Structural FEA-Based Design and Functionality Verification Methodology of Energy-Storing-and-Releasing Prosthetic Feet

Abstract

The prosthetic feet that are most often prescribed to individuals with K3/K4 levels of ambulation are the ESR feet. ESR stands for energy-storing and -releasing. The elastic energy is stored by the elastic elements in composite materials (carbon fiber or glass fiber). ESR feet must be developed and optimized in terms of stiffness, taking into account the loads that a healthy human foot undergoes and its kinematics while walking. So far, state-of-the-art analyses show that the literature approaches for prosthetic foot design are not based on a systematic methodology. With the aim of optimizing the stiffness of ESR feet following a methodological procedure, a methodology based on finite element structural analysis, standard static testing (ISO 10328) and functional verification was optimized and it is presented in this paper. During the path of optimization of the foot prototypes, this methodology was validated experimentally. It includes the following: (i) geometry optimization through two-dimensional finite element analysis; (ii) material properties optimization through three-dimensional finite element analysis; (iii) validation test on physical prototypes; (iv) functionality verification through dynamic finite element analysis. The design and functional verification of MyFlex-$\gamma$, a three-blade ESR foot prosthesis, is presented to describe the methodology and demonstrate its usability.

1. Introduction

Current commercially available prostheses are mostly energy-storing -and -releasing (ESR) feet and they are the most prescribed prosthetic feet for individuals with K3 and K4 levels of ambulation. ESR feet are passive prosthetic devices made of elastic elements, which ensure the ESR feet work as springs that store energy during the mid-stance of the gait cycle and release it for the propulsion during late stance [1,2]. The elastic elements are generally leaf springs, also called blades, of composite materials (carbon fiber-reinforced plastic—CFRP; or glass fiber-reinforced plastic—GFRP). The stiffness of the elastic elements is a crucial characteristic of foot prostheses. Stiffness depends on the geometries and the material properties, especially for ESR feet [3,4], and the choice of the global stiffness category of foot prostheses depends on the weight and the users’ activity levels [5]. Moreover, the few active ankle prosthetic devices on the market are joined to feet by a system of composite leaf springs (see

Table 1. Commercial active and semi-active foot prostheses with composite elastic foot.

Manufacturer	Model	Website (Access Date)	Country
Blatchford	Elan	www.blatchford.co.uk (1 November 2021)	UK
Blatchford	Elan${}^{IC}$	www.blatchford.co.uk (1 November 2021)	UK
Fillauer	Raize	www.fillauer.com (1 November 2021)	USA
Freedom-Innovations	Kinnex 2.0	www.freedom-innovations.com (1 November 2021)	USA
Össur	Proprio Foot	www.ossur.com (1 November 2021)	Iceland
Ottobock	Empower	www.ottobock.com (1 November 2021)	Germany

for remarkable examples). The contribution of the elastic elements is always significant.

The present work aims to propose and apply a systematic methodology that includes elastic elements in the design process of prosthetic feet. The methodology combines numerical and experimental techniques for the design and the functional verification of foot prostheses. The proposed methodology consists of three main phases: (i) the design phase, which includes the stiffness and safety optimization; (ii) the mechanical test phase on a physical prototype; and (iii) the functionality verification phase. The stiffness optimization, carried out during the design phase, is performed using static structural finite element (FE) analyses (FEA). Optimizing a passive foot prosthesis means optimizing its stiffness in such a way that it gives rotations in both dorsiflexion and plantarflexion similar to the rotations of a healthy foot. In the ankle–foot system of a human who has not suffered any amputation to the lower limbs, there are muscles that control every movement giving the desired or necessary rotation. With transtibial amputation, most of the muscles are eliminated or reduced and a passive device such as the ESR foot cannot guarantee active control as muscles do. However, by studying the literature, it is possible to determine the loads to which a healthy foot is subjected while walking and its corresponding rotations. Therefore, an ESR foot can be optimized in terms of stiffness considering the loads it is subjected to and the rotations it must make while walking.

In previous publications, FEA was used with different aims in the design/study of prosthetic feet. Omasta et al. and Bonnet et al. used FEA to analyze the stress-strain behavior of load-bearing components [6,7]. Bonnet et al. also used FEA to analyze the energy stored by an ESR foot [7]. FEAs were also applied to the design of foot prostheses that comply with standards such as ISO 22675 (www.iso.org, accessed on 1 January 2021) [8] or American Orthotic and Prosthetic Guidelines (AOPA, www.aopanet.org, accessed on 1 January 2021) [9]. FEA was used by Prost et al. to carry out the stiffness optimization of the elastic elements of passive [10] prosthetic feet, while Sheperd et al. used it for the same aim but to optimize a quasi-passive [11] foot prosthesis. The profile shapes of prosthetic feet were to obtain an optimal roll-over by carrying structural FEAs [12,13]. Dao et al. designed an ESR foot with elastic elements in the composite materials (glass fiber) [14]. Rigney et al. included FEA in a methodology that concerned ESR prosthetic foot characterization; however, they did not propose it as a tool to design a new energy-storing and -releasing foot [15]. Tryggvason et al. presented a work where the main aim was to create a model which was meant to serve as a platform for iterative modifications of a foot prosthesis design by simulating a standardized dynamic mechanical test (ISO/TS 16955), wherein the foot performed a complete roll-over [16].

Table 2. Aim and type of simulations and material properties used in previous works where finite element analysis is applied to study foot prostheses.

Ref.	Aim	Type	Mat. Prop.
Omasta et al. [6]	analysis	3D Static	linear, isotropic
Bonnet et al. [7]	analysis	3D static	linear, isotropic
Naveed et al. [8]	design	3D dynamic	linear, isotropic
Santana et al. [9]	design	3D static	non linear, orthotropic
Prost et al. [10]	design	3D static	linear, isotropic
Shepherd et al. [11]	design	3D static	linear, isotropic
Mahmoodi et al. [12]	design	3D dynamic	linear, isotropic
Ke et al. [13]	design	3D static	linear, isotropic
Dao et al. [14]	design	3D static/dynamic	linear, isotropic
Rigney et al. [15]	analysis	3D static/dynamic	linear, isotropic
Tryggvason et al. [16]	design	3D dynamic	non linear, orthotropic

summarizes the literature on the use of FEA for the design and analysis of prosthetic feet characterized by the elastic elements in composite materials. So far, state-of-the-art analyses show that the literature approaches relatively to prosthetic foot design are not based on a systematic methodology.

