*3.6. Ankle Manifold Representation*

In this section, we show the results in the software SageMath Manifolds. We load the model and visualize it as a manifold, we show the axis and the sagittal plane intersection. With the model parameters loaded, *r*1, *r*2, *ω*1, *ω*2, and the origin established in the center of the base modules. We apply the equation:

$$
\hat{r}\_1 = \overline{c}\_0 + \hat{n}\_p \cdot d \tag{62}
$$

where *c*<sup>0</sup> is the median center computed from trajectories A, B, and C center fitting, and *n* ˆ *p* is the median planes' normal vectors containing the circles. Table 8 shows values for the TC axis in the PM chart. In Table 9, we show the Plucker coordinates for the TC and ST axes.

Finally, Figure 38a shows the ankle manifold, and Figure 38b, the chart representing the range of movement and angle coordinates.

**Table 8.** Axis estimation data.


**Table 9.** Plucker line coordinates.


**Figure 38.** Ankle joint manifold. (**a**) Manifold for PM, (**b**) chart with ankle axis coordinates.

#### **4. Discussion**

In this work, we addressed the human ankle joint model from an alternative approach. We used statistical measurements for the development of a new device, specially designed to capture the human ankle joint movements. In animal joints, it is difficult to place encoders and linear sensors to measure the range of movement of complex joints in each internal living tissue reference frame. The product of exponential formulas uses only two frames, and it is useful in this case. Furthermore, in our work, we used a trilateration method for finding the device's platform position, which is an analytic method. Therefore we avoid numerical approximations that can diverge and reduce rounding errors. We proposed the ankle joint model as a Riemannian manifold. We can define a chart as a subset of such a manifold with angle coordinates for measuring the range of movement. Our presented device is lightweight, non-invasive, and can be used in remote places, on beds, or on the floor. By characterizing the ankle parameters, we can conduct symmetry studies by

correlating the left and right ankle joints. We can enhance the device configuration in future versions by replacing the draw-wire sensors used from potentiometers to digital encoders connected by a CAN bus, reducing wiring, space, weight, and energy consumption. We will use the model for the synthesis and reconfiguration of an ankle parallel rehabilitation robot, programmed by symmetrical movements at the opposite ankle. By employing the axis location and the screw theory, forces, and torques, we will study the ankle dynamics by using reciprocal screws to the axis location in a re-configurable platform. The robot will be lightweight because of the use of cable-driven actuators, inspired by antagonistic muscles that work with reciprocal inhibition for energy optimization. The robot will reconfigure the structure, considering the ankle joint as a central mast, and referenced it with MMP and MLP markers.

Figure 39 shows a schematic of the re-configurable approach.

**Figure 39.** Re-configurable cable-driven robot concept.

Other applications are, for example, by visualizing the platform trajectories one can explain how the calcaneal Achilles insertion is near to the platform's A point. The platform's normal vector changes abruptly near this region, as was depicted in Figures 24b and 26a,b. Furthermore, Riemannian models have different properties. We will explore diagnosis and treatments based on the model and metrics by employing machine learning algorithms. This approach can be applied to other joints in humans and other animals, by designing specialized re-configurable hardware and software. Tracking the parameters in different ages and weight conditions, and comparing the ankle models in healthy and injured people.

#### **5. Conclusions**

Computer tomography (CT) and magnetic resonance (MR) images have greater precision and accuracy. Measurements in medical imaging will help us compare the errors (RMS) in the HAJ. In biomechanics, we have not found an ideal model for error comparison. Then, we will compare the error with an accurate measurement. The device has limitations regarding mechanical precision and deformation of its parts. We face up to the error through the electronic design system. The calibration process is imperative for enhancing accuracy.

The calibration process is human-dependent. We read the digital measurement and compare it with caliper measurements directly in the sensor. Then, we register the data in a table to find the equivalence. An electronic board with trimmers avoids saturation, bias,

and calibration; a 10 bit ADC and an exponentially weighted moving average (EWMA) filter the noise signals. We have implemented a processing (Software) calibration interface. We avoid adding more specific technical data, such as CMRR, ADC speed, mechanical tolerance, and other issues inherent to the measuring devices.

Digital sensors, communications, and POE function fitting use machine learning techniques.

The ankle is the most commonly injured joint of the lower limb, fundamental to the human body's balance; it is necessary to measure the range of motion by in vivo methods for patients in lying positions in reduced or remote places. The device's development considers ankle anatomy and anthropometry. We propose a Riemannian manifold model based on the device's data readings. Performing simulations enabled us to design the size of the device and the maximal length of the wires. We present a trilateration algorithm, projecting the tetrahedron's sides on the base plane. The sensors are modular and part of the device's lightweight and portable structure. The electronic system is modular, replaced by other single-board computers (SBC) and microcontroller unities. We will also use the TM for ankle characterization and diagnosis for rehabilitation robotics, prosthesis, and orthosis design. The prototype is not a finished product (the TRL is 2). The work's scope is to validate the use of a modern alternative biomechanic representation of the human ankle joint. It is a platform for testing an alternative trilateration method that employs draw-wire sensors (DWS). Such sensors have a constant tension, coiled on a drum attached to a potentiometer, and a flat spiral spring. We also attempted to develop a flexible device design for several foot sizes. We are working on a newer device version with an enhanced attachment system, a more compact design, and digital DWS compatible with a configurable robot. Machine learning and edge computing will assist in disease diagnosis and rehabilitation of patients.

**Author Contributions:** Conceptualization, J.V.-R., Á.V. and Ó.A.-V.; methodology, Á.V. and Ó.A.-V.; software, J.V.-R.; validation, Á.V. and Ó.A.-V.; formal analysis, J.V.-R.; investigation, J.V.-R.; resources, Á.V. and Ó.A.-V.; data curation, Á.V. and Ó.A.-V.; writing—original draft preparation, J.V.-R.; writing—review and editing, Á.V. and Ó.A.-V.; visualization, J.V.-R.; supervision, Á.V.; project administration, Á.V. ; funding acquisition, J.V.-R., Á.V. and Ó.A.-V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially funded by Colciencias-Colfuturo PhD Scholarships Program Educational Credit Forgivable grant number 568, and by Vicerrectorado de Investigación de la Universitat Politècnica de València (PAID-11-21).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** The code and CAD electronics and mechanical designs are available.

**Acknowledgments:** The authors thank the Colfuturo Colciencias Collaboration for supporting this work, as well as the Universitat Politècnica de València and the Universidad de los Llanos.

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

#### **Abbreviations**

The following abbreviations are used in this manuscript:

