 Note that the small positive values 𝜖𝜖 for stiffness and damping were 1e-5 and 1e-6, respectively. The identified impedance parameters were finally validated by reconstructing the target joint moments. Figure 6 shows the resulting stiffness and damping surfaces together with the reconstructed moments, while quantitative reconstruction performance is summarized in Table I. For the LW and ascent models, the reconstructed moments closely matched the target AB trajectories, yielding a high overall squared Pearson correlation coefficient 𝑟𝑟 2 values for both the knee (0.94 ± 0.04) and ankle (0.76 ± 0.15). High ES and LS accuracies indicate that the proposed parameterization preserves the primary biomechanical characteristics of the target moments. The descending models exhibited lower overall 𝑟𝑟 2 values, primarily due to reduced ES reconstruction accuracy. This behavior was expected because the proposed 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 -based yielding strategy intentionally does not reproduce the initial stance phase moment peak during weight acceptance. As a result, the knee and ankle ES models achieved 𝑟𝑟 2 values of 0.41 ± 0.37 and 0.26 ± 0.17, respectively. In contrast, the LS models maintained substantially higher accuracies (0.75 ± 0.21 for the knee and 0.91 ± 0.05 for the ankle), demonstrating that the proposed parameterization accurately reproduces the support and unloading dynamics prioritized by the controller despite the reduction in early stance reconstruction performance.
IV. EXPERIMENTAL METHODS
A. Prosthetic System with Bilateral Sensing The proposed controller was implemented on the secondgeneration Open-Source Leg (OSL), a powered knee-ankle prosthesis platform developed by the University of Michigan
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Fig. 7. Experimental setups are illustrated. (A) The proposed wearable system incorporates prosthetic and intact side sensing system to capture both limb dynamics simultaneously. (B) The proposed FSM was enhanced with the relative geometry of limbs, enabling slopeindependent T1 condition. (C) The relative geometry between both ankles 𝑑𝑑ℎ was computed through forward kinematics using the bilateral sensing system. (D) A representative time series of computed ankle geometry, impedance parameters, and the load cell signal are illustrated. Green arrows indicate computed ankle geometry values, where (+) stands for the intact ankle forward.
[35], [36]. The system incorporated bilateral sensing from both limbs [11], [37], [38], onboard computation, and impedancecontrolled actuation for continuous terrain-adaptive locomotion, as illustrated in Fig. 7A. The prosthesis incorporated powered knee and ankle actuators (Actpack 4.1, Dephy, MA, USA) with embedded joint encoders and a six-axis load cell (M3564F, Sunrise Instrument, MI, USA) mounted between the knee and ankle modules. Four 9-axis IMUs (3DM-GX5-25, HBK, Darmstadt, Germany) were mounted on the prosthetic-side thigh and foot and the intact-side thigh and shank. The intact limb was additionally instrumented with an XSENSOR X4 module (X4, XSENSOR, Calgary, Canada), comprising a foot-mounted IMU and a pressure-
sensitive insole. All onboard sensing and control algorithms were executed on a Jetson Orin Nano under ROS 2 Jazzy at 100 Hz. Motion capture measurements were synchronized with the ROS 2 data stream for post-hoc analysis.
B. Controller Implementation
Our mode-unified framework employs a four-state FSM that is shared across all locomotion modes (Fig. 7B), similar to other studies [18], [22], [23], [25]. Transition conditions (T1-T4) are also summarized in Fig. 7B. T1 was defined from bilateral limb progression, corresponding to the instant when the intact foot advances beyond the prosthetic foot during stance with prosthetic load beyond a threshold. The remaining transitions (T2-T4) followed conventional prosthesis control logic.