This work aims to introduce a methodology that helps the designer develop their initial idea of a foot prosthesis by combining ergonomic and functional requirements with stiffness and strength requirements and choosing the most suitable materials. In the methodology, two different FE models of the foot prosthesis were built during the first phase (design phase, Section 3.1). The first FE model was two-dimensional (2D FE model) and considered the foot prosthesis only in the sagittal plane (Section 3.1.1). The second FE model was three-dimensional (3D FE model); the profile shapes of the elastic elements in the sagittal plane were the ones optimized in the 2D FE analysis (Section 3.1.2). In previously published studies, 3D FEAs have been carried out to perform stiffness optimization of the energy-storing parts [12,13,16]. However, when the analysis aims to study the effect of varying many geometric parameters on the behavior of the prosthesis, 3D FEA is time-consuming. Therefore, the 2D FE model built in the present work is aimed to be used as a tool to determine the effect of the geometry variation on the foot prosthesis behavior in the sagittal plane, reducing the computation time. The 3D FE model was built to perform a more detailed structural analysis, aiming to optimize the stiffness and the strength of the prosthetic device. The optimization was carried out by determining the final material properties to use; therefore, the lamination sequence of the layers of CFRP was used to build the elastic elements. In previous studies, the composite elastic elements have been simplified by simulating them with isotropic properties in 3D FEAs [12,13,14]. Simplifying the laminate composites as isotropic materials could lead to high approximated results, both in terms of stiffness and strength points of view. In Section 3.1.2, a guideline to build the 3D FE model is given, also specifying the constraints and the load conditions of the equivalent of ISO 10328 static tests. The second phase of the present methodology is the FE model validations. The 2D and 3D FE models and analyses were subsequently subjected to validation through an ISO 10328-inspired static test (mechanical test phase, see Section 3.2). The design phase was carried out by optimizing the stiffness of the ESR foot with static loads; thus, the validation phase was carried out with static tests. Nevertheless, the foot prostheses were loaded dynamically during use. Therefore, it is crucial to understand the behavior of the foot prostheses when they are under dynamic conditions. Two approaches to verify the functionality and study prosthetic feet’s behavior when subjected to dynamic loads are proposed. They were both carried out through two-dimensional FE analysis, wherein only the motions in the sagittal plane were considered. The load conditions of the first approach consisted of a simplified version of the dynamic test proposed in the ISO 10328 standard, whereby the foot was loaded at the heel and toe with two platforms, simulating the ground reaction forces (Section 3.3.2). The second approach is based on the ISO 22675 and ISO/TS 16955 standards load conditions, whereby a tilting table simulated the relative rotation between the thigh and the ground and an actuator simulated the ground reaction forces by pushing the thigh–shank–foot system (Section 3.3.3) downwards. As a case study to describe the methodology and demonstrate its usability, the stiffness optimization of an ESR foot is presented: the MyFlex-$\gamma$ foot, size 25, for 60 kg users and K3/K4 as a level of ambulation. The configuration of the composite blades of MyFlex-$\gamma$ is similar to the ESR foot Pro-Flex Pivot by Össur.

2. Requirements

Designing ESR feet means optimizing the stiffness of their elastic elements. The basic definition of stiffness of a loaded structure is stiffness equal to load divided by the deformation. In the present case, the deformation corresponds to the rotation of the foot in the sagittal plane (dorsiflexion and plantarflexion). On the other hand, the load corresponds to the ground reaction forces. In this section, the ankle–foot range of rotation on the sagittal plane (Section 2.1) and the ground reaction forces (Section 2.2) during normal ground walking are given. Normal ground walking is a cyclic movement. The cyclic movement is properly called the gait cycle or stride. It consists of two main phases, the stance phase (when the foot is touching the ground) and the swing phase. For the aim of the design methodology, the work is focused on the stance phase only. The stance phase consists of three sub-phases: the early stance, the mid stance and the late stance. The heel portion of the ESR foot absorbs the impact of the foot prosthesis against the ground in the early stance; the elastic elements store elastic energy during mid stance, then release it during the late stance for the push off.

2.1. The Ankle Rotation on the Sagittal Plane

The ankle range of motion while walking on normal ground is significantly influenced by several factors, such as walking velocity, age and gender [17,18,19,20,21,22,23,24]. However, the curves that describe the behavior of the ankle rotations for different conditions follow the same path. Thus, the healthy human ankle rotation has an initial plantarflexion during early stance (from 8% to 12% of the gait cycle) that goes from −1° to −8°. The maximum dorsiflexion before heel-off occurs from 60% to 70% of the gait cycle and goes from 6° to 16° of ankle rotation.

2.2. The Vertical Ground Reaction Forces during Normal Ground Walking

The gait cycle is influenced by different factors, not only kinematically but also in terms of how the foot impacts the ground at heel strike and toe strike and how it loads the ground during the entire stance phase. The forces exchanged by the foot and the ground are called ground reaction forces. The value of ground reaction forces during normal walking vary under different conditions, for instance, depending on different walking velocities [21]. The vertical ground reaction forces during normal ground walking have two peaks (M-shaped ground reaction force); the first peak is at the early stance and it is between 95% to 130% of the body weight; the second peak is at the end of the mid-stance and it is from 95% to 105% of the body weight.

2.3. The ISO 10328 Standard Static Test

The stiffness characterization and safety verification of ESR feet and foot prostheses, in general, are carried out with static tests regulated by standards, such as the ISO 10328, or the AOPA Guidelines. According to both the standard and the guidelines, the forefoot and the heel are loaded in two separate tests. The force is imposed on the forefoot and the heel with a platform. For the heel test/plantarflexion test, the foot is relatively inclined backwards by 15° to the platform to simulate the angle between the foot and the ground at heel strike. When loaded, the foot prosthesis is subjected to a plantarflexion. For the toe test/dorsiflexion test, the foot is relatively inclined forward by 20° to the platform to simulate the angle between the foot and the ground at the end of the stance phase, at the toe-off. When loaded, the prosthetic device is subjected to dorsiflexion. For the stiffness determination of the foot prosthesis, the settling force (ISO 10328) is used and its maximum value depends on the weight category of the intended users. For the plantarflexion test, the settling force at the heel is from 105% (100 kg weight category) to 125% (60 kg weight category) of the body weight of the user. For the dorsiflexion test, the settling force at the forefoot is from 94% (100 kg user) to 108% (60 kg user) of the body weight of the user. Both the forces are imposed with a rate between 100 N/s and 250 N/s.

2.4. Biomechanical Requirements

Based on the kinematics (Section 2.1), the loads (Section 2.2) and the standard test (Section 2.3), the target is to optimize the elastic elements of the foot prosthesis in order to provide the aimed rotations both in plantarflexion and dorsiflexion during normal ground walking; as objective chosen for the present work, the foot must have a plantarflexion at early stance comprised in the range between −5° and −8° when the heel is loaded with between 95% and 130% of the body weight of the intended users and dorsiflexion in the range between 14° and 18° when the forefoot is loaded between around 95% and 108% of the weight category. The values of dorsiflexion and plantarflexion and of the loads corresponding to them vary according to the end user; therefore, even if using this same methodology, the biomechanical objectives can vary.

