**1. Introduction**

Taking in vivo measurements in the human ankle joint is troublesome because the ankle is a complex mechanism [1]. Deviations in the axis increase the pronation or supination moments, causing instability and enhancing injuries risk. In this work, we present a device intended for the study of the human ankle joint (HAJ). Modeling and measuring this lower limb joint is essential in physiology, biomechanics, and rehabilitation (also in humanoid robotic limb development).

Our primary aim is to develop a device for the two axes model estimation of the human ankle joint. Secondary objectives are: it must be non-invasive, compact, energy-efficient, and easy to set up and transport. It should also be compatible with laying positions, such as with the foot in the elevated position. To accomplish the objectives, we followed a plan, first by understanding the ankle movements. Then, we used statistics for dimensional determination. We also use a modern approach, such as the Product of Exponential (POE) formula. We then designed the structure based on embedded non-invasive distance sensors.

**Citation:** Agudelo-Varela, Ó.; Vargas-Riaño, J.; Valera, Á. Turmell-Meter: A Device for Estimating the Subtalar and Talocrural Axes of the Human Ankle Joint by Applying the Product of Exponentials Formula. *Bioengineering* **2022**, *9*, 199. https://doi.org/ 10.3390/bioengineering9050199

Academic Editors: Farshid Guilak, Christina Zong-Hao Ma, Zhengrong Li and Chen He

Received: 3 February 2022 Accepted: 20 April 2022 Published: 4 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Our contribution to the ankle joint axis localization is the holistic development of a specific device. Draw-wire sensors measure distance, are composed of a wire wound around a drum, and are attached to a potentiometer and a spring. They are retractile with constant tension. For bias correction and gain calibration, we designed a capture system. We adjust the voltage to avoid the maximal value of the analog-to-digital conversion. We calibrated each sensor through direct measuring with a Vernier caliper. Then, we measured the voltage and adjusted the offset and gain by a calibration program in Processing (Software). Limitation measurements are by 10-bit analog-to-digital converters and digitally filtered in the acquisition board. Technology Readiness Level (TRL) is 2.

We highlight our approach over traditional methods because we apply the POE formula to the ankle kinematic model. Furthermore, we estimate the ankle axis localization by a geometric approach, solved algebraically. We computed it from the pseudo-inverse application. For the talocrural and subtalar axes estimation, we use circle fitting. As an alternative ankle joint representation, we propose a Riemannian chart. We have limited the scope to the human ankle joint (HAJ) model. There are applications in physical therapy and HAJ mobility diagnosis.

The state of the art in the ankle localization is detailed in [1–14].

There are different HAJ models in the literature; we focus on the two-axes approach. The approach is recommended by the International Society of Biomechanics (ISB) [15], anatomy and biomechanics books [3,16–19], and simulation software [20]. We found models of the ankle joints in several articles [14,21–26]. Contributions to the study of the ankle joint axes are in [2,8,9,27]. The most cited research about the subtalar axis are in [5,7,10–13]. A literature review of functional representations is in [4].

Draw-wire sensors (DWS) are distance measurement sensors, who use a wire coiled on a drum attached to a potentiometer and a spiral spring that are retractile at constant tension. Similar robotic applications are in [28–30], also in linear position tracking [31], and easy robot programming [32]. Inertial measurement units (IMU) were post-processed and complemented with other sensors [33–37]. We shall employ our device for the HAJ bi-axial measurements and for other models as well [38]. BiodexTM and HumacnormTM are manufacturers of general kinetics machines.

We divide the materials and methods section into two subsections: the motion theory and the mechatronics design. In the first section, we study anatomy, statistics, proportions, and anthropometry to understand the functional HAJ movements and standard dimensions. Then we perform the HAJ simulation using the POE formula. Here, we do not include a deep study of infinitesimal kinematics. We intend to design a device for a healthy HAJ with no singularities with a continuous range of movement. We describe the trilateration method to find the platform pose. It is a geometrical method based on tetrahedrons; we avoid numerical solutions that depend on finite derivative terms. The tetrahedron is a well-defined 3D geometrical structure. Solving tetrahedron geometry is the expansion of planar trigonometry. Knowing the sides allows us to find the height of a tetrahedron. We attach the platform to the foot; the sensors are passive elements and do not support or add high tensile forces. We have selected the first seven sensor configurations 3-2-2 (seven sensors) instead of 3-3-3 (nine sensors) or 3-2-1 (five sensors) for hardware limitations, sensor redundancy, and symmetrical design (for both limb use).

The device's mechatronics design and implementation are in the second subsection. We used Draw-wire sensors to measure the tetrahedron sides. These sensors have a constant tension because they comprise a drum attached to a spiral spring. We limit them to the maximal distance, and the precision depends on the potentiometer and electronics signal conditioning with a high common-mode rejection ratio (CMRR). The calibration process deals with accuracy and precision. First, we made rough adjustments to the acquisition system. Second, the software calibration process makes fine adjustments. Our proposed method avoids numerical errors because it uses geometric formulas. We validate the position through sensor redundancy. We conduct calibration and testing in a healthy

patient and represent the HAJ movements as a manifold chart. The complementary source code was uploaded to [39].

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

This section is grouped in two main subsections, first the motion theory, and second the mechatronic system. For the first part, we show the simulation using anthropometric values and the POE formula. Using the plots, we estimate the DWS maximal length. Next, we present the device's geometrical design and the trilateration method. Finally, we compute the axis position by circle fitting and modeling the ankle joint as a Riemannian manifold chart. In the second subsection, we describe the mechanics design and implementation, we used SolidWorks ® (2017–2018 Student Edition, Dassault Systèmes, Vélizy-Villacoublay, France), KiCad ©(6.0.4, Jean-Pierre Charras and KiCad developers, CERN, Linux Foundation), and FreeCad © (0.19, Jürgen Riegel, Werner Mayer, Yorik van Havre and others) StepUp tools addon.
