2.1. Desired trajectory

The desired trajectory for the end-effector corresponds to the natural flexion-extension movement of the fingers. As an end-effector mechanism, only the fingertip trajectory is considered. The development of the hand prototype presented in ref. [Reference Ceccarelli, Rodríguez and Carbone18] assures that the motion of the finger is performed with a typical motion, with this assumption it can be considered that an end-effector mechanism is enough to reproduce the finger movement.

A set of representative movement curves were obtained from a group of healthy subjects. It was possible to determine, through a principal component analysis, that the flexion-extension movement can be located within a plane, since there is minimal variation in one of the three coordinate axes. The representative curves for each finger vary in amplitude. The curve with the greatest amplitude (Fig. 1), which corresponds to the movement of the middle finger, is considered as the desired trajectory for this work [Reference Zapatero-Gutiérrez, Castillo-Castañeda and Laribi19]. The authors in ref. [Reference Orlando, Dutta, Saxena, Behera, Tamei and Shibata20] found an important correlation ship among the index, middle, and thumb for rotational movement control of small objects; the results suggest that middle and thumb have a preponderant participation which can be a useful guide for designing new mechanisms.

Figure 1. Real flexion-extension trajectory.

Although only one curve is considered, the mechanism has been designed to modify the amplitude of the desired trajectory, as will be explained in Section 3.

2.2. Five-bar kinematic

The design of this rehabilitation device can be classified within the end-effector-type devices. Point $C$ of a symmetric five-bar mechanism generates a natural flexion-extension trajectory of the fingers, when the actuated joints $\theta _{1}$ and $\theta _{2}$ move from $\theta _{{1_{o}}}$ , $\theta _{{2_{o}}}$ to $\theta _{{1_{f}}}$ , $\theta _{{2_{f}}}$ , respectively; as seen in Fig. 2(a).

Figure 2. (a) Five-bar mechanism positions. (b) Five-bar mechanism configuration.

In Fig. 2(b), $L_{1}$ , $L_{2}$ , $L_{3}$ , and $L_{4}$ represent the lengths of the links, with $L_{1}=L_{4}$ and $L_{2}=L_{3}$ . The length $L_{0}$ is the distance between the fixed points $A$ and $E$ . Point $C$ is a rotational joint that joins the links $L_{2}$ and $L_{3}$ [Reference Zapatero-Gutiérrez, Laribi, Castillo-Castañeda, Zeghloul, Laribi and Arsicault17], its kinematic equation is given by Eq. (1) [Reference He and Lu21].

(1) \begin{equation}C=\left[\begin{array}{c} C_{x}\\[5pt] C_{z} \end{array}\right]\end{equation}

where

\begin{equation*} C_{x}=L_{1}\cos \theta _{1}+\frac{1}{2}\left[L_{0}+L_{1}\!\left(\cos \theta _{2}-\cos \theta _{1}\right)\right]-\left[L_{1}\!\left(\sin \theta _{2}-\sin \theta _{1}\right)\right]\left[\sqrt{\frac{{L_{2}}^{2}}{H^{2}}-\frac{1}{4}}\;\right] \end{equation*}

\begin{equation*} C_{z}=L_{1}\sin \theta _{1}+\frac{1}{2}\left[L_{1}\!\left(\sin \theta _{2}-\sin \theta _{1}\right)\right]+\left[L_{0}+L_{1}\!\left(\cos \theta _{2}-\cos \theta _{1}\right)\right]\left[\sqrt{\frac{{L_{2}}^{2}}{H^{2}}-\frac{1}{4}}\;\right] \end{equation*}

\begin{equation*} H^{2}={L_{0}}^{2}+2L_{0}L_{1}\!\left(\cos \theta _{2}-\cos \theta _{1}\right)+2{L_{1}}^{2}\left[1-\left(\cos \theta _{1}-\cos \theta _{2}\right)\right] \end{equation*}

$\theta _{1}$ is the sum of the angles $\beta$ and $\gamma$ as shown in Eq. (2), while the difference between the $\omega$ and $\sigma$ angles compute $\theta _{2}$ given by Eq. (3). The $\gamma$ and $\omega$ angles come from the real flexion-extension trajectory as analyzed in ref. [Reference Zapatero-Gutiérrez, Laribi, Castillo-Castañeda, Zeghloul, Laribi and Arsicault17] and is shown in Fig. 1. The five-bar mechanism point $C$ follows from $i$ point to $N$ point of the real trajectory expressed by D.