2.5. Foot Prosthesis Configuration

The ESR foot considered to demonstrate the usability of the design methodology is MyFlex-$\gamma$, whose design is inspired by Pro-Flex Pivot by Össur. MyFlex-$\gamma$ can be subdivided into three functional groups (see Figure 1), i.e., the ankle group (ankle frame and tube connector), the tendon group (link and spring holder) and the foot group (lower blade, middle blade and upper blade).

When the foot is loaded at the heel (early stance, before toe strike), the heel portion of the lower blade, as well as the middle blade, is subjected to a deflection. On the other hand, when the foot is loaded at the forefoot (late stance, after heel-off), the upper blade and the middle blade are deflected and deformed in respect to each other—see Figure 2. The deflection of each of the above-mentioned elastic parts depends on their elasticity, intended as material properties, and their geometries, which define the contact—thus, how they deflect each other. Given this, the geometry and material optimizations are fundamental to reaching the prosthetic foot’s targeted stiffness.

3. Materials and Method

The design and functionality verification methodology proposed consists of a procedure made of three main phases: (i) the design phase, (ii) the mechanical test phase to validate the design and (iii) the functional verification phases. In Figure 3, the flowchart of the design and functionality verification is depicted.

Once the requirements were defined and once the foot prosthesis configuration was chosen, an initial geometry of the foot prosthesis (two-dimensional CAD model) was drawn and used in the design phase. The FE models built and used to optimize the stiffness of the foot prosthesis were then validated through static tests, loading a physical prototype of the prosthesis. Functional verification was subsequently made through two 2D dynamic FEAs built upon the 2D FE model of the design phase.

3.1. Design Phase

The most important locomotion’s degree of freedom of the foot is the rotation in the sagittal plane. Therefore, to simplify the preliminary stiffness optimization of the prosthesis, based mainly on the modification of the geometry of the prosthetic device, a 2D FE model was built to simulate the motion of the foot in the sagittal plane (Section 3.1.1). Once the geometry in the sagittal plane was defined, a 3D FE model was built to carry out a more detailed simulation, wherein the composite elastic elements were optimized by modifying the laminate material properties, working mainly on the type, orientation and numbers of layers of CFRP prepregs (Section 3.1.2). Before explaining in detail the various stages of the proposed methodology, it is reminded that optimizing a passive foot prosthesis means optimizing its stiffness in such a way that it gives rotations in both dorsiflexion and plantarflexion similar to the rotations of a healthy foot, when such prosthesis is subjected to loads (ground reaction forces) similar to those to which a healthy foot is subjected. For the present application, as already mentioned in Section 2.4, the aim is to optimize the prosthesis’ stiffness in such a way that, when it is loaded at the heel with a load between 95% and 130% of the weight of the user, it gives a plantarflexion between −5° and −8°; when it is loaded at the toe with a load between 95% and 108% of the body weight, it must give a dorsiflexion between 14° and 18°.

3.1.1. Geometry Optimization: 2D FE Model

In this first step of the design phase, geometry optimization was carried out through a 2D FEA. For example, this step can be used to optimize the profile shapes of an already defined configuration of elastic elements, to configure the energy-storing parts of a new ESR foot, or to investigate the behavior of an existing prosthetic device after modifications or additions of other functional components, such as dampers and actuators.

Optimizing the geometry of a foot prosthesis is very important, especially with regard to ESR feet and their elastic components. Although the result of the 2D FE model in terms of stiffness is not definitive, this model is essential to optimize the shapes of the various elastic components to ensure the full range of motion needed to walk. A non-optimization of the shapes of the elastic components can bring a range of motion too reduced or too wide. Regardless of the stiffness of the elastic components, if the range of motion is too small, the foot rotates around the ankle joint up to a certain angle and then, a sudden sensation of increase in the equivalent rotational stiffness around the ankle joint follows. Due to this sudden increase in stiffness of the ankle joint, the foot no longer rotates around the ankle and the heel comes off the ground before the desired moment. This situation can be perceived by the user as a sudden disappearance of the support, which can cause discomfort or even a fall of the user. If the range of motion is too wide, the foot may continue to rotate in dorsiflexion beyond the desired limit by delaying the hell-off. Even this situation can generate discomfort or even a fall of the user.

For MyFlex-$\gamma$, geometry optimization was focused on the profile shape of the middle blade and the upper blade, aiming to meet the biomechanical requirements previously defined (Section 2.4). The lower blade used for the present application was provided by Össur and it was already optimized for the chosen weight category. Then, an initial 2D geometry of the foot was drawn bearing in mind the dimensional parameters defined by the human anatomy such as length L of the foot, the height h of the ankle joint from the ground and the distance d of the ankle joint from the heel, which is around 1/3 of the foot length. For the present application, the length of the foot was 250 mm and the height of the ankle joint was between 80 mm and 100 mm, as shown in Figure 1.

Load conditions. The CAD model was drawn in the x–y plane and imported into ANSYS Workbench. The FE model was simulated in a 2D static structural analysis. Following the ISO 10328 standard, the dorsiflexion test was simulated by loading the FE model with a platform at the forefoot. For the plantarflexion test, the FE model was loaded at the heel. The platforms were moved by imposing 10 mm of displacement for the plantarflexion test and 50 mm for the dorsiflexion test. The shank was constrained with a fixed support at the top of the tube connector. For the present application, the loads and the constraints are depicted in Figure 4.

The geometry optimization was carried out by varying the geometric parameters. These parameters depend on the foot prosthesis architecture and the designers can choose them. For MyFlex-$\gamma$, the parameters are depicted in Figure 5 and listed in

Table 3. Geometric parameters of MyFlex-$\gamma$ varied in the 2D FE model of the design phase—see Figure 5.

Parameter	Min Value	Max Value
upper blade thickness UB${}_t$ (mm)	6.00	8.50	defined in step 2
upper blade curvature 1 UB c${}_1$ (deg)	1.00	3.00
upper blade curvature 2 UB c${}_2$ (deg)	1.00	3.00
upper blade curvature 3 UB c${}_3$ (deg)	1.00	3.00
upper blade curvature 4 UB c${}_4$ (deg)	1.00	3.00
upper blade curvature 5 UB c${}_5$ (deg)	3.00	5.00
middle blade length MB${}_L$ (mm)	150	175
middle blade thickness MB${}_t$ (mm)	7.00	10.00	defined in step 2

. The upper blade was defined in the sagittal plane by 6 parameters. The parameter UB${}_t$ is the thickness, while the parameters c${}_1$, c${}_2$, c${}_3$, c${}_4$ and c${}_5$ define the curvatures of the curved profile of the upper blade, which was composed by 6 straight sections—starting from the metatarsus (front), c${}_1$ is the relative inclination between the first and the second section, c${}_2$ is the relative inclination between the second and third section, etc. The middle blade, which had a straight profile in the sagittal plane, was defined only by its length (MB${}_L$) and thickness (MB${}_t$). The lower blade, provided by Össur, had been already optimized, in terms of shape and material properties, for specific users’ weight categories.