10 This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. The intact and prosthetic limb geometries were reconstructed ramps (± 9 °), and stairs (± 24 °). After acclimation, swingfrom wearable IMU orientations and subject-specific segment phase stiffness and damping were adjusted by up to 10% from lengths, as illustrated in Fig. 7C. Let 𝐺𝐺 𝐩𝐩ℎ denote the hip joint their nominal values to achieve comfortable limb advancement. position in the global frame. The segment direction vectors 𝑑𝑑𝑡𝑡,𝑖𝑖, In addition, constant heel-contact angle offsets of up to 10° 𝑑𝑑𝑠𝑠,𝑖𝑖, 𝑑𝑑𝑡𝑡,𝑝𝑝, and 𝑑𝑑𝑠𝑠,𝑝𝑝, together with the corresponding segment were introduced during descending locomotion to promote a lengths 𝐿𝐿𝑡𝑡,𝑖𝑖, 𝐿𝐿𝑠𝑠,𝑖𝑖, 𝐿𝐿𝑡𝑡,𝑝𝑝, and 𝐿𝐿𝑠𝑠,𝑝𝑝 were used to reconstruct the more extended knee and reduced ankle plantarflexion at initial contact, improving heel-contact stability while preserving the intact and prosthetic ankle positions 𝐺𝐺 𝐩𝐩𝑎𝑎,𝑖𝑖 and 𝐺𝐺 𝐩𝐩𝑎𝑎,𝑝𝑝 through continuous slope-dependent trajectories. This adjustment was the forward kinematic relationships, introduced because direct reproduction of the AB heel-contact 𝐺𝐺 𝐩𝐩𝑎𝑎,𝑖𝑖 = 𝐺𝐺 𝐩𝐩ℎ + 𝐿𝐿𝑡𝑡,𝑖𝑖 𝑑𝑑𝑡𝑡,𝑖𝑖 + 𝐿𝐿𝑠𝑠,𝑖𝑖 𝑑𝑑𝑠𝑠,𝑖𝑖 resulted in excessive knee flexion and ankle plantarflexion (31) 𝐺𝐺 𝐩𝐩𝑎𝑎,𝑝𝑝 = 𝐺𝐺 𝐩𝐩ℎ + 𝐿𝐿𝑡𝑡,𝑝𝑝 𝑑𝑑𝑡𝑡,𝑝𝑝 + 𝐿𝐿𝑠𝑠,𝑝𝑝 𝑑𝑑𝑠𝑠,𝑝𝑝 which was destabilizing in prosthesis users. The relative ankle position was then computed as The proposed controller was subsequently evaluated during level walking, ramp walking, and stair ambulation, as illustrated Δ𝐩𝐩𝑎𝑎 = 𝐺𝐺 𝐩𝐩𝑎𝑎,𝑖𝑖 − 𝐺𝐺 𝐩𝐩𝑎𝑎,𝑝𝑝 (32) The prosthetic forward direction 𝐟𝐟𝑡𝑡,𝑝𝑝,ℎ was obtained by in Fig. 1C. Level walking and ramps (0 ° - 13 °) were performed projecting the prosthetic thigh IMU local positive 𝑧𝑧-axis onto on an instrumented incline/decline treadmill at 0.4 - 0.5 m/s, the horizontal plane and normalizing the resulting vector. The whereas stair ascent and descent were evaluated over slopes limb-crossing metric was subsequently computed by projecting ranging from 15 ° to 35 °. The intermediate 13 ° ramp and 15 ° stair conditions were not represented in the underlying AB the inter-ankle vector onto this forward direction, 𝑇𝑇 biomechanics dataset and were included to evaluate controller 𝑑𝑑ℎ = Δ𝐩𝐩𝑎𝑎 𝐟𝐟𝑡𝑡,𝑝𝑝,ℎ (33) where 𝑑𝑑ℎ ≥ 0 indicate that the intact ankle has advanced performance in terrain regions absent from the reference data. beyond the prosthetic ankle in the prosthesis-referenced walking Due to experimental time constraints, the 13 ° ramp and 15 ° stair conditions were completed by 9 and 6 of the 10 direction and was used to trigger the T1 transition (Fig. 7C). Within the unified FSM, the slope-dependent impedance participants, respectively. Participant inclusion in these parameters derived in Section III were executed using the load- conditions was determined by available testing time rather than reactive impedance strategy proposed by Simon et al. [25]. K-level or task performance. Whole-body kinematics and During level walking and ascending locomotion, stance phase ground reaction forces were recorded using a motion-capture stiffness and damping were continuously interpolated according system synchronized with an instrumented treadmill and stair to the normalized prosthetic side vertical load, while the force plates for post-hoc analysis. During stair ascent, equilibrium angle followed the prescribed trajectory shown in participants initiated the first step with the intact limb, whereas Fig. 5B. During descending locomotion, the same load-reactive the prosthetic limb served as the leading limb during descent in interpolation