**1. Introduction**

Humanoid robotics has fascinated and challenged scientists for decades [1]. The design of humanoid robots has evolved from the serial design of the first humanoids, such as WABOT (WAseda roBOT) [2], to more refined architectures [3–12]. However, despite decades of research, the mobility of most of these robots is still limited to the legs, arms, hands, and head only, with very few examples including torso [10] and foot [13–23] mobility, despite the feet and torso playing a key role in human motion [24]. The reduced mobility of humanoid robots in those regions hinders their capability to achieve balance and to perform complex dynamic tasks. Human foot architecture is characterized by three main segments—namely the heel, mid-foot, and toes—that fulfill distinct roles during bipedal locomotion [25].

However, several humanoid robots use flat, one-segmen<sup>t</sup> feet [2–12]. Some famous examples include Honda's ASIMO (Advanced Step in Innovative Mobility) [3], the iCub design and its variants from the Italian Institute of Technology [7], and Softbank Robotics NAO [9]. Several research groups have tried to improve the foot performance by compensating a limited mechanical design with an enhanced sensing/control capability, which can be obtained by adding force/pressure sensors to the plant of the foot [26–30], with mixed results. Others kept a one-segmen<sup>t</sup> design, but moved from a flat rigid sole to a compliant one [13–15], obtaining significant results and increased performance (e.g., Boston Dynamics' Atlas [14]). Soft options have been also explored through designs with a highly elastic foot sole directly connected to the ankle joint [16].

Some research groups investigated the possibility of using two-segment feet, with a heel segmen<sup>t</sup> and a toe segmen<sup>t</sup> [16–20]. Foot designs with three segments have been proposed to better mimic human-like motion [21,22], but are usually characterized by active mechanisms that require a complex control and motion coordination with the rest of the body [21], or by planar rigid-body mechanisms [22] that achieve balance through spring-loaded joints only. While successful, the lack of compliance in these three-segment designs limits their behavior when obstacles introduce 3D torques and loads.

**Citation:** Russo, M.; Chaparro-Rico, B.D.M.; Pavone, L.; Pasqua, G.; Cafolla, D. A Bioinspired Humanoid Foot Mechanism. *Appl. Sci.* **2021**, *11*, 1686. https://doi.org/10.3390/ app11041686

Academic Editors: Manuel Armada and Giuseppe Carbone

Received: 21 January 2021 Accepted: 9 February 2021 Published: 13 February 2021

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In this paper, an innovative passive mechanism for robotic feet is presented, which is able to adapt to the environment without the need of external actuation thanks to joint elasticity. First, the mobility requirements to achieve human-like motion are extracted from medical literature and experimental gait analysis using a motion capture system. Then, the proposed design is introduced, along with its main characteristics and technical details. The foot mechanism is validated through numerical simulation with both Finite Element Analysis (FEA) and Multibody Dynamics, and a prototype is manufactured with 3D printing in order to prove the feasibility of the proposed mechanism on a range of terrains and obstacles.

#### **2. Materials and Methods**

#### *2.1. Requirements for Humanoid Foot Mechanisms*

In order to obtain the mobility requirements for humanoid foot mechanisms, the human gait was analyzed in terms of absolute and angular motion [31–35] looking retrospectively at clinical outcomes from patients who outcome a healthy gait cycle. In accordance with the medical standards, a variation of the Davis protocol [31]—with a marker on the second metatarsus instead of fifth—was used to acquire the motion of 11 subjects (details in Table 1) over 10 distinct gait sessions of four walking steps each. The motion was acquired through a VICON Motion Capture system, with the following retroreflective marker set configuration and alignment (Figures 1 and 2):




The BTW Gaitlab software (BTS Bioengineering Corp., Quincy, MA, USA) [32] was used for the data processing, according to the following steps:


• The remaining dataset was used to obtain a normalized average gait in terms of ankle angle, which is illustrated by the black line in Figure 3.

**Figure 1.** Example of the marker placements on the left foot (CAD model), with the left ankle (LA), left toe (LT) and left heel (LQ) markers.

**Figure 2.** Example of the marker placements on the right foot of a patient, with the right ankle (RA), right toe (RT) and right heel (RQ) markers.

**Figure 3.** Normalized average gait as the ankle angle, in normalized time.

The resulting average angular displacement for the whole motion is approximately 25 degrees, compared to a peak value of 45 degrees of the ankle angular displacement for some acquisitions. By expressing the angular motion range of the heel part of the foot, the values in Figure 3 identify all of the possible foot configurations in which the foot must be

in stable contact with the ground in order to support the body, and will be used both for dimensioning and simulating the proposed humanoid foot mechanism design.

#### *2.2. An Underactuated Humanoid Foot Mechanism*

The proposed foot mechanism aims to enhance robot mobility and efficiency by enabling body support in all of the stances acquired in Section 2. In order to do so, a three-segment design was considered, based on the diagram in Figure 4. During the gait, the foot motion is ruled by four main centers of rotation, which correspond to the ankle joint, the heel bone (or calcaneus), the navicular tuberosity, and the metatarsophalangeal (MTP) joint, as shown in Figure 4. The first segmen<sup>t</sup> of the foot can be associated to the calcaneus-talus complex, which includes the heel and ankle, and can be approximated to a rigid body. The motion of the central segmen<sup>t</sup> of the foot is enabled by the active stiffening behavior of the intrinsic foot muscles in the plantar arches, which connect the calcaneus to the MTP joint. The plantar fascia and several ligaments support the plantar arches, which store mechanical energy during weight bearing by deforming. This behavior can be reproduced with the elasticity of a compliant body. The motion of the last segmen<sup>t</sup> of the foot can be approximated to a flexion/extension movement of the toes, which rotate around the MTP joint and the interphalangeal joints. The MTP joint is usually described as a condyloid joint, while the interphalangeal joints resemble revolute joints.

**Figure 4.** Human foot mobility [23], with a focus on the main joints of the foot: the ankle joint ( **A**), the heel bone (or calcaneus (**B**)), the navicular tuberosity ( **C**), and the metatarsophalangeal (MTP) joint ( **D**/**E**).

In order to replicate the behavior of the human foot, a new compliant foot mechanism based on the kinematic scheme in Figure 5 is here presented. The proposed mechanism is based on a compliant mechanism in the plant, which can be approximated with an RRPR (revolute-revolute-prismatic-revolute) architecture, with revolute joints in points B, C, and D, and a link of varying length between B and D. Furthermore, a second compliant mechanism is added in the MTP joint in point E, with a rigid revolute joint controlled by a torsional spring.

**Figure 5.** Kinematic scheme of the proposed foot mechanism based on the corresponding points (A to E) in the foot mobility diagram in Figure 4: (1) hindfoot; (2) midfoot; (3) forefoot; (4) compliant body.

The mechanism works mainly on the sagittal plane, even if the compliant part can be designed to balance 3D loads as well. When analyzing the planar behavior of the proposed foot design, its degrees of freedom can be evaluated from its rigid mechanism equivalent with the Chebychev-Grübler-Kutzbach criterion as two, controlled respectively by the plantar compliant element and by the torsional spring of the MTP joint. An exemplar design of the proposed foot mechanism is shown in Figure 6.

**Figure 6.** CAD model of the proposed foot mechanism, based on the equivalent kinematic scheme of Figure 5.