Mesh modeling. The FE model was built in ANSYS Workbench. PLANE183 (ANSYS) elements were used for mesh building; these were two-dimensional 8- and 6-node elements with quadratic displacement behavior and two translations at each node as degrees of freedom. The final 2D FE model of MyFlex-$\gamma$ was meshed (Figure 6), presenting a total of 10,000 nodes corresponding to 20,000 degrees of freedom.

Contacts modeling. The contacts between parts had to be modeled differently according to the real conditions. For parts that worked bending toward each other with slight sliding, the contacts were modeled as frictional using pure penalty formulation and 0.1 was set as the normal stiffness factor. For contacts that could be modeled as glued/bonded, the bonded contact type was used with augmented Lagrange formulation and 1 was set as the normal stiffness factor.

In the present work, the upper blade, middle blade and lower blade bent toward each other with slight sliding when the foot was loaded. Therefore, the contacts between the upper blade and middle blade and between the middle blade and the lower blade were modeled as frictional contact using pure penalty formulation. The ankle frame and the upper blade were joined together with two M8 bolts along the longitudinal direction; for this condition, the contact between the ankle frame bottom surface and the upper blade top surface could be modeled as bonded to simplify the simulation, using augmented Lagrange formulation. A summary of the contacts is presented in

Table 4. Contacts’ properties. See also Figure 7. AF = ankle frame; UB = upper blade; MB = middle blade; LB = lower blade; SH = spring holder; TC = tube connector.

Surface 1	Surface 2	Type	Formulation	Frict. Coeff.	Norm. Stiff. Fact.
AF top	UB bottom	bonded	augm.Lagrange	-	1.00
UB bottom	MB top	frictional	pure penalty	0.20	0.01
MB bottom	LB top	frictional	pure penalty	0.20	0.01
MB bottom	SH top	frictional	pure penalty	0.20	0.01
AF ankle	TC ankle	no separation	augm.Lagrange	-	1.00

.

For the 2D FE model of MyFlex-$\gamma$, a no-separation contact type between the tube connector and the ankle frame (see Figure 7) was used to model the ankle joint; a no-separation contact type allows frictionless motions to be performed without separation of the parts.

Joints: Pretensioned bolts were modeled as preloaded springs, where the preload given as force was equal to the bolt pretension of the corresponding bolt. More specifically, the bolts were modeled as longitudinal springs body-to-body joints (longitudinal COMBIN14, ANSYS). Both the nodes of the COMBIN14 element were applied as direct attachment to the nodes of the connected parts. The elastic elements were joined together with two pretensioned M8 bolts in the physical MyFlex-$\gamma$ that could be seen as a single bolt in the sagittal plane. In addition, the middle blade and the spring holder were joined with preloaded M6 screws. Bolt pretension was simulated as a normal force given to the longitudinal springs body-to-body joints, which corresponded to the standard bolt pretensions for M6 (7.4 kN, 8.8 class, frictional coefficient = 0.20) and M8 (13.7 kN, 8.8 class, frictional coefficient = 0.20). Connection links with hinge joints at both extremities were modeled with body-to-body beam joints (BEAM3, ANSYS). BEAM3 is a 2D uniaxial element with tension, compression and bending capabilities and it has the translations in both directions of the x–y plane and the rotation around the z-direction at each node. In MyFlex-$\gamma$, the link part, characterized by two hinge joints that connected the tube connector to the spring holder, was modeled by BEAM3.

Simulation conditions: ESR feet elastic elements are subjected to high deflections. Therefore, the 2D FE model was simulated in a nonlinear static structural analysis environment.

Simulation outputs: The behavior of the foot can be evaluated in two different modalities by plotting the reaction force at the fixed constraint against the platform displacement in the $y_{pf}$ direction (see Figure 4) or against the foot rotation. The foot rotation is calculated by using two virtual markers (G and M in Figure 8). The positions of the virtual markers were chosen considering the marker-set protocol [21,25]. Since the shank was fixed, the J marker was considered as (0,0).

The displacements of the virtual markers (given in the global x–y reference) are given as direct output of the simulation and they can be considered as functions of the platform displacement $v_{pf}$ in the vertical direction $y_{pf}$ of the same platform (see Figure 4). Therefore, the displacements of the G marker in the global x- and y-directions are

$$u_G=u_G\left(y_{pf}\right)$$

(1)

$$v_G=v_G\left(y_{pf}\right)$$

(2)

The displacement of the M marker in the global x- and y-directions are

$$u_M=u_M\left(y_{pf}\right)$$

(3)

$$v_M=v_M\left(y_{pf}\right)$$

(4)

If ($x_{G0}$;$y_{G0}$) and ($x_{M0}$;$y_{M0}$) are the initial positions of G and M markers, referred to J (0;0), the coordinates of G and M are:

$$x_G\left(y_{pf}\right)=x_{G0}+u_G\left(y_{pf}\right)$$

(5)

$$y_G\left(y_{pf}\right)=y_{G0}+v_G\left(y_{pf}\right)$$

(6)

$$x_M\left(y_{pf}\right)=x_{M0}+u_M\left(y_{pf}\right)$$

(7)

$$y_M\left(y_{pf}\right)=y_{M0}+v_M\left(y_{pf}\right)$$

(8)

The foot rotation $\Delta$$\alpha$ is calculated as the variation of the angle between the GM line and the horizontal direction (Figure 9). The initial angle ${\alpha }_0$ between the GM line and the horizontal direction and the rotation $\Delta$$\alpha$ of the foot are calculated as:

$${\alpha }_0=arctan \left(\frac{y_{0G}-y_{0M}}{x_{0G}-x_{0M}}\right)$$

(9)

$$\Delta \alpha =\alpha -{\alpha }_0$$

(10)

In Equation (10), $\alpha$ is determined as

$$\alpha =arctan \left(\frac{y_G-y_M}{x_G-x_M}\right)$$

(11)

3.1.2. Material Properties Optimization: 3D FE Model

With the 2D FE model, the profile of the elastic elements and the general geometry of the foot prosthesis in the sagittal plane were defined. The 3D CAD model for the 3D FE model was built upon the geometry obtained in the previous step of the design phase methodology (Section 3.1.1). The elastic elements of ESR feet, in general, are laminate composite and they are built by stacking, in sequence, layers of CFRP (carbon fiber-reinforced plastic) or GFRP (glass fiber-reinforced plastic) prepregs, which are sheets of pre-impregnated fibers (generally pre-impregnated with resin) and they can be unidirectional or woven. The stiffness of each of the elastic elements of the ESR feet is defined by lamination sequence, type, number, orientation and order of each layer. For MyFlex-$\gamma$, unidirectional and woven CFRP prepregs were used. The lamination sequence of CFRP prepregs layers was optimized to reach the targeted stiffness, thus meeting the biomechanical requirements defined in Section 2.4. The elastic properties of the CFRP prepregs used are gathered in

Table 5. Orthotropic elasticity of CFRP prepregs used to manufacture the upper blade, middle blade and lower blade.