(2) \begin{equation}\theta _{1}=\beta +\gamma\end{equation}

(3) \begin{equation}\theta _{2}=\omega -\sigma\end{equation}

where

\begin{equation*} \gamma =atan\;2\!\left(D_{z},D_{x}\right) \end{equation*}

\begin{equation*} \omega =\pi -atan\;2\!\left(D_{z},L_{0}-D_{x}\right) \end{equation*}

To obtain the angles $\beta$ and $\sigma$ , defined by Eqs. (4) and (5), it is necessary to calculate the $\varphi$ and $\alpha$ angles. The $\varphi$ and $\alpha$ angles are obtained by applying the law of cosines and the trigonometric properties of sine and cosine [Reference He and Lu21].

(4) \begin{equation}\beta =atan\;2(L_{2}\sin \!\left(\pi -\varphi \right),L_{1}+L_{2}\cos \!\left(\pi -\varphi \right)\!)\end{equation}

(5) \begin{equation}\sigma =atan\;2(L_{3}\sin \!\left(\pi -\alpha \right),L_{4}+L_{3}\cos \!\left(\pi -\alpha \right)\!)\end{equation}

where

\begin{equation*} \varphi =atan\;2\!\left(\sin \varphi,\cos \varphi \right) \end{equation*}

with

\begin{equation*} \cos \varphi =\frac{{L_{1}}^{2}+{L_{2}}^{2}-\left({D_{x}}^{2}+{D_{z}}^{2}\right)}{2L_{1}L_{2}} \end{equation*}

\begin{equation*} \sin \varphi =B\sqrt{1-\cos \varphi ^{2}} \end{equation*}

and

\begin{equation*} \alpha =atan\;2\!\left(\sin \alpha,\cos \alpha \right) \end{equation*}

with

\begin{equation*} \cos \alpha \frac{{L_{3}}^{2}+{L_{4}}^{2}-\left[\left(L_{0}-D_{{x_{i:N}}}\right)^{2}+{D_{{z_{i:N}}}}^{2}\right]}{2L_{3}L_{4}} \end{equation*}

\begin{equation*} \sin \alpha =F\sqrt{1-\cos \alpha ^{2}} \end{equation*}

There are four possible configurations for a five-bar mechanism, depending on the configuration of its elbows $B$ and $F$ . The prototype presented in this document explores only one of the four configurations when elbows $B$ and $F$ are both up, and they are represented by a magnitude of positive one [Reference Zapatero-Gutiérrez, Laribi, Castillo-Castañeda, Zeghloul, Laribi and Arsicault17].

2.3. Five-bar optimal solution

The geometric parameters of the five-bar mechanism were obtained using an optimization algorithm based on the generation of the real flexion-extension trajectory (Fig. 1). An objective function, given by Eq. (6), is used to minimize the error function $\boldsymbol{{E}}(\boldsymbol{{I}})$ [Reference Zapatero-Gutiérrez, Laribi, Castillo-Castañeda, Zeghloul, Laribi and Arsicault17].

(6) \begin{equation}\boldsymbol{{E}}\!\left(\boldsymbol{{I}}\right)=\frac{1}{N}\sum _{i=1}^{N}\sqrt{({C_{{x_{i}}}}-{D_{{x_{i}}}})^{2}+({C_{{z_{i}}}}-{D_{{z_{i}}}})^{2}}\end{equation}

$\boldsymbol{{E}}(\boldsymbol{{I}})$ is defined as the sum of the squared root of the $i^{th}$ position of the desired trajectory and the coordinates of point $C$ , which represents the end-effector of the mechanism, and it is computed by Eq. (1). Through the gradient method, implemented in MATLAB with the fmincon function, the optimal parameters of the design vector $\boldsymbol{{I}}$ , as summarized in Table I, were found. The values of the input angles, as summarized in Table I, were computed using the Eqs. (2) and (3) considering the first point of the real flexion-extension trajectory, given by Fig. 1, and the last point of the real flexion-extension trajectory.

