Adaptive Ankle-Foot Prosthesis with Passive Agonist-Antagonist Design

Adaptive Ankle-Foot Prosthesis with Passive Agonist-Antagonist Design Matteo Crotti1,2, Anna Pace1, Giorgio Grioli1,2, Antonio Bicchi1,2, Manuel G. Catalano1 Abstract— The development of prosthetic feet that closely replicate the natural biomechanics of the human foot remains a significant challenge in prosthetics engineering. This paper presents the design and testing of a novel agonist-antagonist architecture for the ankle joint of a passive prosthetic foot featuring an adaptive sole. The ankle mechanism, inspired by the dynamics of the human leg-ankle-foot complex, utilizes compliant elements in an agonist-antagonist configuration to passively achieve an ankle torque close to that of a sound ankle without the need for external actuation. Concurrently, the adaptive sole adjusts its shape in response to different terrains, potentially improving stability and comfort for the user. The theoretical model underlying the proposed design is presented, followed by a preliminary validation through simulations. Finally, a prototype based on the new architecture is tested by a healthy subject using customized walking boots, demonstrating its potential to improve the functional performance of prosthetic feet in diverse environments. I. INTRODUCTION Prosthetic foot design has advanced significantly, aiming to restore mobility and improve lower-limb prosthetic users’ (LLPUs) quality of life. Despite these advancements, designing a prosthetic foot that closely mimics the complex dynamics of the human foot remains challenging. Consequently, LLPUs usually show gait asymmetries, reduced walking speed, and rapid fatigue, and their stability is jeopardized when walking on unevenness with the associated risk of falling [1], [2], [3], [4]. Energy-storing-and-returning (ESR) prosthetic feet aim to lower the metabolic cost of walking by mimicking the human ankle’s energy-recycling function. Made of carbon composite springs, these passive devices store energy when loaded during the stance phase of walking and release it during push-off unloading, but only until the foot goes back to its unloaded configuration, lacking an active plantarflexion and power generation. This results in limited energy return and higher metabolic costs [5], [6]. Various passive designs use clutch-and-spring mechanisms to enhance push-off (e.g. [6], [7]). Only active devices that inject energy with motors can provide the net positive energy required during walking. Nonetheless, they offer only slight improvements over ESR feet [8], with persistent joint asymmetries [9], and they also tend to be costly, heavy, complex, and energy-intensive [10]. Simulations suggest elastic structures may suffice for efficient walking without active ankle actuation [11]. Some This research has received funding from the European Union’s Horizon 2020 Research, ERC programme Grant No.810346 (Natural BionicS). (1)Soft Robotics for Human Cooperation and Rehabilitation, Fondazione Istituto Italiano di Tecnologia, Genoa 16163, Italy. (2)Department of Information Engineering and Research Center “E. Piaggio,” University of Pisa, Pisa 56122, Italy. Fig. 1. The Agonist-Antagonist SoftFoot: a prosthesis with the passive agonist-antagonist ankle mechanism. research prototypes embody compliant elements as series or parallel elastic actuators to reduce actuation weight and improve performance. They maximize the energy stored during stance to be released during push-off by actively controlling the locking mechanism, pre-loading elastic elements, or combining these two principles [12], [13], [14], [15]. Despite the advantages of active ankle control and advanced energy storage mechanisms, ankle-foot prostheses still use rigid, flat foot soles made of carbon fiber, limiting ground adaptation on uneven surfaces [16]. This rigidity increases instability, fall risk, and long-term issues like osteoporosis [16]. A few years ago, we developed the SoftFoot [17], an entirely passive artificial foot with a flexible sole for enhanced ground adaptation in robotic applications. The SoftFoot adjusts its shape based on the ground profile, mimicking human adaptation and filtering obstacles to reduce fall risk. Tests on the HRP-4 robot showed superior obstacle negotiation compared to its original rigid feet [18]. A preliminary prosthetic foot prototype with an adaptive sole and ankle joint, similar to the