Development of a Nonmotorized Mechanism for Ankle Rehabilitation ^†

Abstract

The objective of this paper is to present the development of a novel nonmotorized mechanism for ankle rehabilitation. The mechanism is based on the four-bar linkage. The mechanism transmits angular movement generated by the patient’s hand to an oscillatory movement on the ankle joint. The design of the device used a differential evolution algorithm to find the mechanism dimensions. To validate the system, a virtual tridimensional model was constructed to verify the angular outputs, check the existence of singularities, and execute movements with a virtual wooden test dummy.

1. Introduction

Robotics has a wide area of application which it can include the development of equipment for the rehabilitation of the human body for different types of patients with neurological problems [1,2]. Brain injuries can impair the necessary movements of the lower limb, influencing human gait [3,4]. The use of robotic structures can generate benefits such as reducing costs with active labor for movement-based rehabilitation treatments, as well as expanding the range of exercises performed, thus helping patients to maintain mobility through continuous therapy [5].

The objective of this paper is to develop a non-motorized device for the rehabilitation of the human ankle. Equivalent devices are available on the market but are motorized [6]. The system proposed in this paper aims to explore a purely mechanical mechanism and hopes to obtain a system easily operated by patients and health professionals. Being a purely mechanical mechanism, the costs of the equipment with actuators and control systems are reduced. It is noteworthy that being a purely mechanical device, the system can be more robust and easier to handle.

The actuation of the mechanism is performed by the patient’s upper limb force. So, it is possible to stimulate both the upper and lower limbs. In [7,8], it was shown that self-operated systems can generate positive effects in terms of mobility in the treatment of brain injuries.

In this way, this paper seeks to associate the advantages of using a nonmotorized and self-operated structure to accelerate the recovery process of patients with possible mobility gains in the upper and lower limbs applied to the ankle.

This work is divided into a brief review of the kinesiology of the human ankle followed by mathematical modeling of the device and three-dimensional simulations in CAD/CAE software, finishing with the conclusions.

2. Ankle Kinesiology

The movements of the ankle joint are: plantar flexion and dorsiflexion, occurring in the sagittal plane; abduction/adduction, occurring in the transverse plane; and inversion/eversion, occurring in the frontal plane [9,10].

The ankle ranges of motion are characterized by a significant variability between individuals in function of geographical/cultural differences, anatomical structures, and different ways to measure the angles [11].

In this paper the limit to the dorsiflexion movement goes from 0° to 30°, as shown in Figure 1a, and the plantar phase goes from 0° to 50°, Figure 1b. The abduction/adduction has a range of ±10° and the inversion/eversion of ±12° [11].

The main action of the ankle joint is to allow dorsiflexion and plantar flexion of the foot, and this information was used to design the proposed novel ankle device [12].

3. Mathematical Model of the Ankle Device

To design the mechanism for ankle rehabilitation, an analytical method was used to find the dimensions that make it possible to obtain the angular output of the ankle flexion joint, as well as to ensure that the stresses produced are supported by the components.

The method based on static balance for planar four-bar mechanisms [13] was used considering that the movement performed has low velocity and acceleration due to the nature of the rehabilitation exercises.

The objective of the designed mechanism is to convert a 360° angular input turn from the subject’s hand to an oscillatory angle output in the ankle joint to be rehabilitated. One device that satisfies these conditions is a crank–rocker four-bar mechanism.

The links and angles of a crank–rocker model are shown in Figure 2. The angle ${\theta }_2$ needs to turn 360° in a complete rotation, and the angle ${\theta }_4$ needs to have the angular displacement of the ankle joint, as shown in Figure 1.

