**3. Results**

#### *3.1. Simulation and Results*

The behavior of the humanoid foot was checked through a nonlinear, dynamic study using FEA. Iterative schemes to solve Equation (1) through the Newton–Raphson method (NR), already integrated in the software, were used for each node (i).

$$\left[ \left[ M \right]^{t+\Delta t} \{ \mathcal{U}'' \} \right]^{(i)} + \left[ \mathcal{C} \right]^{t+\Delta t} \{ \mathcal{U}' \}^{(i)} + \prescript{t+\Delta t}{} \left[ K \right]^{(i)} \mathop{\mathbb{I}}^{t+\Delta t} \{ \Delta \mathcal{U} \}^{(i)} = \prescript{t+\Delta t}{} \{ R \} - \prescript{t+\Delta t}{} \{ F \}^{(i-1)} \tag{1}$$

where:


In nonlinear static analysis, equations have to be solved at any time step *t*+Δ*t*. Because the internal nodal forces *<sup>t</sup>*+Δ*<sup>t</sup>*{*F*}(*i*−1) depend on the nodal displacements at time *t*+Δ*t*, *<sup>t</sup>*+Δ*<sup>t</sup>*[Δ*U*](*i*), an iterative method must be used to find a converging solution. The abovementioned iterations have several methodologies in place. The NR scheme was used for this simulation. Within this method, the tangential stiffness matrix is formed and decomposed in a certain step in every iteration. The NR method using Newmark integration is used because it has a high rate of convergence, and is quadratic. However, each iteration generates and decomposes the tangential rigidity, which is prohibitively costly for larger models. Thus, a different iterative approach may be advantageous.

The humanoid foot was placed above a solid body, simulating the ground. The ground was fixed, giving it 0 degrees of freedom. A gravity of 9.807 m/s<sup>2</sup> was set, acting on the entire assembly. The temperature loads were included, setting a constant temperature of 298 Kelvin. The mean angular displacement of the ankle acquired from the subject was used as the input motion for the nonlinear dynamic simulation. The definition of the time curve for the angle displacement application was set to simulate the mean step duration for 0.63 s, using a force control technique. The input given to the simulation was extracted by the gait analysis data, and it corresponds to the average gait of Figure 3. The input motion was applied to the ankle joint axis, which is highlighted in blue in Figure 7.

**Figure 7.** Ankle joint axis of the CAD model, on which the input motion is applied.

The humanoid foot is composed of four bodies, three of which are in ABS rigid plastic (Table 2), and the middle of which is set to thermoplastic polyurethane (TPU) for his flexible mechanical properties, simulating the muscle part. Another key component is the spring—shown in Figure 8—in the forefoot, which passively improves the ability of the foot to adapt to the ground; the spring was modelled with a 2 mm diameter, five coils, and an 8 mm length, and alloy steel was chosen as material in order to simulate a commercial part. After setting all of the required parameters, a mesh was generated with the parameters in Table 3 to perform the FEA.



**Figure 8.** Detailed view of the forefoot torsional spring.



Figure 9 shows the FEM nonlinear dynamic analysis stress results computed by the use of the Von Mises formulation. A maximum value of 2.57 × 10<sup>6</sup> N/m<sup>2</sup> is observed in the upper part of the rigid midfoot, next to the connection to the compliant body. The maximum value is significantly smaller than the yield limit of the material, thus validating the proposed humanoid foot for an average walking gait. The behavior of the impact of the foot on the ground is obtained by applying the mean angle displacement taken from the analyzed subjects (see Section 2). The results can be observed in the step-by-step effect of the load on the different bodies of the humanoid foot, as is shown in the linear displacement distribution reported in Figure 10:


**Figure 9.** Von Mises stress distribution.

**Figure 10.** Displacement values in the humanoid foot.

These results validate the behavior of the proposed humanoid foot, showing that it acts similarly to the human foot in terms of stress while walking. Furthermore, the results prove that the proposed design can safely withstand the loads associated with an average human gait, including the impact of the foot on the ground.

#### *3.2. Experimental Tests*

Due to the promising simulation results, a preliminary prototype was built at the Biomechatronics Lab of IRCSS Neuromed in Pozzilli, Italy. The prototype is composed of six parts: five were manufactured by 3D printing, using the rapid prototyping techniques explained by Cafolla et al. [36], whereas the last one is a commercial mechanical component. The printing materials were chosen from among a wide variety of commercial resins and plastics, in order to ensure the desired foot behaviour: the four rigid bodies were printed in PLA filament; the flexible component, representing the midfoot of the mechanism, was manufactured using a TPU filament with properties matching the ones of the FEA. The properties of both materials are reported in Table 4, for reference. Different infill percentages and patterns were tested for the midfoot, and the final infill was selected for its optimal TPU stiffness. Both the value of the infill and the 3D support pattern with which the infill was printed were optimized according to the surface curve of the compliant component through an FEA with constrained motion (as per the acquired gait).

The correct relative positioning of the components is defined by male and female rectangular pins. Then, the components are locked together with interlocking shapes. Additionally, epoxy glue was added to the coupled surfaces in order to improve the strength of the connection, as it is the most loaded component of the system, as shown in Figure 9. The assembly of the entire foot is shown in Figure 11. The prototype of the humanoid foot is composed of a rigid hind foot, a flexible mid foot, and a rigid forefoot, which is made of three 3D printed rigid parts, namely the connection pin, the sole and the toe, and a commercial part, the spring. The prototype was scaled down from human size in order to fit a service humanoid robot design [10–12], and thus the whole foot can fit into a box of (146 × 52 × 41) mm; it weighs 57.80 g.