robotic design, showed good obstacle negotiation in virtual simulations but poor push-off power generation. The Achilles tendon’s stretch and recoil cycle is essential for human gait, as it reduces metabolic cost, propels the body forward, and contributes nearly 84% of total ankle push-off power [19], [20]. We addressed the power limitation by adding two compliant elements to the ankle joint, replicating the agonist-antagonist muscles and tendons in the human ankle. This paper introduces a novel prosthetic ankle-foot device with an adaptive, passive design for natural gait on all terrains (Fig. 1), featuring an agonist-antagonist ankle system that stores and releases energy like in the human foot, and an 2025 IEEE International Conference on Robotics and Automation (ICRA) May 19-23, 2025. Atlanta, USA 979-8-3315-4139-2/25/$31.00 ©2025 IEEE 967 2025 IEEE International Conference on Robotics and Automation (ICRA) | 979-8-3315-4139-2/25/$31.00 ©2025 IEEE | DOI: 10.1109/ICRA55743.2025.11127721 Authorized licensed use limited to: University of Michigan Library. Downloaded on September 09,2025 at 15:46:57 UTC from IEEE Xplore. Restrictions apply. adaptive sole for continuous ground adaptation. The passive ankle provides shock absorption at heel strike and braking torque during stance while storing energy for propulsion at push-off. In the following sections, we will present a detailed design description of the foot-ankle prosthesis, including the principles underlying its functioning, mechanisms, and testing in simulation and in the real environment. II. AGONIST-ANTAGONIST ANKLE ARCHITECTURE We introduce a mathematical model describing a simplified ankle-foot complex scheme (represented in Fig. 2) to show how the elastic elements at the ankle joint replicate the human energy-recycling behavior. The model builds on the one presented in [17], which includes the foot sole, an anterior arch, and a posterior arch. We added a shank link to accommodate the elastic elements (see Fig. 2 (a) and Section IV-A for a detailed explanation). The shank is rigidly connected to the ankle joint, while the anterior and posterior arches are hinged at the ankle, allowing free relative motion. The anterior arch is linked to the rear side of the shank via a posterior spring and to the sole at the metatarsal level via a revolute joint. Similarly, the posterior arch connects to the front of the shank via an anterior spring and to the sole at the heel level. The elastic elements at the ankle joint, in series with the arches (see Fig. 2(a)), provide torque and energy recycling similar to human agonist-antagonist muscles. The anterior spring activates during plantarflexion. It absorbs shock by stretching during load acceptance and recoils, returning the ankle to its neutral position. It also stretches during push-off, recoiling in the swing phase for better foot-ground clearance. The posterior spring activates during dorsiflexion. It provides a resistive torque as the ankle dorsiflexes during mid-stance, braking the user and storing energy, allowing a smooth forward body progression. This energy is released during push-off, aiding in plantarflexion and forward propulsion, replicating the Achilles tendon’s role. A. Mathematical Model The mathematical model presented in [17] describes the relation between foot configuration, external forces applied on the foot, and the foot constructive parameters. Here, we expand that model with the new agonist-antagonist scheme (Fig. 2), including the action of the compliant elements on the ankle when the ankle-foot complex is in static balance. In this section, we will refer to the variables, the reference frames, and the rigid bodies defined in Fig. 2 (b). The variable q= [q0...qn+2]∈Rn+3collects the configuration of the foot sole and the phalanges. Imposing the force and torque equilibria of the foot as a whole yields FP=F1+F2+F3 FP=bCβ+aCα xH F2+bCβ+aCα+L(Cn+Cn+1) xH F3(1) Where F1,F2,F3are the three ground reaction forces, Fp is the force applied on the foot by the LLPU, xHis the projection of the application point, a,bare, respectively, the length of the posterior arch and the anterior arch, αand βare geometric constraints (see Fig. 2 (b)). For the sake of compactness, we use the notation Cα=cos(α),Cβ=cos(β). We can define the force exerted by the two compliant elements with Hooke’s law:  Fa=ka ∆la  Fb=kb ∆lb(2) Where  Fa, Fbare the forces exerted by, respectively, the