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

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

Figure 1

Reference points from anthropometric values K, L, O, and P.

Figure 2

Q, W, and w distances from lateral and transverse views.

Figure 3

Mean relative position of the ST and TC axis.

Figure 4

Planes, axes, and points of corresponding references.

Figure 5

Vectors and points on the sagittal plane.

Figure 6

Forward kinematics for (a) initial position and (b) ${\theta }_{1range}={\theta }_{2range}=[-{15}^{\circ },{15}^{\circ }],{\theta }_1={\theta }_2={10}^{\circ }$ and (c) ${\theta }_{1range}={\theta }_{2range}=[-{10}^{\circ },{10}^{\circ }],{\theta }_1={\theta }_2=5^{\circ }$.

Figure 7

Geometric design: (a) is the platform center, base, and $r_1,r_2$; and (b) platform vertices with talocrural and subtalar axis.

Figure 8

Geometric design of the DWS arrays.

Figure 9

Tetrahedron trilateration flowchart.

Figure 10

Finding the apex ${\mathrm{A}}_{\mathrm{p}}$.

Figure 11

Rotation of $\alpha$ angle about the axis $B_1-B_3$: (a) original tetrahedron, $T_B$ (b) rotated tetrahedron.

Figure 12

Finding the apex ${\mathrm{B}}_{\mathrm{pr}}$.

Figure 13

Adjusting the foot, the shank and the Most Medial Point reference.

Figure 14

Computed positions from sensor lengths at portable configuration: (a) the rest position, (b) apex A, (c) apex B, and (d) apex C.

Figure 15

Simulation of the platform central point with variations in the mean statistical values: (a) platform’s center point manifold, (b) manifold chart and a geodesic.

Figure 16

Draw-wire sensor design.

Figure 17

Mechanical attachment: (a) calf support and (b) aluminum tube structure.

Figure 18

Base and platform: (a) DWS modules support (b) platform with foot’s size adjustment.

Figure 19

Two Op. Amp. instrumentation amplifier.

Figure 20

Power system with backup, BMS, boost, and buck converters.

Figure 21

Modular electronics casing.

Figure 22

Complete prototype.

Figure 23

Rendered image with a 175 cm height patient.

Figure 24

Simulation of all points: (a) platform’s central point, (b) attachment a, (c) attachment b, and (d) attachment c.

Figure 25

Simulation of the platform central point with variations in the mean statistical values: (a) 10% below, and (b) 10% over.

Figure 26

Simulation of the platform’s attaching point A: (a) mean values plus 10%, (b) mean values minus 10%.

Figure 27

Attaching point B simulation: (a) adding 10% to the statistic mean values, (b) subtracting 10%.

Figure 28

Simulation results for C: (a) mean values plus 10%, (b) mean values minus 10%.

Figure 29

Interactive simulation example: (a) sliders, (b) rendering.

Figure 30

Connections and electronics.

Figure 31

Assembled structure.

Figure 32

Processing calibration interface.

Figure 33

Measuring in SolidWorks (2017–2018 Student Edition, Dassault Systèmes, Vélizy-Villacoublay, France)^®: (a) sensor 1, (b) sensor 2, (c) sensor 3.

Figure 34

First two trilateration results: (a) position 1, (b) position 2.

Figure 35

Latest two trilateration results: (a) position 3, (b) position 4.

Figure 36

TC axis circle fitting: (a) trajectory A, (b) trajectory B, (c) trajectory C, (d) trajectory PM.

Figure 37

ST axis circle fitting: (a) trajectory A, (b) trajectory B, (c) trajectory C, (d) trajectory PM.

Figure 38

Ankle joint manifold. (a) Manifold for PM, (b) chart with ankle axis coordinates.

Figure 39

Re-configurable cable-driven robot concept.