Type	Gramm.	Thick.	${\mathit{E}}_1$	${\mathit{E}}_2$	${\mathit{E}}_3$	${\mathit{G}}_{12}$	${\mathit{G}}_{23}$	${\mathit{G}}_{13}$	${\mathit{ϵ}}_{12}$	${\mathit{ϵ}}_{23}$	${\mathit{ϵ}}_{13}$
	g/m${}^{\mathbf{2}}$	mm	GPa	GPa	GPa	GPa	GPa	GPa	-	-	-
UD	150	0.151	112.5	7.4	7.4	4.3	2.6	4.3	0.33	0.44	0.33
UD	250	0.251	112.5	7.4	7.4	4.3	2.6	4.3	0.33	0.44	0.33
W	200	0.234	61.3	61.3	6.9	3.3	3.3	2.7	0.04	0.30	0.30

.

Loads and constraints: The 3D FE model is intended to be validated with a mechanical test on a physical prototype of a foot prosthesis. In the ISO 10328 standard, the foot is loaded with an inclined platform, which means two inclined actuators are required, one for the heel and one for the toe, to compress the foot prosthesis. A dedicated test setup was designed and built (Figure 10) to avoid the necessity of two inclined actuators. The ISO 10328-equivalent test setup was characterized by a vertical piston that pushed a platform upward. The relative inclination between the foot and the platform was created by two different adapters that inclined the foot backwards by 15° and forwards by 20°. The platform, free to move along the longitudinal direction of the foot, compressed the foot prosthesis, fixed at the top. A vertical displacement was imposed to the platform to compress the foot at the heel and at the toe in two different tests to simulate the ground reaction forces at both the early and the late stance, respectively. The total displacement was 10 mm and 50 mm at the heel and the toe load conditions, respectively, because of the different range of motion of the foot during plantarflexion and dorsiflexion. For the toe load, the platform was moved upward linearly at a rate from 3 mm/s to 4 mm/s. The platform moved linearly along the vertical direction for the heel load to compress the foot at a rate from 0.6 mm/s to 0.8 mm/s. These rate values, both for the plantarflexion and dorsiflexion tests, were chosen for a reason of convergence of the simulations. It was verified that the results did not change if the dorsiflexion simulation was performed at a rate of 3 mm/s or at a rate of 4 mm/s. The exact same applies to the plantarflexion simulation. This is justified by the fact that the simulation was still in rate values under static conditions.

The results from the design phase FEAs can be plotted as reaction force against the rotation, or reaction force against the platform displacement.

Components modeling: The 3D CAD Model was imported into ANSYS by setting 3D as the analysis type. All isotropic parts were provided as solid to ANSYS, while composite parts were surfaces—Figure 11. All parts involved in contacts were modeled as flexible elements (platform, foot shell, lower blade, middle blade, upper blade, ankle frame and spring holder), while the parts connected with joints were modeled as rigid bodies (link and tube connector)—see Figure 12 and Figure 13.

Based on the results obtained from the 2D FE model in the previous step, the initial values of thicknesses and elastic properties of the composite parts were predefined and used as a reference to define the lamination sequences of layers of CFRP prepregs. Then, the types, the numbers, the orientations and the order of the layers of CFRP prepregs were changed until the targeted thickness and elastic properties (calculated with the Classical Theory of Laminates) were reached.

The 3D CAD model of the foot prosthesis was provided to ANSYS in Geometry. The lamination sequences of the composite parts were modeled inside ANSYS Composite PrePost (ACP). The main fiber direction for each of the composite parts is depicted in Figure 14 and it corresponds to the longitudinal direction of the foot prosthesis. Then, the composite parts were exported from ACP to Static Structural modeler as solid, with the final thicknesses.

Mesh modeling: The flexible components (platform, foot shell, lower blade, middle blade, upper blade, ankle blade and spring holder) were meshed. The elastic elements, including the lower blade and the foot shell, were mainly modeled with SOLID186 elements, which were 20-node elements with 3 degrees of freedom at each node and with quadratic behavior. SOLID187 elements (10-node elements with 3 degrees of freedom per node) were used for irregular areas. Ankle frame, spring holder and platform were modeled with SOLID185 elements, 8-node elements with 3 degrees of freedom at each node and linear behavior. The final 3D FE model of MyFlex-$\gamma$ consisted of around 1,800,000 degrees of freedom (600,000 nodes) (Figure 12 and Figure 13).

Contact modeling: As in the 2D FE model, for parts that bend to each other with slight sliding, the contacts were modeled as frictional using pure penalty formulation and 0.1 was set as the normal stiffness factor (the factor was set to 1 for normal dominated conditions). For contacts that could be modeled as glued/bonded, the bonded contact type was used with augmented Lagrange formulation and 1 was set as the normal stiffness factor. In MyFlex-$\gamma$, the upper blade, the middle blade and the lower blade bend to each other with slight sliding when the foot is loaded. The contacts between the upper blade and the middle blade and between the middle blade and the lower blade were modeled as frictional contact using pure penalty formulation. The contact between the ankle frame bottom surface and the upper blade top surface was modeled as bonded to simplify the simulation, using augmented Lagrange formulation. The contacts’ properties used in the present 3D model are summarized in Table 6.

Table 6. Contacts’ properties. See also Figure 15. AF = ankle frame; UB = upper blade; MB = middle blade; LB = lower blade; SH = spring holder; TC = tube connector.

Surface 1	Surface 2	Type	Formulation	Frict. Coeff.	Norm. Stiff. Fact.
AF top	UB bottom	bonded	augm.Lagrange	-	1.00
UB bottom	MB top	frictional	pure penalty	0.20	0.01
MB bottom	LB top	frictional	pure penalty	0.20	0.01
MB bottom	SH top	frictional	pure penalty	0.20	0.01

Table 6. Contacts’ properties. See also Figure 15. AF = ankle frame; UB = upper blade; MB = middle blade; LB = lower blade; SH = spring holder; TC = tube connector.

Surface 1	Surface 2	Type	Formulation	Frict. Coeff.	Norm. Stiff. Fact.
AF top	UB bottom	bonded	augm.Lagrange	-	1.00
UB bottom	MB top	frictional	pure penalty	0.20	0.01
MB bottom	LB top	frictional	pure penalty	0.20	0.01
MB bottom	SH top	frictional	pure penalty	0.20	0.01

Bolt modeling: In MyFlex-$\gamma$, the elastic elements were joined together with two preloaded M8 bolts. In addition, the middle blade and the spring holder were joined with preloaded M6 screws. The screws were modeled as preloaded spring joints (ANSYS; COMBIN14) and with specific axial stiffness and pretensions that corresponded to M8 (front bolts) and M6 (spring holder bolts). The extremities of the springs were applied, as remote attachments, on the portions of the upper-blade top surface and on the bottom of the lower-blade surface to simulate the washers’ section.