anterior and posterior spring, ka,kbare their stiffness, and  ∆la, ∆lbare the length variation from the rest condition, which depends on the foot configuration:  ∆la=a1cos(α−α0) sin(α−α0)  ∆lb=b1cos(β−β0) sin(β−β0)(3) Where α0and β0are the αand βangles values when the foot is at rest with no force applied, and a1,b1are the length of the two arches from the ankle joint to the spring attachment. Through the force balance along xof the foot rigid bodies, we obtain ∀i∈ {1,...,n}: Rx 1=Rx T Rx i=Rx i+1 Rx M+Rx n=Rx n+1 Rx n+1=Rx n+2 Rx n+2=0 Rx T=Rx a+Fx a Rx M=Rx b+Fx b Rx a+Fx a=Rx b+Fx b(4) where Rx iis the horizontal reaction force exerted on the ith link by the ith−1 link. Rx Tand Rx Mare the horizontal reaction forces due to the interaction between the sole and the arches. Rx aand Rx bare the horizontal reaction forces due to the arches and ankle joint interaction. Simple algebraic manipulations allow to transform (4) in Rx n+1=Rx n+2=0 Rx T=Rx M Rx i=−Rx M Rx a=Rx T−Fx a Rx b=Rx M−Fx b(5) With the same steps, the balance along yis ∀i∈ {1,...,n}: Ry 1=F1−Ry T Ry i=Ry i+1 Ry M+Ry n+1=Ry n+F2 Ry n+1=Ry n+2 Ry n+2=−F3 Ry T=Ry a+Fy a Ry M=Ry b+Fy b Ry a+Fy a+Ry b+Fy b=Fp(6) 968 Authorized licensed use limited to: University of Michigan Library. Downloaded on September 09,2025 at 15:46:57 UTC from IEEE Xplore. Restrictions apply. Fig. 2. Simplified model of the agonist-antagonist foot-ankle complex. (a) Main elements of the system. The plantar fascia is implemented by the set of links (4-5-6) and the tendon (in green), which runs from the calcaneus to the tip of the toe. Bodies (4-5-6-7-8) are connected through a spring of stiffness, e. Bodies (2-4) are also connected through a spring of stiffness e0. Links (7-8) represent the phalanxes of the toes. (b) Forces acting on the foot: Fpis the force weight of the user, F1,F2,F3are the three considered contact forces. The other parameters represent the geometric values defining the static configuration of the system. Where Ry i,Ry T,Ry M,Ry a,Ry bare the vertical reaction forces for the interaction in the foot components as for the xaxis. Solving (6) yields Ry n+1=Ry n+2=−F3 Ry T=−Ry M+F1+F2+F3 Ry i=Ry M−F1−F2 Ry a=Ry T−Fy a Ry b=Ry M−Fy b(7) Finally, we evaluate the torque balance for the bodies 4, 5, 6, 7, 8 mn+2=−F3LCn+1 mi−mi+1+Rx i+1LSi−Ry i+1LCi=0 mn−mn+1+Rx nLSn−Ry nLCn=0 mn+1−mn+2+Rx n+1LSn+1−Ry n+1LCn+1=0 (8) where miis the torque exerted on the ith link. We used the abbreviations Si=sin(∑i 1qj),Ci=cos(∑i 1qj). By substituting (5), (7) in (8), we get ∀i∈ {0,...,n} mn+2=−F3LCn+2 mn+1−mn+2=−F3LCn+1 mi−mi+1=Rx MLSi+(Ry M−F2−F3)LCi=0 (9) As in [17], this can be expressed in matrix form as Mn+3m+Lx+Ly=0 (10) where m∈Rn+3collects the terms mi Mm=         1−1 0 ... 0 0 1 −1.... . . . . ..........0 0... 0 1 −1 0... 001         ∈Rmxm (11) Lx∈Rn+3collects the terms Rx MLSi, and Ly∈Rn+3 collects the terms (Ry M−F2−F3)LCiand −F3LCn+2. The torque mis due to elastic effects and the tendon coupling and can be explicitly evaluated to be ∀i∈ {1,...,n+2} m0=−e0(q0−θ)+r0T,mi=−eiqi+riT(12) Where eiis the elastic constant of the spring, Tis the tendon tension, riis the transmission ratio (i.e., the pulley radius) on the ith joint, θis the pretension of the first spring. As in [17], we express (13) in matrix form m=−Eq −eEθ+RTT(13) Where eE= [e0,0,...,0]Tmaps the effect of the spring connected to the structure, E∈Rn+3xn+3collects the elastic elements ei, and R∈R1xn+3the transmission ratios ri. Finally, by torque balancing the two arches and the shaft Ry bb2Cβ−Rx bb2Sβ+Fy bb1Cβ−Fx bb1Sβ=0 Rx aa2Sα−Ry aa2Cα−Fy aaCα−Fx aaSα−e0(q0−θ) = 0 (Rx a−Rx b)hp−(Ry a+Ry b)xp−Fy bxb+Fy axa=0 (14) Where b2,a2are the length of the arches segment connecting the ankle joint to the sole. xp,hp,xb,xaare geometric constraints relative to the shaft link. Considering the relations in (5) and (7) we can retrieve the components of RM: Rx M=Fx a−Fx b 2−Fy a xa 2hp −Fy b xp+xb 2hp −Fp xp 2hp Ry M=Fx a tan(β) 2+Fx b 2b1−b2 2b2 tan(β)−Fy a xa+xb 2hp tan(β) −Fy b(b1 b2 +xp+xb 2hp tan(β))+Fp xp 2hp tan(β) Now, if we include the constraint imposed by the ground structure L∑n+2 0Si=δ, and by the tendon Rq=σ, where σis the tendon length and δthe terrain height, we can collect (1), (10) and (13) to obtain the same set of n+7 969 Authorized licensed use limited to: University of Michigan Library. Downloaded on September 09,2025 at 15:46:57 UTC from IEEE Xplore. Restrictions apply. nonlinear