was applied to stiffness and damping, whereas the accordance with how stair negotiation is commonly trained in equilibrium angle followed the proportional yielding strategy clinical practice. described in Section III. During swing flexion and extension, D. Statistical Analysis constant stiffness and damping values were used, while the The weighted squared Pearson correlation coefficient (𝑟𝑟 2 ) equilibrium angles were assigned to the slope-dependent was used to quantify how closely the experimentally measured maximum swing-flexion and heel-contact angles, respectively. prosthetic outcomes followed the slope-dependent trends of the To improve robustness against variations in user loading and corresponding model predictions and AB reference data. To gait-state transition timing, the normalized vertical load at the account for the non-uniform spacing of terrain slopes, each ES-to-LS transition was stored and subsequently used as the slope was weighed according to the interval it represented reference during the unloading phase. A representative within the slope domain, based on the spacing to adjacent slope implementation of the unified controller is illustrated in Fig. 7D, values. showing the synchronized evolution of the FSM states, prosthetic side vertical load, and continuously varying
V. RESULTS
impedance parameters throughout the gait cycle. The experimentally measured prosthesis kinematics and
C. Experimental Protocol
kinetics are compared with the target RBF-unified Ten individuals with unilateral transfemoral amputation biomechanics across the continuous terrain slope domain in Fig. (eight male and two female; 5 right and 5 left amputations or 8. Notably, the proposed controller generated smooth limb difference; age: 39.7 ± 11 years; body mass: 85.9 ± 13.9 biomechanical trajectories at the intermediate 13° ramp and 15° kg; height: 1.74 ± 0.13 m; years since amputation: 17.9 ± 11.7 stair conditions (★), although neither terrain was represented in years) participated in this study. All participants provided the underlying AB biomechanics dataset. The measured written informed consent in accordance with the approval by the trajectories remained consistent with the continuous slope Georgia Institute of Technology Institutional Review Board progression predicted by the RBF-unified model, demonstrating (H21117). successful interpolation to terrain regions absent from the Before testing, the prosthesis was fit and aligned by a reference dataset. certified prosthetist during quiet standing. Participants then Overall, the proposed controller reproduced the intended completed a static calibration trial to initialize the wearable slope-dependent joint behaviors across both the knee and ankle. IMUs and establish a common global reference frame, followed During ascending locomotion, knee flexion and ankle by an acclimation period consisting of level walking (0 °), dorsiflexion increased progressively with terrain slope
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Fig. 8. Experimental results on inter-participant average (n = 10) (A) kinematics and (B) kinetics are illustrated, by comparing with the RBFunified biomechanics models derived in Section II. The enlarged panels on the left highlight representative early stance knee angle and torque to illustrate the observed slope-dependent scaling. Red star (★) in the legend indicates the intermediate 13 ° ramp (lines with circles, n = 9) and 15 ° stair (lines with squares, n = 6) conditions, which were not included in the AB dataset used to construct the partitioned polynomial models.