Loads and constraints: The FEA was carried out in a nonlinear static structural analysis by enabling the large deflection option. The vertical displacement was imposed on the platform, free to move along the x-direction and fixed along y (transverse)-direction (see Figure 10 and Figure 13). The platform compressed the ESR foot and the reaction force was measured where the fixed support constraint was set.

3.2. Mechanical Test Phase

In the equivalent test setup, a hydraulic press machine (INSTRON 8033) was used as an actuator to compress the physical prototype of the foot prosthesis. The relative inclination between the foot and the platform was obtained by specifically designed adapters (Figure 16). As in the 3D FE model of the design phase (Section 3.1.2), the platform was moved under displacement control; it moved upwards linearly with the same rate used in the design phase, i.e., 0.6–0.8 mm/s in the plantarflexion test and 3–4 mm/s in the dorsiflexion test, to replicate the same conditions. Thanks to a linear guide, the platform that pushed the foot prosthesis upward was free to move along the longitudinal direction of the same foot prosthesis. The platform was mounted on the linear guide, attached to a frame that could freely move when the hydraulic piston was actuated. The results of the mechanical test can be given as displacement–reaction force, where the displacement was the vertical motion of the platform and the reaction force was measured at the top of the inclination adapter with a load cell. By applying markers to the prosthesis of the foot, the rotations could also be determined; therefore, the stiffness curve of the prosthesis could be plotted as rotation–reaction force. Knowing, from the simulations, the ratio of the platform displacement and rotation of the foot, the results from the mechanical tests could be plotted as rotation–reaction force even without the use of markers.

3.3. Functionality Verification Phase

The functionality verification of the foot prosthesis was conducted in two different modalities, both through transient structural FEAs. The first modality was a simplified dynamic test inspired by the cyclic tests from ISO 10328 (Section 3.3.2). The cyclic tests proposed in ISO 10328 relate to normal walking activities where loads occur regularly with each step. In the second modality, the roll-over task of the foot prosthesis was simulated and it is inspired by the dynamic tests from ISO 22675 and ISO/TS 16955 (Section 3.3.3).

3.3.1. Two-Dimensional FE Model Adjustment

The 2D FE models used to simulate the two dynamic tests were built upon the 2D FE model developed in Section 3.1.1. The 2D FE model was characterized by speeding up the simulations in the case of geometry optimization, carried out by changing specific geometric parameters. Nevertheless, two-dimensional FE models are less precise than three-dimensional FE models. The first reason is, in the 2D FE model, the width of each part cannot be set as variable, as depicted in Figure 17. The second reason is geometric and concerns the holes, which are not considered in the 2D FE model, which is depicted again in Figure 17.

The third reason regards the material properties. The orientation of the fibers in composite materials is fundamental, in terms of both stiffness and strength. In the 3D FE model, the fibers were modeled following the curvatures of the components (middle blade and upper blade in MyFlex-$\gamma$). The same situation did not occur in the 2D FE model. Therefore, the 2D FE model built as described in Section 3.1.1 was simulated again. The widths of the elastic parts were slightly adjusted to obtain the same displacement force curves obtained in the static mechanical test (Section 3.2).

3.3.2. ISO 10328 Cyclic Tests

The re-adapted 2D FE model of the foot prosthesis was used to simulate the cyclic tests proposed in the ISO 10328 standard. Constraints, contacts, mesh modeling, joints, materials and width (dimension in the transverse direction) were the same for the foot prosthesis, while the load conditions were changed. The foot was loaded at the heel and forefoot with two platforms (Figure 18). The two platforms compressed the foot prosthesis with two forces (Figure 19), simulating the ground reaction forces during the gait cycle. The heel and the forefoot platforms were inclined by −15° and 20°, respectively. An initial space between the forefoot platform and the foot prosthesis was given (Figure 18). The forefoot touched the front platform when the forefoot force started to increase from 0 N (around 22% of the gait cycle, Figure 19). The first contact between the forefoot and forefoot platform simulated the toe strike. By considering the values of the ground reaction forces specified in Section 2 and the 60 kg weight category of users, the heel force peak was 764 N, while the forefoot force was 635 N. The same virtual markers used in Section 3.1.1 were used to calculate the rotation of the ankle–foot. For the dynamic analysis, an entire gait cycle of 1 s was considered.

3.3.3. ISO 22675–ISO/TS 16955 Dynamic Test

The re-adapted 2D FE model of the foot prosthesis was used to build the present model to simulate the ISO 22675–ISO/TS 16955 dynamic/roll-over test. Other parts were added, such as the shank, the thigh and the tilting table (Figure 20). The contacts, mesh modeling, joints, materials and width (dimension in the transverse direction) were the same used in the first step of the design phase. The connection between the shank and the ankle–foot system was modeled as a bonded contact. The contact between the knee portion of the thigh and the knee portion of the shank was modeled in the FE model as a no-separation contact to create the knee joint. The thigh remained vertical and was free to move along the vertical direction only. The tilting table was rotated according to ISO 22675–ISO/TS 16955, while the M-shaped force was imposed from the top of the thigh (Figure 20). The angle between the foot and the tilting table ranged from −20° to 40° (Figure 20). The rotation of the foot during the roll-over test was calculated using the virtual markers G, M, J and K. First of all, the initial coordinates of each marker were determined from the CAD model as G ($x_{G0}$;$y_{G0}$), M ($x_{M0}$;$y_{M0}$), J ($x_{J0}$;$y_{J0}$) and K ($x_{K0}$;$y_{K0}$). The displacements of the virtual markers during the roll-over test were direct outputs of the simulation—$u_G$, $u_M$, $u_J$ and $u_K$, in the x-direction, and $v_G$, $v_M$, $v_J$ and $v_K$, in the y-direction. The coordinates of the markers during the roll-over test were: G($x_{G0}$ + $u_G$;$y_{G0}$ + $v_G$), M($x_{M0}$ + $u_M$;$y_{M0}$ + $v_M$), J($x_{J0}$ + $u_J$;$y_{J0}$ + $v_J$) and K($x_{K0}$ + $u_K$;$y_{K0}$ + $v_K$). The rotation $\Delta$$\alpha$ of the foot was given as the variation of the angle between the shank axis and the GM line. The angle $\alpha$ between the shank axis and the GM line was given as follows:

$$\alpha ={\alpha }_s-{\alpha }_{GM}$$

(12)

where ${\alpha }_s$ is the angle between the shank axis and the horizontal direction, calculated as

$${\alpha }_s=arctan \left(\frac{(y_{K0}+v_K)-(y_{J0}+v_J)}{(x_{K0}+u_K)-(x_{J0}+u_J)}\right)$$

(13)

whereas ${\alpha }_{GM}$ is the angle between the GM line and the horizontal direction, which is calculated as follows:

$${\alpha }_{GM}=arctan \left(\frac{(y_{G0}+v_G)-(y_{M0}+v_M)}{(x_{G0}+u_G)-(x_{M0}+u_M)}\right)$$

(14)

The rotation of the foot was then calculated in the following way:

$$\Delta \alpha =\alpha -{\alpha }_0$$

(15)

with ${\alpha }_0$ as

$${\alpha }_0={\alpha }_{s0}-{\alpha }_{GM0}$$

(16)

where ${\alpha }_{s0}$ and ${\alpha }_{GM0}$ are the initial angle between the shank axis and the horizontal direction and the initial angle between the GM line and the horizontal line, respectively. For the dynamic analysis, an entire gait cycle of 1 s was considered.

