*Article* **Turmell-Meter: A Device for Estimating the Subtalar and Talocrural Axes of the Human Ankle Joint by Applying the Product of Exponentials Formula**

**Óscar Agudelo-Varela 1,† , Julio Vargas-Riaño 2, \* and Ángel Valera 2,†**


**Abstract:** The human ankle is a complex joint, most commonly represented as the talocrural and subtalar axes. It is troublesome to take in vivo measurements of the ankle joint. There are no instruments for patients lying on flat surfaces; employed in outdoor or remote sites. We have developed a "Turmell-meter" to address these issues. It started with the study of ankle anatomy and anthropometry. We also use the product of exponentials' formula to visualize the movements. We built a prototype using human proportions and statistics. For pose estimation, we used a trilateration method by applying tetrahedral geometry. We computed the axis direction by fitting circles in 3D, plotting the manifold and chart as an ankle joint model. We presented the results of simulations, a prototype comprising 45 parts, specifically designed draw-wire sensors, and electronics. Finally, we tested the device by capturing positions and fitting them into the bi-axial ankle model as a Riemannian manifold. The Turmell-meter is a hardware platform for human ankle joint axes estimation. The measurement accuracy and precision depend on the sensor quality; we address this issue by designing an electronics capture circuit, measuring the real measurement with a Vernier caliper. Then, we adjust the analog voltages and filter the 10-bit digital value. The Technology Readiness Level is 2. The proposed ankle joint model has the properties of a chart in a geometric manifold, and we provided the details.

**Keywords:** human ankle model; product of exponentials formula; anthropometry; biomechanics; coordinate measuring machines; kinematics; pose estimation; position measurement; biomedical informatics