throughout stance, while larger swing-phase flexion and clearance angles were observed on steeper terrains. During descending locomotion, the controller generated progressively greater knee yielding together with increased ankle dorsiflexion during late stance. Although subject-dependent timing differences were present, the experimental trajectories preserved the continuous slope-dependent progression predicted by the RBF-unified biomechanics. An additional quantitative evaluation was performed using representative kinematic landmarks and peak moments (Fig. 9). The experimentally measured quantities closely followed both the RBF-unified model and the underlying AB reference across the terrain domain. During ascent, heel-contact knee flexion, heel-contact ankle dorsiflexion, swing clearance angles, and peak knee support moments increased monotonically with terrain slope. During descent, increasing terrain steepness produced progressively greater knee toe-off angles and yielding behavior while maintaining relatively constant ankle toe-off, consistent with the proportional-yielding strategy introduced in Section III. For the ankle, the controller also preserved the continuous slope-dependent progression of heel-contact dorsiflexion, swing clearance, and toe-off angle, demonstrating that the unified parameterization successfully reproduced the
intended kinematic adaptation across both ascending and descending locomotion. Figs. 9A and 9B summarize the slope-dependent scaling performance using 𝑟𝑟 2 for each kinematic landmark and peak torque. Here, kinematic landmarks refer to joint angles at specific gait events, including heel contact during both ascending and descending locomotion, swing clearance during ascent, and toe-off during descent. Knee-related metrics showed high agreement between the proposed unified controller and the RBF-unified model (𝑟𝑟 2 = 0.82 - 0.99), while ankle kinematic metrics ranged from 0.78 to 0.96. In contrast, ankle stance moment peaks showed rather low agreement (𝑟𝑟 2 = 0.08 during ascent and 0.25 during descent). High agreement was also observed between the unified controller and the AB reference, with knee-related metrics ranging from 0.75 to 0.99 and ankle kinematic metrics from 0.85 to 0.97, indicating that the proposed controller successfully reproduced the AB biomechanical characteristics encoded by the unified model. The ankle stance torque peaks likewise exhibited a medium level of agreement with the AB reference (𝑟𝑟 2 = 0.47 during ascent and 0.51 during descent).
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Fig. 9. Representative (A) kinematic landmarks (B) peak torques, and (C) torque-relevant kinematic peaks across terrain slope. Experimental results from the proposed unified controller are compared with the RBF-unified biomechanics model and the underlying AB reference. The left and right 𝑟𝑟 2 values in each subplot denote squared Pearson correlation coefficients between the experimental results and the RBF-unified model and AB data, respectively. Bars represent mean ± 1 SD. Red star (★) in each subplot indicates the intermediate 13 ° ramp and 15 ° stair conditions, which were not included in the AB biomechanics dataset. In the posture schematics, orange arrows in (A) indicate prosthetic limb progression, whereas red arrows in (C) indicate the direction of joint torque.
13 This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. Further inspection of the torque-relevant kinematic peaks in introducing separate mode-specific parameter sets or additional Fig. 9C was conducted to examine how the realized joint FSM-based transitions. This reduces controller complexity kinematics contributed to the stance-torque outcomes. Under the while preserving continuous scaling in biomechanical behavior impedance relationship in (13), the stiffness-generated joint across the modeled terrain domain. The proposed RBF-unified biomechanics addresses a torque depends on the angular displacement from the modeled fundamental limitation of polynomial-based locomotion models equilibrium angle �𝜃𝜃 − 𝜃𝜃𝑒𝑒𝑒𝑒 �; therefore, insufficient realization by reconciling local biomechanical fidelity with global of the corresponding joint excursion can reduce the achievable continuity. While partitioned polynomial models accurately stance torque. Averaged across terrain slopes, the reconstructed each locomotion subset, directly concatenating experimentally realized knee peak stance torques reached them introduced discontinuities near the shared level-walking 101.4% and 79.3% of the RBF-model values during ascent and condition, whereas a single global polynomial excessively descent, respectively, while peak knee flexion reached 80.0% smoothed the local biomechanical characteristics. The RBF and 85.6% of the corresponding model values. In comparison, blending preserved the locally optimized biomechanics while ankle peak stance torques reached only 74.8% and 56.2% of the providing a continuous representation suitable for real-time RBF-model values during ascent and descent, respectively, controller parameterization. Importantly, the objective of the accompanied by late-stance (LS) peak dorsiflexion reaching continuous biomechanical representation is not to reconstruct only 73.7% and 66.2% of the model values. Thus, the greater unknown AB biomechanics exactly, but to provide a continuous under-realization of ankle torque coincided with insufficient LS biomimetic prior for controller design. This distinction is dorsiflexion relative to the RBF-unified biomechanics, particularly relevant within the transition region between steep providing a mechanical explanation for the lower ankle stance- ramps and shallow stairs, where changes in foot-ground contact geometry may produce substantially different ankle torque agreement observed in Fig. 9B. biomechanics despite similar terrain inclinations. Therefore, the successful implementation at the 13 ° ramp and 15 ° stair
VI. DISCUSSION
conditions should be interpreted as validation of the continuous The primary contribution of this work is the demonstration controller parameterization rather than confirmation of the that lower-limb prosthesis control can be formulated using a interpolated AB biomechanics themselves. continuous biomechanical representation spanning level Although unified biomechanics provide continuous reference walking, ramps, and stairs, rather than as a collection of trajectories, directly reproducing AB joint kinematics and independently designed locomotion modes. Existing biomimetic kinetics is neither feasible nor necessarily desirable for powered prosthesis controllers generally derive impedance parameters prostheses. The unified biomechanics served as a design prior from discrete AB locomotion datasets and therefore inherit the rather than a direct control objective, requiring transformation same mode-specific structure, requiring separate controller into impedance parameters that balance biomimetic behavior designs and transition logic for each ambulation task. In with practical implementation. Rather than optimizing all three contrast, the proposed framework represents joint biomechanics impedance parameters simultaneously [6], [7], [39], the as continuous functions of terrain slope and gait phase, enabling proposed framework derived stiffness and damping after both biomechanical modeling and controller parameterization designing equilibrium angles (Fig. 5B) based on the biowithin a unified domain. This formulation shifts controller inspired support angle �𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 � definition (Fig. 4). The 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 design from defining discrete locomotion modes to parameterizing a continuous environmental variable, providing formulation, constrained optimization, and proportional yielding a more scalable foundation for future terrain-adaptive prosthesis strategy [22], [23], [25] collectively reduced the controller control and real-world mobility. The results support two dimensionality while preserving the biomechanical behaviors hypotheses introduced in Section I. First, the RBF-unified most relevant to prosthesis function, including limb support, biomechanics preserved the reconstruction accuracy of the controlled yielding, and swing clearance. Interestingly, the constrained LSQ optimization consistently locally optimized ascent and descent models while converged to trapezoid-like stiffness and damping profiles outperforming the global polynomial representation, supporting throughout stance (Figs. 6A and 6B). These profiles emerged H1. Second, experiments with ten individuals with transfemoral naturally from the optimization while closely following the amputation demonstrated that the unified controller reproduced prescribed arctangent upper-bound envelopes, indicating that the principal slope-dependent biomechanical trends of healthy the constraints actively regulated the resulting impedance individuals, including intermediate terrain conditions absent parameters. An exception was observed for knee stiffness from the reference dataset, supporting H2. However, H2 was during ascending locomotion, where the optimized values only partially supported for ankle stance kinetics, where the remained below the allowable upper bound despite accurate experimental torque trends exhibited lower agreement both with moment reconstruction (Fig. 6C). This behavior suggests that the unified model and AB reference. slope-dependent knee support was achieved primarily through The proposed continuous formulation provides two modulation of the equilibrium angle trajectory rather than advantages. First, it enables direct interpolation to intermediate increasing joint stiffness, highlighting the complementary roles terrain conditions not present in the reference dataset, as of stiffness and equilibrium angle within the proposed demonstrated experimentally for the 13 ° ramp and 15 ° stair conditions. Second, because the controller is parameterized parameterization. The transformation from AB biomechanics to prosthesis continuously with respect to terrain slope, a single controller impedance control required deliberate departures from reference architecture can span the entire modeled terrain domain without