The geometric relations are obtained using the law of cosines and are described in (1) to (5).

$$\begin{array}{c}s=\sqrt{r_1^2+r_2^2-2r_1{r}_2\cos {\theta }_2}\end{array}$$

(1)

$$\begin{array}{c}\beta ={\cos }^{-1}\left[\frac{r_1^2+s^2-r_2^2}{2r_{1}s}\right]\end{array}$$

(2)

$$\begin{array}{c}\psi ={\cos }^{-1}\left[\frac{r_3^2+s^2-r_4^2}{2r_{3}s}\right]\end{array}$$

(3)

$$\begin{array}{c}\lambda ={\cos }^{-1}\left[\frac{r_4^2+s^2-r_3^2}{2r_{4}S}\right]\end{array}$$

(4)

$$\begin{array}{c}\gamma =\pm {\cos }^{-1}\left[\frac{r_4^2+r_3^2-s^2}{2r_4{r}_3}\right]\end{array}$$

(5)

Using the relations from (1) to (5), and angles ${\theta }_3$ and ${\theta }_4$, the output of the system can be obtained by applying (6) to (9).

$$\begin{array}{c}{\theta }_3=\psi -\beta ,\hspace{1em}0°\le {\theta }_2<180°\end{array}$$

(6)

$$\begin{array}{c}{\theta }_3=\psi +\beta , \hspace{1em}180°\le {\theta }_2<360°\end{array}$$

(7)

$$\begin{array}{c}{\theta }_4=180°-\lambda -\beta ,\hspace{1em}0°\le {\theta }_2<180°\end{array}$$

(8)

$$\begin{array}{c}{\theta }_4=180°-\lambda +\beta ,\hspace{1em}180°\le {\theta }_2<360°\end{array}$$

(9)

The length of links $r_1$, $r_2$, $r_3$, $r_4$ and the angle ${\theta }_4$ are set on an evolutional algorithm, and the values for each link vary until the relations (1) to (5) are complacent. We constrained the variables to obtain a mechanism suitable for a 1.80 m height person. The evolutional algorithm [14] uses a population of 50 individuals, with link lengths ranging from 100 to 300 mm, converging into the minimal link lengths, respecting the constraints (1) to (5)

The calculated link lengths are r₁ = 273 mm; r₂ = 104 mm; r₃ = 235 mm and r₄ = 164 mm.

The mechanism obtained has no singularities, and the mathematical model development to obtain the singularities is detailed in [6].

Generally, in the rehabilitation exercises, the speed and acceleration are considered low, and a static model for calculating the forces on each link and to select the transversal section is considered. We used an individual of 1.80 m height with a 150 kg weight as the external load, and a factor of safety equal to 1.5 was used to compensate simplification factors in the model such as link frictions, load variations, and orthosis weight.

The links are made of aluminum alloy with a strength of 276 MPa [15].

The used transversal sections with the maximum stress values of each link of the proposed mechanism are given in

Table 1. Maximum stress calculated on the links of the proposed device.

Stress on Bar ${\mathit{r}}_1$ (MPa)	Stress on Bar ${\mathit{r}}_2$ (MPa)	Stress on Bar ${\mathit{r}}_3$ (MPa)	Stress on Bar ${\mathit{r}}_4$ (MPa)	Transversal Section (Width/Height/Thickness)
0.86	0.93	1.02	18.60	1/2” × 1” × 2 mm

.

To transmit the input movement from the subject’s upper arm to the link $r_2$ and to decrease the necessary torque to an acceptable level, a chain drive is used, as shown in Figure 3.

4. CAD/CAE Simulations and Results

The Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE) models of the proposed device were built and submitted to simulations of the complete cycle of motion.

Figure 4 shows the CAD/CAE model of the proposed device. To simulate the movement of the structure, we used a 3D wooden puppet of 1.8 m height.

Figure 5 presents the variation of the output angle ${\theta }_4$ and the angle ${\theta }_3$ found for the link lengths obtained for the ankle.

The angular amplitude obtained for the ankle mechanism is 78.89°, against 80° for the joint, with a relative error of 1.39%.

Figure 6 presents the Finite Elements Analysis of the proposed device at the angle ${\theta }_2$= 44°. The used aluminum transversal section, Table 1, resisted the food weight.

5. Conclusions

In this paper, a simple and innovative device for ankle rehabilitation using a crank–rocker mechanism is proposed, combined with a chain drive transmission. This novel device will be a tool for the simultaneous rehabilitation of the upper and lower limb, focusing on the ankle joint.

The proposed device was modeled mathematically, and the dimensions were obtained with the aid of an evolutional algorithm.

Static analysis showed that the device can be built using easy-to-find aluminum materials, leading to a low-cost device.

The CAD/CAE simulations showed that the designed device can make the ankle joint movements in function of the upper limb action.

The next step will be the construction of the prototype and to realize of experimental tests with patients.