Table I. Design vector I values.

The gradient method, also known as non-linear programming, is used to find the minimum of a scalar function of several variables that begin with an initial estimate [22]. A maximum of 150 iterations with 2000 allowed evaluations of the function were chosen. The iterative process to minimize the objective function considers an upper and lower limit for each component of the design vector $\boldsymbol{{I}}$ . The design vector provides the length of the links and the initial position angles for $L_{1}$ and $L_{4}$ [Reference Zapatero-Gutiérrez, Laribi, Castillo-Castañeda, Zeghloul, Laribi and Arsicault17].

The optimization problem considers four constraint equations, Eqs. (7a), (7b), (7c), (7d), that ensures viable solutions for $H$ and guarantees that the position of the end-effector, described in Eq. (1), is within real numbers.

(7a) \begin{equation}R_{1}=\left| \frac{{L_{1}}^{2}+{L_{2}}^{2}-\left({D_{{x_{i:N}}}}^{2}+{D_{{z_{i:N}}}}^{2}\right)}{2L_{1}L_{2}}\right| \lt 1\end{equation}

(7b) \begin{equation}R_{2}=\left| \frac{{L_{3}}^{2}+{L_{4}}^{2}-\left[\left(L_{0}-{D_{{x_{i:N}}}}\right)^{2}+{D_{{z_{i:N}}}}^{2}\right]}{2L_{3}L_{4}}\right| \lt 1\end{equation}

(7c) \begin{equation}R_{3}=\left| H_{i}\right| \lt 1\end{equation}

(7d) \begin{equation}R_{4}=\left| \sqrt{\frac{{L_{2}}^{2}}{{H_{i}}^{2}}-\frac{1}{4}}\right| \lt 1\end{equation}

The constraint Eqs. (7a)–(7b) ensures always to obtain a viable solution for the coordinates of the point $C$ of the mechanism. These constraints are computed inside the optimization algorithm for every iteration which implies that the vector design not only provides the optimal dimensions of the mechanism but also ensures that these dimensions can pass for every point of the reference trajectory, avoiding singularities.

Especially, the constraint Eqs. (7c) and (7d) restrict the solution to ensure that only feasible values are taken; therefore, the imaginary part derived from $C_{x}$ and $C_{z}$ is zero. The algorithm only finds an optimal solution when the constraints are met. In this way, it is conditioned to the fact that the determinant of the Jacobian matrix is different from zero for the entire trajectory of the end-effector.

The design parameters obtained in Table I affect the mechanism’s performance since the dimensions of the links determine the position that the end-effector can reach, which is strictly related to the real flexion-extension trajectory. The set of points of the real trajectory with which the coordinates of the end-effector are computed define the workspace of the mechanism; it must be ensured that for each coordinate within the workspace, a singularity is not reached.

The input angle values were computed by evaluating the inverse kinematics and lengths of the links provided by the optimal solution.

3.1. Design and assembly

The prototype has two similar five-bar mechanisms that act in parallel. One of the mechanisms works as a master mechanism because it moves directly through the motors attached to its links $L_{1}$ and $L_{4}$ . The second mechanism acts as a slave because it follows the movements of the master mechanism through guide bars and a connecting bar located at the junction of links $L_{2}$ and $L_{3}$ , as shown in Fig. 3. This bar also has the connections to adjust the length of the thimbles (Fig. 4). A pair of brackets have been designed for the motors and the slave mechanism to be able to fix them to a base.

Figure 3. Rehabilitation prototype design.

Figure 4. Rehabilitation prototype design, view with thimbles.

3.2. Materials, components, and control

The links of the mechanism are 6 mm thick and were printed in 3D using polylactic acid. The connecting bar, between the master mechanism and the slave mechanism, is designed to rotate freely to adapt to the position of the finger and has a diameter of 7 mm. Four 25 mm protruding bars are distributed along the connecting bar, which, in turn, are assembled with extension bars that allows the position of the thimbles to be adjusted (Fig. 5).

Figure 5. Connecting bar and thimbles adjustment.

The prototype movement is executed using two 6V DC motors (Pololu, 34:1 Metal Gearmotor with 48 CPR Encoder) controlled via a RoboClaw 2x7A (Motion Control) control card. The control interface was carried out using LabView (National Instruments) running on a PC (Intel Core i5-6300HQ, 8.00 GB RAM, 2.30 GHz).

The LabView program (VI) controls the motor via a point-to-point position control. A set of fifth-order polynomial, given by the set of Eqs. (8a)–(8e), generates the trajectory in a period of ten seconds. From Eq. (8a), $i$ takes the value 1 for the motor M1 and 2 for the motor M2. The joint trajectory considers zero initial and final values for the velocities and accelerations of both motors, as well as two via points. Figure 6 shows the flowchart of the VI. The positions obtained through every encoder are received by an Action-Status module, which is integrated in the control card. The module compares the encoders’ positions with the computed positions by the polynomials. A pair of subVIs (LabVIEW subprograms) contains the position equations given by (8b) and (8f). There is one subVI for every motor due to the different coefficients of the polynomials of each one. The coefficients of the polynomials represent the inputs of the block, and the desired position is the out. For simplicity, the value of the coefficients was obtained by MATLAB and added to the subVIs as constants. The output changes depending on the elapsed time, programmed from 0 to 10 s. The baud rate is 9600 and a 100-ms data reading-writing delay.

Figure 6. Flowchart of the LabView programming.

Fifth-order polynomials were chosen for the trajectory in the joint space to ensure continuity in position, velocity, and acceleration [Reference Porawagama and Munasinghe23]. Equations (8e) and (8i) correspond to the rate of change of acceleration (jerk); computing the jerk can assure a smooth movement and allows to obtain the coefficients of the polynomials.

The design of the rehabilitation mechanism considers that the movement of the fingers starts from the flexed finger position to the extended finger position. In this article, a cycle is defined as the finger trajectory from the flexion position to the extended position and vice versa. In such a way, the trajectory begins and ends in the zero position as shown in Fig. 7.

Figure 7. A cycle of movement.

(8a) \begin{equation}\theta _{Mi}(t)=\begin{cases} \begin{array}{l} \theta _{1i}\!\left(t\right) \quad if\;0\leq t\leq t_{v1}\\[5pt] \theta _{2i}\!\left(t\right) \quad if\; t_{v1}\leq t\leq t_{f1} \\[5pt] \theta _{3i}\!\left(t\right) \quad if\; t_{f1}\leq t\leq t_{v2} \\[5pt] \theta _{4i}\!\left(t\right) \quad if\; t_{v2}\leq t\leq t_{f2} \end{array} \end{cases}\end{equation}

For $k \epsilon \left\{1,2\right\}$ :

(8b) \begin{equation}\theta _{ki}\!\left(t\right)=a_{0ki}+a_{1ki}t+a_{2ki}t^{2}+a_{3ki}t^{3}+a_{4ki}t^{4}+a_{5ki}t^{5}\end{equation}

(8c) \begin{equation}\dot{\theta }_{ki}\!\left(t\right)=a_{1ki}+2a_{2ki}t+3a_{3ki}t^{2}+4a_{4ki}t^{3}+5a_{5ki}t^{4}\end{equation}

(8d) \begin{equation}\ddot{\theta }_{ki}\!\left(t\right)=2a_{2ki}+6a_{3ki}t+12a_{4ki}t^{2}+20a_{5ki}t^{3}\end{equation}

(8e) \begin{equation}\dddot {\theta }_{ki}\!\left(t\right)=6a_{3ki}+24a_{4ki}t+60a_{5ki}t^{2}\end{equation}

and, where $k \epsilon \left\{3,4\right\}$ :

(8f) \begin{equation}\theta _{ki}\!\left(t\right)=b_{0ki}+b_{1ki}t+b_{2ki}t^{2}+b_{3ki}t^{3}+b_{4ki}t^{4}+b_{5ki}t^{5}\end{equation}

(8g) \begin{equation}\dot{\theta }_{ki}\!\left(t\right)=b_{1ki}+2b_{2ki}t+3b_{3ki}t^{2}+4b_{4ki}t^{3}+5b_{5ki}t^{4}\end{equation}

(8h) \begin{equation}\ddot{\theta }_{ki}\!\left(t\right)=2b_{2ki}+6b_{3ki}t+12b_{4ki}t^{2}+20b_{5ki}t^{3}\end{equation}

(8i) \begin{equation}\dddot {\theta }_{ki}\!\left(t\right)=6b_{3ki}+24b_{4ki}t+60b_{5ki}t^{2}\end{equation}

Considering that, for $k=1$ :

(8j) \begin{equation}\theta _{ki}\!\left(t_{v1}\right)=\theta _{(k+1)i}\!\left(t_{v1}\right)\end{equation}

(8k) \begin{equation}\dot{\theta }_{ki}\!\left(t_{v1}\right)=\dot{\theta }_{(k+1)i}\!\left(t_{v1}\right)\end{equation}

(8l) \begin{equation}\ddot{\theta }_{ki}\!\left(t_{v1}\right)=\ddot{\theta }_{(k+1)i}\!\left(t_{v1}\right)\end{equation}

and for $k=3$ :

(8m) \begin{equation}\theta _{ki}\!\left(t_{v2}\right)=\theta _{(k+1)i}\!\left(t_{v2}\right)\end{equation}

(8n) \begin{equation}\dot{\theta }_{ki}\!\left(t_{v2}\right)=\dot{\theta }_{(k+1)i}\!\left(t_{v2}\right)\end{equation}

(8o) \begin{equation}\ddot{\theta }_{ki}\!\left(t_{v2}\right)=\ddot{\theta }_{(k+1)i}\!\left(t_{v2}\right)\end{equation}

The Eq. (8b) evaluated in $k = 1$ provides the flexion (zero) position to the first via point. The Eq. (8b) evaluated in $k = 2$ provides the position from the first via point to the extension (final) position of the finger. $\theta _{1i}$ and $\theta _{2i}$ correspond to the half of a cycle.

Equation (8g) evaluated in $k = 3$ , expresses the movement from the extended position to the second via point. Finally, Eq. (8g) evaluated in $k = 4$ provides the position from the second via point to flexion position of the finger, completing a cycle.

The coefficients of the polynomials corresponding to the first half of the cycle ( $C_{fe}$ ) are calculated by multiplying the inverse matrix $Mz_{1i}$ that considers Eqs. (8b)–(8i) by the vector of defined conditions $q_{1i}$ , as shown in Eq. (9a).

(9a) \begin{equation}\left[C_{fe}\right]_{12\times 1}={\left[Mz_{1i}\right]^{-1}}_{12\times 12}\left[q_{1i}\right]_{12\times 1}\end{equation}

where

\begin{equation*} q_{1i}=\left[\begin{array}{c} \theta _{1i}\!\left(0\right)\\[5pt] \dot{\theta }_{1i}\!\left(0\right)\\[5pt] \ddot{\theta }_{1i}\!\left(0\right)\\[5pt] \dddot {\theta }_{1i}\!\left(0\right)\\[5pt] \theta _{1i}\!\left(t_{v1}\right)\\[5pt] \theta _{2i}\!\left(t_{v1}\right)\\[5pt] 0\\[5pt] 0\\[5pt] \theta _{2i}\!\left(t_{f1}\right)\\[5pt] \dot{\theta }_{2i}\!\left(t_{f1}\right)\\[5pt] \dot{\ddot{\theta }}_{2i}\!\left(t_{f1}\right)\\[5pt] \dddot {\theta }_{2i}\!\left(t_{f1}\right) \end{array}\right] \end{equation*}

The coefficients of the polynomials corresponding to the second half of the cycle ( $C_{ef}$ ) are calculated by multiplying the inverse matrix $Mz_{2i}$ that considers Eqs. (8j)–(8q) by the vector of defined conditions $q_{2i}$ , as shown in

Eq. (9b).

(9b) \begin{equation}\left[C_{ef}\right]_{12\times 1}={\left[Mz_{2i}\right]^{-1}}_{12\times 12}\left[q_{2i}\right]_{12\times 1}\end{equation}

where

\begin{equation*} q_{2i}=\left[\begin{array}{c} \theta _{3i}\!\left(t_{f1}\right)\\[5pt] \dot{\theta }_{3i}\!\left(t_{f1}\right)\\[5pt] \dot{\ddot{\theta }}_{3i}\!\left(t_{f1}\right)\\[5pt] \dddot {\theta }_{3i}\!\left(t_{f1}\right)\\[5pt] \theta _{3i}\!\left(t_{v2}\right)\\[5pt] \theta _{4i}\!\left(t_{v2}\right)\\[5pt] 0\\[5pt] 0\\[5pt] \theta _{4i}\!\left(t_{f2}\right)\\[5pt] \dot{\theta }_{4i}\!\left(t_{f2}\right)\\[5pt] \dot{\ddot{\theta }}_{4i}\!\left(t_{f2}\right)\\[5pt] \dddot {\theta }_{4i}\!\left(t_{f2}\right) \end{array}\right] \end{equation*}

An MPU-6050 IMU was placed on the connecting bar to measure the end-effector angle on the coordinate axes through an Arduino UNO. The connection is independent of the motor control program; it means that it does not close the control chain because it is used as an external sensor to verify the behavior of the connecting bar. The prototype instrumentation is shown in Fig. 8.

Figure 8. Mechanism instrumentation.

3.3. Experimental evaluation

To use the mechanism, the patient must sit in front of it and place his/her arm on the fixed forearm support. The arm is tied through a pair of straps which restricts wrist movement, as shown in Fig. 9. The fingers are tied to the thimbles in a flexed position. The trajectory along which the mechanism will guide the fingers begins with the flexed position and ends in the extended position.

Figure 9. (a) Finger flexed position. (b) Finger extended position.

The movement begins once the therapist initiates the movement cycle from a computer screen running the VI. If the therapist needs to stop the mechanism in any part of the trajectory, he/she can also do it from the VI. The hand position does not change while the mechanisms guide the finger through the desired trajectory. Figure 9 is only a representation; no hand experiments have been performed using the mechanism.

The flexion-extension trajectory is carried in ten seconds and represents one cycle of motion. The encoders were initialized to zero for 153° in the $L_{1}$ link connected to the $M_{2}$ motor, and 83° in the $L_{4}$ link connected to the $M_{1}$ motor, approximately, as shown in Fig. 10. The encoder resolution is 48.14 counts per turn at the output of the reducer. Table II indicates the relationship between the articular positions and the positions of the actuated links, at the initial, vias, and final points, obtained through inverse kinematics. The via points were computed for $t_{v1}=2.5\,{\rm s}$ and $t_{v2}=7.5\,{\rm s}$ . Figure 11 shows the profiles of the flexion-extension trajectory of both motors in radians.

Figure 10. L₁ and L₄ links orientation.

Figure 11. Joint movement profiles.

The first experimentation step was to decouple the actuated links from the underactuated links. This made it possible to ensure that the range of motion of $L_{1}$ link and $L_{4}$ link will be consistent with the angles described by Table II. Once the range of movement of the actuated links corresponded to the desired movement, $L_{3}$ link and $L_{4}$ link were placed.

At the second experimentation step, the IMU was directly coupled to the connecting bar to measures the rotation of the bar with respect to the x, y, and z axes during four cycles. The connecting bar rotates freely and adjusts to the position of the finger as the mechanism performs the movement.

The angles obtained by the rotation of the IMU in the x axis ( $\zeta _{x}$ ), the y axis ( $\zeta _{y}$ ), and the z axis ( $\zeta _{z}$ ) during the four cycles are shown in Fig. 12. The IMU angles are given in degrees and represent the rotational position of the connecting bar, which in turn positions the fingers from flexion to extension and vice versa. It is worth mentioning that the displacement carried out by the connecting bar involves both a rotational movement in y axis and a translational movement in the xz plane. For this reason, it is to be expected that the greatest dispersion is found in the y axis, while the x and z axes show little rotational dispersion. It was observed that the connecting bar presents a little friction when rotating almost to the end of the trajectory.

Table II. Position setting.

Figure 12. Comparison between the motor position and rotation angles obtained by the IMU.

The reason for using an IMU on the connector bar is because we are interested in measuring the rotation of the end-effector, and we suppose that this rotation could then be correlated with the rotation of the fingers and establish a pattern that helps measure the recovery of the patient. This assumption will be explored in future work.