14 This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. joint dynamics. This was particularly important during without requiring full reproduction of the modeled dorsiflexion. descending locomotion, where the initial knee moment peak Such modifications may be particularly important during late obtained from inverse dynamics represents the net mechanical stance, consistent with prior findings highlighting the moment at the biological joint rather than a direct actuator biomechanical importance of push-off timing and power torque requirement. Such net moments reflect the combined generation [42], [43], [44]. effects of active muscle forces and passive musculoskeletal A further difference was observed during the swing structures [40] and therefore cannot be interpreted as one-to-one extension, where the experimental trajectories exhibited plateau motor commands. Specifically, during stair descent, the regions immediately before initial contact (Fig. 8A). This biological knee extensors dissipate energy eccentrically while behavior is consistent with terminal-swing stabilization the knee flexes to lower the body [41]. Therefore, directly strategies previously reported in unilateral prosthesis users. reproducing the initial moment peak with a large prosthetic Prior studies have identified a prolonged latency between full support torque would oppose the controlled knee yielding prosthetic knee extension and initial contact, interpreted as a required to “ride” the prosthesis during descent [25]. Beyond period during which users ensure a stable position before this departure from the AB knee moment during descent, loading [45], and have reported corresponding adaptations in several additional design choices intentionally prioritized terminal-swing muscle activation [46]. More recent work prosthesis functionality over one-to-one reproduction of natural further demonstrated that when prosthetic knee extension was biomechanics. The ankle support equilibrium angle was completed earlier, users tended to preserve or increase this bounded to 10 ° dorsiflexion to remain within the prosthesis latency rather than initiate ground contact earlier [47]. Thus, the operating range and preserve sufficient plantarflexion torque. observed plateau may reflect user-selected preparation for stable Stance biomechanics were represented through a three-region initial contact rather than a limitation of the continuous equilibrium angle structure, with stiffness and damping biomechanical representation. constrained to physically realizable bounds and progression Beyond the impedance parameter formulation, successful patterns rather than directly reproducing the full AB moment controller implementation also relied on the proposed bilateral trajectory. During swing, the controller similarly targeted sensing architecture (Fig. 7). Compared with conventional functional landmarks (e.g. maximum flexion for limb clearance mode-specific controllers, this architecture eliminates modeand heel-contact angle for preparation for loading) rather than dependent gait state definitions while preserving a single FSM tracking the complete continuous AB kinematic trajectory. across all modeled terrain conditions. The relative limb Collectively, these design choices preserved the biomechanical geometry was estimated using a simplified sagittal-plane functions considered most relevant to prosthesis control while forward kinematics model based on wearable IMUs, without intentionally relaxing exact replication of natural joint behavior. explicitly accounting for out-of-plane pelvic motion or hip tilt. Remaining discrepancies primarily reflect implementation Consequently, the computed geometry should be interpreted as choices rather than limitations of the underlying biomechanics. a robust gait state variable rather than an anatomically exact The largest kinematic differences occurred at heel-contact for estimate of bilateral limb configuration. Despite this both joints, consistent with the constant heel-contact angle simplification, the experimental results demonstrated reliable offsets introduced to improve prosthetic stability during initial gait state transitions across all evaluated terrain conditions, contact (Section IV-C). The ankle kinetic discrepancy highlights suggesting that the proposed geometry provides sufficient a tradeoff introduced by the mode-unified state transition information for unified controller implementation. strategy. In conventional mode-specific control, the ES-to-LS For additional reference, the biomechanical replication results transition can incorporate an ankle angle threshold, ensuring of the proposed controller were examined alongside previously that sufficient dorsiflexion develops before entering late stance reported results using a mode-specific, hand-tuned controller and thereby preserving the angular displacement �𝜃𝜃 − 𝜃𝜃𝑒𝑒𝑒𝑒 � [24], as detailed in Appendix and Fig. 10. Although both studies required for modeled torque. Such thresholds, however, vary evaluated common terrain conditions (± 5 ° ramps and ± 24 ° substantially across different modes and are difficult to retain stairs), they involved different participant