**Table 4.** Mechanical properties of the materials of the 3D printed prototype.

**Figure 11.** Humanoid foot prototype assembly.

As a preliminary test, the adaptability of the foot to different step heights was tested. Figures 12 and 13 show the foot adapting to a 22 mm high step and a 54 mm high step, respectively. The combination of the forefoot spring and the compliant midfoot actions allow the proposed foot design to balance on step heights up to approximately 35% of the length of the foot.

**Figure 12.** Humanoid foot prototype assembly on a 22 mm step: (**a**) unloaded test; (**b**) loaded test.

**Figure 13.** Humanoid foot prototype assembly on a 54 mm step: (**a**) unloaded test; (**b**) loaded test.

Further experimental tests were conducted in order to measure the reaction force between the foot and the ground, and to validate the FEA results of Figures 9 and 10. By using the experimental setup shown in Figure 14, a load cell was calibrated with a set of calibration weights (accuracy 1%) before the experimental test, and was then used to measure the force, according to the scheme in Figure 15. A camera was used to measure the foot motion by tracking the marker in Figure 14 through the Kinovea software. By using the real dimension of the marker, i.e., 10 × 10 mm, the motion in pixels is converted to real-world one with a semi-automated tracking process, which has an estimated accuracy of 5% (measured through a checkerboard calibration).

**Figure 14.** Experimental setup for weight and displacement acquisition.

**Figure 15.** Experimental setup for weight and displacement acquisition.

A 1-second stance cycle was manually reproduced, moving according to the motion in Figure 16, which was measured with the camera during the experiment. During the load cycle, the marker moved 9.5 mm, with a comparable motion to the corresponding point in the analysis in Figure 10. The action of the foot on the ground, measured with the load cell, was reported in Figure 17.

**Figure 16.** Ankle angle, as acquired with motion capture by the camera during a gait cycle.

**Figure 17.** Force (normalized by body weight of 3.6 kg), as acquired by the load cell during a gait cycle.

The vertical ground reaction force applied to the foot is bimodal, with an initial impact peak, followed almost immediately by a propulsive peak as the foot pushes off against the ground. In a conventional humanoid robotic foot, this impulsive force is transmitted fully to the ankle, with the risk of disrupting the robot's balance. Conversely, the proposed foot is able to dampen this impact through the spring and the compliant body, significantly reducing the stress on the ankle joint.

Hall [37] and Munro [38] report that for a gait speed between 3.0 m/s to 5.0 m/s, the impact forces range from 1.6 to 2.3 times the body weight, and the propulsive forces range from 2.5 to 2.8 times the body weight. In order to compare their results to the experimental ones, an overall body weight of 3.6 kg was estimated from humanoid robots of similar sizes and proportions [10]. The acquired results, reported in Figure 16, are in a comparable range (6% error on displacement) of the FEA simulation, and show a comparable behaviour to that of the human locomotion system, with a propulsive peak of approximately 3.0 times body weight.

#### **4. Discussion and Conclusions**

In this paper, an innovative robotic foot design was introduced, based on the mobility of the human foot in walking gaits. Whereas most other humanoid robot feet use a singlebody or two-segment design, the proposed system is characterized by three underactuated segments that replicate the behavior of the human foot. The motion of the human foot was studied with gait analysis, which was performed on a sample of 110 human gaits in order to acquire the motion requirements for the proposed design. A passive foot mechanism with compliant elements was then introduced in order to improve humanoid robots' impact absorption and stability during walking gaits. The proposed design was validated through a simulation with FEA and Multibody Dynamics, which demonstrated its functionality. Experiments with a 3D printed prototype of the proposed foot mechanism were also reported to prove the feasibility of this study.

Overall, this work introduced a novel design that increases foot mobility thanks to a three-segment foot mechanism, without introducing additional degrees of freedom and control complexity into the system. In future works, the foot will be assembled on a humanoid robot for full validation. In addition, the proposed humanoid foot is feasible for future developments and studies in the field of prosthetics and neurorehabilitation.

**Author Contributions:** Conceptualization, M.R. and D.C.; methodology, B.D.M.C.-R. and D.C.; validation, B.D.M.C.-R. and D.C.; formal analysis, B.D.M.C.-R. and D.C.; investigation, M.R., B.D.M.C.-R. and D.C.; resources, L.P. and D.C.; data curation, M.R., B.D.M.C.-R., L.P., G.P., D.C.; writing—original draft preparation, M.R. and D.C.; writing—review and editing, B.D.M.C.-R. and D.C.; visualization, M.R. and D.C.; supervision, D.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by a gran<sup>t</sup> from Ministero della Salute (Ricerca Corrente 2021).

**Institutional Review Board Statement:** This is a retrospective observational study where data were used from clinical outcomes of 11 patients who outcome a healthy gait cycle. The patients were diagnosed and treated at IRCCS NEUROMED—the Mediterranean Neurological Institute (Italy), according to the national guidelines and agreements that govern its hospital center. All data were collected as part of routine diagnosis and treatment. This study does not report on the use of experimental or new protocols. Although the rehabilitation program carried out by the 11 selected patients is described in this study, the rehabilitation program was neither designed nor modified for the purposes of this study.

**Informed Consent Statement:** Patients entering IRCCS NEUROMED gave a generic consent to use their data for future scientific research purposes according to GDPR (General Data Protection Regulation) regulation.

**Data Availability Statement:** The data presented in this study are available on request at the discretion of the corresponding author. The data are not publicly available due to privacy reasons.

**Conflicts of Interest:** The authors declare no conflict of interest.