equation in the n+7 unknown quantities q∈ Rn+3,F1,F2,F3∈Rand T∈R, resulting in [17]. Similar to the SoftFoot, the mathematical model can be numerically solved to accurately describe the system’s static behavior. However, since our goal is to develop a tool for modelbased design of this adaptive agonist-antagonist system, we introduce some approximations to obtain a closed-form solution for the nonlinear system. Introducing the small angle hypothesis and applying the same mathematical steps done in [17], we finally obtain: q=− I−E−1[RTdT]R cE−1[RTdT]−1R c! ×E−1mE +E−1[RTdT]R cE−1[RTdT]−1σ δ L(15) This equation provides the closed-form relationship between the floor configuration q, external forces, and system parameters, such as spring characteristics and arch geometry. All these factors influence the final equilibrium point. External forces determine the equilibrium point once the mechanical properties are fixed, allowing the user to control the device’s stability. Conversely, with external forces fixed, equilibrium is set by adjusting the foot and spring mechanics. III. PRELIMINARY SIMULATIONS To preliminary evaluate the device’s performance featuring the architecture described in Section II, we conducted simulations in the MuJoCo physics engine before its physical implementation. Simulations employed a simplified humanoid model (whose anthropomorphic data comes from [21]) with PID-controlled joints and lumped upper body mass, following reference trajectories from human gait data [22]. A model of the prosthetic device (described in Section IV) was realized in MuJoCo, with the same process illustrated in [23], and used to replace the right foot of the humanoid model. Through simulations, we were able to assess the torque and power profile of the ankle joint of the prosthesis during the gait cycle, focusing on push-off power generation for forward body propulsion. Moreover, by simulating the interaction between the adaptive sole and the ground, we could analyze its ground adaptation, gaining valuable insights into stability and obstacle negotiation. We also compared the results obtained with those of the same device deprived of the compliant elements to shed light on the potential of the agonist-antagonist scheme. We first simulated the humanoid model attempting to maintain an upright stance by actively moving the upper body to shift its center of mass (see Fig. 3(a)). On level ground, results showed that the agonist-antagonist system stabilized faster and with fewer oscillations (Fig. 4). This improved stability was due to the compliant elements, which minimized the movement required in the center of mass for balance. In contrast, without these elements, the model relied more on the upper body movement to reach stability. On an arch-shaped obstacle (see Fig. 3(b)), a stable equilibrium Fig. 3. Simulations in MuJoCo with the humanoid model with a prosthetic device: (a) standing equilibrium on level ground, (b) standing equilibrium on an obstacle, (c) level walking. Fig. 4. Equilibrium analysis in MuJoCo simulation with (right) and without compliant elements (left). The picture illustrates the transient motion of the humanoid center of mass (zmp) when asked to stand still (zmp des). was only possible with compliant elements, as without them, the model could not stabilize and fell. Then, we conducted simulations of level ground walking (see Fig. 3(c)). The agonist-antagonist system, which will be referred to as the Agonist-Antagonist SoftFoot (AA SoftFoot) from now on, showed clear advantages during push-off, achieving an ankle torque peak of 1.25 Nm/kg - nearly double that achieved without springs, and a power peak of 2 W/kg, compared to only 0.15 W/kg without springs (see Fig. 5). IV. A PASSIVE AGONIST-ANTAGONIST ADAPTIVE FOOT Given the encouraging results from the simulation for the agonist-antagonist mechanism, we developed a prototype of the proposed ankle-foot prosthesis starting from the robotic SoftFoot [17]. A. Agonist-Antagonist Ankle The AA SoftFoot features three main rigid links: the anterior and posterior arches and the shank link. The two arches replicate the longitudinal arches of the human foot. They are pivotally coupled with the shank component through a revolute joint representing the ankle joint, and they host one end of the anterior and posterior springs (see Fig. 6). In the robotic SoftFoot, the anterior arch was hinged to the posterior arch (rigidly connected to the robot leg) via the shaft acting as the ankle joint. In this design, the shank link is fixed to the ankle’s revolute joint, enabling connection to the user’s residual limb for load transmission. Additionally, it serves as the attachment point for one end of both the posterior and anterior springs, as depicted in Fig. 6. The shank design in 970 Authorized licensed use limited to: University of Michigan Library. Downloaded on September 09,2025 at 15:46:57 UTC from IEEE Xplore. Restrictions apply. Fig. 5. Ankle torque (a) and power (b) comparison between foot with (in red) and without (blue) compliant elements, during a simulated gait cycle in MuJoCo. Fig. 6. Exploded view of the AA SoftFoot. A detailed view of the sole (1) shows the tendon (in red) and the elastic bands (in blue). The anterior and posterior arches (2,5) are hinged to the ankle joint, which is, in turn, rigidly connected to the shank link (7). The anterior (3) and posterior (6) springs connect the arches to the shank link. The heel (4) connects to (5) with a revolute joint. the proposed prototype was dictated by the need to fit a sixaxis load cell at the interface between the prosthesis and the user’s leg. B. Adaptive Sole The adaptive sole of the prototype is the result of a redesign of the robotic SoftFoot sole [17], which consists of five parallel chains made of basic modules coupled with a revised version of Hillberry’s joint (see Fig. 6). The design of the basic module has been simplified, removing the geared coupling interface. Elastic bands in the module couplings generate torque to maintain the modules’ resting positions. Chains connect the heel to the anterior arch at the metatarsal level, extending to the phalanges forming the toes. Each chain houses a tendon running from the heel to the toe tip, exhibiting a stiffening-by-compression effect, meaning the sole’s stiffness changes based on the load and environmental conditions. In this way, the prototype sole conforms to ground irregularities but can also stiffen to act as a rigid lever to propel the body forward. C. Biomechanical Characterisation The AA SoftFoot requirements are based on kinematic and dynamic data from level walking studies ([24], [22], [25]), focusing on torque and power profiles. The target peak values were 1.6 Nm/kg and 4.1 W/kg [22]. The prototype was designed for a 75 kg person, representing the 50th percentile of the adult male population [21]. As a consequence, the specifications to be satisfied with our design are: •120 Nm ankle torque peak at 10◦dorsiflexion angle right before push-off. •Null ankle torque at about 5◦of dorsiflexion for foot clearance during the swing phase. •Minimize the torque when there is no ankle rotation (i.e., 0◦) to attenuate unbalancing torques while standing. •An ankle range of motion of approximately −20◦(plantarflexion) to 15◦(dorsiflexion) [25]. •Comply with the average anthropometric measures of the ankle-foot complex [21]. Given those targets, we sized the arches and determined the spring’s characteristics accordingly, and assessed the design with FEM analysis. However, considering their biomechanical requirements, the sizing and characterization could be adjusted for different users. The final results obtained for the device are close to the aimed goals: •120 Nm ankle torque peak at 10◦of dorsiflexion. •Null ankle torque for a 4◦dorsiflexion angle. •14.9 Nm ankle torque when there is no ankle rotation. •An ankle joint range of motion from −10◦to 10◦. •The AA Softfoot is 261 mm long, has a mass of 1.65 kg and a height of 220 mm. The limits in the range of motion are achieved through mechanical stops on the arches. V. EXPERIMENTAL VALIDATION A. Experimental Setup and Protocol An experimental session was conducted to assess the adaptability of the AA SoftFoot. An 84 kg healthy subject, wearing modified walking boots fitted with prosthetic devices (as in [26]), was asked to walk over an arc-shaped wooden obstacle (see Fig. 8). Tests were performed with three device pairings: the left leg used a Vari-Flex ( ¨ Ossur, Reykjavik, Iceland), while the right leg tested the Vari-Flex, AA SoftFoot, and AA SoftFoot without springs. The boots added 3.7 kg to 5.1 kg to the user mass, depending on the device. The participant walked at a self-selected speed and stepped onto the obstacle, starting from a distance identified to keep a natural cadence throughout the trial, repeating the task six times for each device pairing. A 6-DoF iPecs Lab load cell (RTC Electronics, Dexter, MI, USA), positioned between the boot of the right leg and the prosthetic device (Fig. 8), was used to record forces and torques at a 500 Hz sampling rate. B. Data Processing Raw data were processed in Matlab (v. 2020b, MathWorks Inc., Natick, MA, USA) using a moving average filter 971 Authorized licensed use limited to: University of Michigan Library. Downloaded on September 09,2025 at 15:46:57 UTC from IEEE Xplore. Restrictions apply. Fig. 7. Forces recorded by the load cell for the three tested prostheses over the gait cycle: mean values during level walking (solid lines) and when stepping onto an obstacle (dotted lines). Forces are given in the load cell reference frame: the x-axis corresponds to the mediolateral axis pointing in the lateral direction, the y-axis to the anteroposterior axis pointing in the anterior direction, and the z-axis to the vertical axis pointing upwards. Fig. 8. Close-up view of the three prosthetic devices tested when walking on an arched obstacle. From left to right: Vari-Flex, AA SoftFoot without springs, AA SoftFoot. The adaptability of the sole is highlighted. The load cell placed between the prosthesis and the boot is also visible. applying a 30 ms window and subsequently normalized to the participant’s body mass and weight for torques and forces respectively. Mean values and standard deviations were calculated across six gait cycles. C. Experimental Results In Fig. 7 we reported the force measurement in the three axes of the iPecs comparing the three device pairings when stepping on an obstacle and during level walking. Fig. 8 shows the different behavior of the prosthetic device when dealing with an obstacle. See the attached video for a complete and detailed view of the performed experiments. VI. DISCUSSION The effect of the adaptive sole is evident on the three axes (Fig. 7) when walking on an obstacle. Specifically, the difference between the forces recorded for the same device during level walking and obstacle stepping is reduced due to the increased contact surface (see Fig. 8). The AA SoftFoot appears to be the least affected by the occurrence of a perturbation. The forces recorded in the presence of the obstacle are closer to those measured during level walking in all three axes compared to the spring-less version. This is likely due to the balancing torque of the agonist-antagonist system, which could mean improved stability for the user. Fig. 7 shows that the peak force in the z-direction, which occurs at the end of the load acceptance phase, is higher for both the AA SoftFoot and the SoftFoot compared to the Vari-Flex. This difference may be due to the adaptive sole. As the contact surface with the obstacle increases, the user may feel more stable and confident in loading the prosthetic limb. Despite these promising results, several limitations should be acknowledged. Firstly, the ankle-foot prototype was designed based on the torque and power profile of level walking. In other situations, it may be less effective or unfavorable with the current mechanical tuning. Furthermore, the design’s optimality depends on user mass, and in this study, it was tested on a single healthy subject with a mass higher than that considered for the design. The current range of motion is limited compared to the physiological range of the human ankle. This could constrain the ability to replicate natural gait patterns, potentially affecting the user’s gait. Moreover, the current results may have been influenced by the variability introduced by the subject’s step adjustment to step on the obstacle. Future works will focus on testing the prototype with prosthetic users to better evaluate its performance and impact on the whole users’ gait and stability on various grounds. VII. CONCLUSION This paper introduced an agonist-antagonist architecture for adaptive ankle-foot passive prostheses. After mathematically describing the ankle-foot architecture and demonstrating that equilibrium depends on the user load and the mechanical characteristics of the system, we conducted preliminary simulations to evaluate its ability to generate ankle power and torque. We then presented the first prototype of the Agonist-Antagonist SoftFoot, an ankle-foot prosthesis featuring an adaptive sole and a passive ankle joint based on this architecture. 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