groups, experimental within a single mode-independent transition rule. The proposed protocols, and hardware capabilities. In specific, the previous bilateral limb-progression criterion removes this dependence but study employed the first-generation OSL [35], with ± 15 ° of decouples the ES-to-LS transition from the desired peak ankle motion, whereas the proposed study used the seconddorsiflexion angle. Consequently, LS may be initiated before generation OSL [36], with an expanded range of ± 30 °. This sufficient dorsiflexion develops, limiting the angular hardware difference is particularly relevant to stair locomotion, displacement and the resulting ankle torque. This consideration where the AB-referenced ankle heel-contact and swing is particularly relevant during steep stair descent, where the clearance angles exceeded the physical limit of the earlier OSL. RBF-unified ankle dorsiflexion trajectory can exceed the Therefore, the observed differences in biomechanical mechanical range of the current prosthesis (Fig. 9C, Ankle LS replication cannot be attributed solely to the control strategies, Peak Dorsiflexion for -20 °, -24 °, -30 °, and -35 °), making and the comparison should be interpreted as descriptive rather direct reproduction of the reference kinematics infeasible. than a direct assessment of controller superiority. Detailed Accordingly, strategies such as prescribing a more plantarflexed comparison methods and numerical results are provided in equilibrium angle or accelerating the progression of impedance Appendix. parameters may help generate greater stance ankle torque
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Fig. 10. Descriptive comparison of biomechanical outcomes from the previous reported mode-specific controller [24] and the proposed modeunified controller relative to the AB reference. (A) Knee and ankle kinematic landmarks and peak stance torques across common terrain conditions (± 5 ° ramps and ± 24 ° stairs). Bars and error bars denote cross-subject mean ± 1 SD for the previous study (blue, n=9) and the proposed unified controller (red, n = 10), while dashed black lines indicate the corresponding AB reference values. Green circles indicate lower MAEs with the proposed controller, whereas red circles indicate no improvement. Green-boxed annotations and arrows highlight the six statistically significant between-study differences favoring the proposed controller, along with the corresponding MAE reductions relative to the AB reference. Significance was assessed using Welch’s two-sample t-tests (*𝑝𝑝 < 0.05, **𝑝𝑝 < 0.01, ***𝑝𝑝 < 0.001, ****𝑝𝑝 < 0.0001). (B) Weight- and stance phase-normalized vertical ground reaction force (vGRF) profiles of the prosthetic and intact sides during 24 ° stair ascent and descent. Solid lines and shaded regions denote cross-subject mean ± 1 SD.
Nevertheless, the proposed framework offers a distinct advantage in controller parameterization. A conventional twojoint, four-state impedance FSM requires at least 24 independently specified stiffness, damping, and equilibriumangle parameters per locomotion mode [24], [25]. In contrast, the proposed framework derives these parameters continuously from terrain slope and gait phase using a unified biomechanical
representation, without independently hand-tuning a complete parameter set for each mode. This formulation reduces the reliance on mode-specific controller design while retaining the principal slope-dependent biomechanical characteristics demonstrated in proposed framework. Beyond the AB-replication analysis, Fig. 10B provides additional insight into how participants loaded the prosthesis
16 This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. during the matched overground stair conditions. During stair biomimetic control framework that reduces mode-specific ascent, the proposed controller was associated with greater tuning while enabling users to address varied terrain without prosthetic-side vGRF throughout stance while intact-side relying on separately parameterized controllers for each loading remained comparatively similar between controller locomotion modes. groups. This pattern suggests greater weight acceptance through the prosthetic limb rather than a compensatory increase in APPENDIX – COMPARISON WITH PREVIOUS STUDY intact-side loading. Stair descent loading profiles were The comparison in Fig. 10 examined biomechanical comparatively similar between controllers. Because the present outcomes from the previously reported mode-specific controller analysis was limited to stance-phase vGRF profiles and did not (n = 9) [24] and the proposed unified controller (n = 10) across include complementary measures such as vertical impulse or four common terrain conditions (± 5 ° ramps and ± 24 ° stairs). loading symmetry, these findings should be interpreted as The same biomechanical landmarks evaluated in Figs. 9A and evidence of altered limb-loading behavior rather than a direct 9B, including heel-contact, swing-clearance, and toe-off angles, measure of functional improvement [48], [49]. together with peak stance torque, were extracted from both The present study has several