4. Results and Discussion

The results obtained from the design, the mechanical test and the functional verification phases are presented and discussed in the following sections. In addition, the comparison of the time elapsed to carry out the simulations is presented.

4.1. Design Phase Results

4.1.1. Geometry Optimization: 2D FE Model Results

Several combinations of the parameters listed in Table 3 and depicted in Figure 5 were used to carry out more than 1000 simulations. From these 1000+ simulations, the final and pre-optimized values of the parameters were obtained and are listed in

Table 7. Final geometric parameters that define the profile shape of MyFlex-$\gamma$ elastic elements, for the 60 kg weight category. See Figure 5.

Parameter	Final Value
upper blade thickness UB${}_t$ (mm)	6.80
upper blade curvature 1 UB c${}_1$ (deg)	2.00
upper blade curvature 2 UB c${}_2$ (deg)	2.00
upper blade curvature 3 UB c${}_3$ (deg)	2.20
upper blade curvature 4 UB c${}_4$ (deg)	2.50
upper blade curvature 5 UB c${}_5$ (deg)	4.20
middle blade length MB${}_L$ (mm)	163
middle blade thickness MB${}_t$ (mm)	7.60

. The tabulated values are the final values from the 2D model for the weight category that was chosen to be developed (60 kg), while they are not the final values for the 3D model. In fact, of these values, the values of the thickness of the two blades in the 3D model change based on their stack-up sequence (which is defined in the 3D model, during the optimization of the properties of the materials).

The choice of the final parameters was based on the rotation–reaction force. However, different configurations of the geometry could give similar results in terms of stiffness. Thus, stress and approximated final weight of the device were also used as evaluation parameters. However, among the 1000+ simulations, a configuration was considered optimal if the curve foot rotation–reaction force fell into the optimal area, highlighted in the graphs in dark grey, as the intersection of the two lighter grey areas in Figure 21. These areas, both for the dorsiflexion and for the plantarflexion, were defined considering the biomechanical requirements previously reported in Section 2.4. Among the 1000+ combinations, the results from one of the non-optimal combination were taken into account to be compared with the results coming from the final combination of parameters that was considered pre-optimized. The dashed curve falls only partially into the optimal plantarflexion area (Figure 21), although the same combination of parameters already gives a curve that falls into the optimal dorsiflexion area. On the other hand, the black straight line is the curve that describes the stiffness of the combination of parameters considered pre-optimized. By varying the combinations of the parameters, similar stiffness curves can be obtained in dorsiflexion and different stiffness curves in plantarflexion or vice versa. This situation depends on the configuration of the elastic elements of each foot prosthesis to be optimized. In this application, one or more combinations of geometric parameters were considered to be those sought for biomechanical objectives if the stiffness curves fell within the ranges of rotations and loads defined in Section 2.4. However, for future works, exploiting the proposed methodology, optimization functions can be implemented in such a way that the choice of the final configuration is not based only on the evaluation of the stiffness curves.

The stiffness curves, plotted as foot rotation–reaction force in Figure 21, are also provided as platform displacement–reaction force, as shown in Figure 22.

4.1.2. Material Properties Optimization: 3D FE Model

The final lamination sequences of the upper blade and the middle blade were obtained after varying the types, the orientations and the numbers of layers of CFRP prepregs and are reported in

Table 8. Final lamination sequences of the upper blade and the middle blade. The properties of the three types of CFRP prepregs are listed in Table 5. The directions of the fibers and the stack up for the upper blade and middle blade are depicted in Figure 14.

Part	Type	Orientation (deg)	No. of Layers	Total Thickness (mm)
	Woven 200 g/m${}^2$	0	3	0.702
	Woven 200 g/m${}^2$	45	2	0.468
Upper blade	Unidir. 250 g/m${}^2$	0	18	4.518
	Woven 200 g/m${}^2$	45	2	0.468
	Woven 200 g/m${}^2$	0	3	0.702
			total =	6.858
	Woven 200 g/m${}^2$	0	3	0.702
	Unidir. 250 g/m${}^2$	0	5	1.255
	Unidir. 150 g/m${}^2$	0	10	1.510
Middle blade	Woven 200 g/m${}^2$	0	3	0.702
	Unidir. 150 g/m${}^2$	0	10	1.510
	Unidir. 250 g/m${}^2$	0	5	1.255
	Woven 200 g/m${}^2$	0	3	0.702
			total =	7.636

.

This prototype was made with the aim of validating the two 2D and 3D models of the design phase. For the present application, no fiber orientation other than 0° and 45° was used, to simplify the manufacturing of the prototype. Other orientations, such as 30 degrees, may also be considered for further optimization. As can be seen in Table 8, 45° oriented fibers were only found in the upper blade, while, in the middle blade, they were absent. Based on the result from the point of view of stress/strength, fibers woven oriented at 45° led to too-high stress values when the prosthesis was subjected to operating loads and this can be justified by the shape of the middle blade, shown in Figure 23. The two parts into which the middle blade was divided into its fork shape were dimensionally too narrow; further, 45° oriented fibers reduced the strength of the entire laminate.

The stiffness in dorsiflexion and plantarflexion of the foot prosthesis (modeled with the final lamination sequences) are presented as foot rotation–reaction force in Figure 24 and as platform displacement–reaction force in Figure 25.

Considering the strength properties of the materials used to realize the elastic components (

Table 9. Orthotropic strength of CFRP prepregs used to manufacture the upper blade, middle blade and lower blade. T${}_1$, T${}_2$ and T${}_3$ are the tensile strength; C${}_1$, C${}_2$ and C${}_3$ are the compressive strength; S${}_{12}$, S${}_{23}$ and S${}_{13}$ are the shear strength.

Type	Gramm.	Thick.	T${}_1$	T${}_2$	T${}_3$	C${}_1$	C${}_2$	C${}_3$	S${}_{12}$	S${}_{23}$	S${}_{13}$
	g/m${}^{\mathbf{2}}$	mm	MPa	MPa	MPa	MPa	MPa	MPa	MPa	MPa	MPa
UD	150	0.151	2200	29	29	−1082	−100	−100	60	30	60
UD	250	0.251	2200	29	29	−1082	−100	−100	60	30	60
W	200	0.234	805	805	50	−509	−509	−170	125	65	65

), the stress–strength ratio analysis for these components was carried out. The Tsai–Wu criterion was used as the assessment criterion of the composite parts (the upper blade and the middle blade). When the foot was loaded with the 220% of the weight category, the inverse reserve factor was under 0.5 (Figure 26) in the most critical area. An inverse reserve factor of 0.5 corresponded to a safety factor of 2.

4.2. ISO 10328 Equivalent Static Test Results

Once the stiffness optimization of its elastic components was performed, a prototype of MyFlex-$\gamma$ was built. The composite elements were manufactured using the lamination sequences for the upper blade and the middle blade listed in Table 8. In addition, the other components were designed with the same safety factor. The assembled physical prototype was tested with the ISO-10328 equivalent static test setup, as presented in Section 2.3. The results are plotted as rotation–reaction force in Figure 24 and as displacement–reaction force in Figure 25.

The dorsiflexion curves, which are reported as rotation–reaction force (Figure 24), or as displacement–reaction force (Figure 25), are quite similar. Therefore, it can be said that both the 2D FE and 3D FE models were validated with the mechanical tests carried out on the physical prototype of MyFlex-$\gamma$.

If the curves of dorsiflexion are very similar to each other, the same cannot be said for those of plantarflexion. In fact, 2D FEA and 3D FEA curves diverge from the test curve obtained for the physical prototype. The reasons for the divergence could be the following: (i) the not-perfect mounting of the foot prosthesis on the test set-up; (ii) the not perfect CAD and FE modeling of the foot shell; (iii) manufacturing defects that can cause slight variations in the geometry of the physical prototype; (iv) the joints, such as the ankle joint, hinge joints between the link and tube connector, and between the link and spring holder, modeled without friction—thus, approximated; (v) the heel of the foot shell not fully covered by the platform, as depicted in Figure 27.

The plantarflexion displacement–reaction force curve obtained from the 3D FEA resulted stiffer than the curve obtained from the mechanical test after around 6–7 mm of the vertical displacement of the platform. After 6–7 mm of vertical displacement, the heel was not fully supported by the heel platform. The vertical support caused this situation (Figure 27), which was not considered in the 3D FEA. During the heel test, the contact area between the heel and the platform shifted forward. The distance between the point of application of the force and the center of the ankle joint was reduced. This condition did not occur in the mechanical test after 6–7 mm; the contact between the foot and the heel platform did not evolve as it should have. With the same torsional stiffness of the foot prosthesis in the sagittal plane, the force necessary to generate a plantarflexion was higher in the 3D FEA due to the shorter lever arm. It has to be specified that the platform was built according to standards. However, the space for the heel portion of the foot was not enough. For a more appropriate assessment of the plantarflexion stiffness, the heel platform (see Figure 27) should provide more space, or the vertical face should be removed. In the case of the dorsiflexion test, the test setup did not present a space issue. It is interesting to note that the two dorsiflexion stiffness curves are in good agreement. This condition should provide confidence in the 3D FE model presented in the design phase (Section 3.1.2).

4.3. Functionality Verification Results

4.3.1. ISO 10328 Cyclic Tests

Figure 28 shows how the elastic elements were deflected when the foot was subjected to the ISO 10328 cyclic test.

The foot prosthesis had its maximum plantarflexion around −6.4° and maximum dorsiflexion around 15°. Both the values are comprised in the ranges specified in the Requirements section (Section 2.4), as also highlighted in Figure 29.

4.3.2. ISO 22675–ISO/TS 16955 Roll-over Test

The behavior of the of the leg prosthesis during ISO 22675 is shown in Figure 30. The maximum plantarflexion of MyFlex-$\gamma$ was −5°, while the maximum dorsiflexion was 15°. The maximum plantarflexion obtained during early stance is comprised in the range of the targeted angles during normal walking. In addition, the maximum dorsiflexion during mid-stance is inside the aimed range (Figure 31).

4.3.3. Elapsed Time of Calculation Comparison: 2D FE Model vs. 3D FE Model

One of the advantages of the present methodology is to reduce the time of calculation during geometry optimization, where the effects of the variation in several geometric parameters are investigated. In previously published works on foot prostheses, geometry optimization has been carried out through 3D FE models [12,13]. The 2D FE model aims to provide a design tool to study the influence of the geometry on the foot prosthesis behavior, giving the result in a shorter time than performing 3D FEAs. The time elapsed to carry out a structural FEA depends on the computing power of the calculator used. However, in similar conditions (same computing power), the time necessary to perform the FEAs depends on several factors, such as the number of degrees of freedom of the FE models. In

Table 10. Number of nodes and number of degrees of freedom from the 2D FE model (Section 3.1.1) and the 3D FE model (Section 3.1.2).

Step	Type of Simulation	Number of Nodes	Number of Deg. of Freedom
1	2D Static	10,000	20,000
2	3D Static	600,000	1,800,000

, the number of nodes and number of degrees of freedom from the 2D FE model (Section 3.1.1) and the 3D FE model (Section 3.1.2) are listed. In the present specific application, the 2D static FEA that simulated the dorsiflexion test required a time of calculation of 45 s, while the equivalent 3D FEA required a time of calculation of 4 h and 19 min.

5. Conclusions

A methodology based on finite element (FE) analysis (FEA) and experimental techniques for designing the elastic elements of prosthetic feet and their functionality verification is presented. The present methodology can be exploited for different purposes. It can be used to design new configurations of energy-storing and -releasing (ESR) feet for different weight categories. It can be applied as a tool to study new systems that can be added to an existing ESR foot to change its stiffness/damping properties. Furthermore, it can be used to study the behavior of active prosthetic feet where their working principle relies on actuators combined with the elastic foot. The methodology is subdivided into three main phases: the design phase, the validation phase and the functionality verification phase. The design phase consists of two steps based on FEA. In the validation phase, the FE models are validated through a static test on physical prototypes. In the functionality verification phase, two standard dynamic tests from ISO 10328 and ISO 22675 are simulated. In the case study of the ESR foot MyFlex-$\gamma$, the static 2D FE model from the first step of the design phase resulted to be a reliable and fast-response tool to predict the influence of geometric parameters on the behavior of the ESR foot. In addition, the 2D FE model resulted to be a reliable instrument to find the optimal values of these geometric parameters. With the static 3D FE model (the second step of the design phase), it was possible to optimize the material properties of the composite elastic elements. By optimizing the geometry and the material properties, it was possible to obtain the correct stiffness of the foot prosthesis for a specific weight category of users, providing a device with a safety factor of 4. The stiffness curves obtained from the 3D model resulted comparable to the results from the ISO 10328 equivalent mechanical test (validation phase). Therefore, the FE models can be considered validated. The rotation angles of the foot obtained in the model based on the dynamic test ISO 10328 and the model based on the dynamic test ISO 22675 are within the target range of motion. The two dynamic tests can be considered alternatives to each other. They can be both considered helpful for a preliminary assessment of the prosthesis dynamic behavior before testing the prototype with amputee users.