limitations that motivate future datasets. Subject-level errors relative to the AB references were research. First, the proposed controller assumed continuous calculated, and between-study differences in these errors were terrain slope information as a high-level input. Although the assessed using Welch’s two-sample t-tests to account for controller architecture is independent of how this information is unequal size and potentially unequal variances (*𝑝𝑝 < 0.05, obtained, integrating onboard slope estimation methods based **𝑝𝑝 < 0.01, ***𝑝𝑝 < 0.001, ****𝑝𝑝 < 0.0001). on wearable sensing [6], [9], [10] or environment perception Figure 10 presents the corresponding experimental outcomes [26], [50], [51] will be necessary for fully autonomous alongside the AB reference, with annotations highlighting the operation. differences in mean absolute error (MAE) between studies. Second, the proposed continuous biomechanics were derived Among the 24 evaluated comparisons, six exhibited from AB datasets collected under self-selected or relatively significantly lower MAEs in the present study. Knee swing slow walking speeds (up to 0.6m/s). Walking speed was not clearance MAE during ramp ascent decreased from 27.1 ° to 6.6 explicitly included as a model parameter, and the resulting ° (𝑝𝑝 < 0.0001). Ankle heel-contact MAE decreased from 14.2 impedance representation is expected to be most applicable ° to 2.2 ° during stair ascent (𝑝𝑝 < 0.01) and from 19.3 ° to 5.8 ° within similar speed ranges. Extending the framework to during stair descent (𝑝𝑝 < 0.0001). Ankle swing-clearance incorporate continuous speed adaptation represents an important MAE during stair ascent decreased from 12.6 ° to 2.4 ° (𝑝𝑝 < direction for future work and could enable unified control 0.01), while ankle toe-off MAE during ramp descent decreased across both terrain and walking speed variations [6], [15], [52]. from 16.3 ° to 2.5 ° (𝑝𝑝 < 0.0001). Knee peak stance-torque Third, the unified biomechanics were constructed by MAE during stair descent decreased from 0.52 to 0.28 Nm/kg blending independently modeled ramp and stair datasets. The (𝑝𝑝 < 0.05). interpolated biomechanics within the transition region between Several additional outcomes showed lower mean MAEs with steep ramps and shallow stairs should be interpreted as the proposed controller, although the between-group differences controller parameterization rather than true AB biomechanics. did not reach statistical significance. For example, knee toe-off Despite these limitations, the proposed framework establishes angle MAE decreased from 8.81 ° to 4.38 ° during ramp descent a continuous biomimetic foundation for terrain-adaptive and from 11.60 ° to 6.28 ° during stair descent, while ankle prosthesis control that unifies biomechanical modeling, heel-contact angle MAE during ramp ascent decreased from impedance parameterization, and controller implementation 5.08 ° to 2.55 °. Knee peak stance-torque MAE also decreased within a single representation. Future integration of autonomous from 0.33 to 0.18 Nm/kg during ramp descent. Lower errors terrain estimation, continuous speed adaptation, and user- were additionally observed for knee swing clearance angle and specific optimization has the potential to further extend this knee peak stance torque during stair ascent. In contrast, ankle framework toward practical daily-life prosthesis control. peak stance torque did not show a consistent improvement across terrain conditions, consistent with the comparatively
VII. CONCLUSION
weaker reproduction of ankle kinetics identified for the This paper presented a mode-unified biomimetic control proposed controller in Fig. 9B. To further examine whether differences in prosthetic joint framework that represents lower-limb biomechanics continuously as functions of terrain slope and gait phase. The behavior were accompanied by changes in limb loading, RBF-unified models preserved the accuracy of locally weight-normalized vGRF profiles were visually compared optimized biomechanics while enabling continuous impedance during the matched overground stair conditions (Fig. 10B). parameterization across level walking, ramps, and stairs. These comparisons were restricted to stairs, for which both Experiments with ten prosthesis users demonstrated continuous controller groups were tested overground, to characterize terrain adaptation, including 13 ° ramp and 15 ° stair conditions differences in limb loading. During stair descent, prosthetic- and absent from the reference AB dataset. The proposed controller intact-side loading remained comparable between controllers. reproduced the principal slope-dependent trends (𝑟𝑟 2 = 0.82-During stair ascent, however, the proposed controller produced 0.99 for all knee-related measures and 0.78 - 0.96 for ankle a higher prosthetic-side vGRF than the previously reported kinematics), while ankle stance torque remained less accurately controller, whereas intact-side loading remained comparable. reproduced. Together, these findings support our unified
17 This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. Detroit, MI, USA: IEEE, Oct. 2023, pp. 2146–2151. doi: