2.1. Mechanism design and control system

The mechanism design of the $\underline{R}-2\underline{U}PS/RR$ ankle rehabilitation robot (size: 546 × 420 × 376 $\textrm{mm}^3$ ) is as in Fig. 1, which is a hybrid linkage mechanism composed of a rotating pair and a parallel mechanism in series. For the hybrid mechanism, the underlined letter denotes the actuated joint, and others denote passive joints. The mechanism consists of a series $\underline{R}$ mechanism and a $2\underline{U}PS/RR$ parallel mechanism, to ensure that the mechanism can meet the 3-DOFs of ankle rehabilitation training, namely DO/PL, IN/EV, and AD/AB, respectively. In addition, to ensure that patients of different sizes and different needs can be accommodated, we designed an angle adjusting mechanism to adjust the pitch angle of the mechanism.

Figure 1. Mechanism design of the $\underline{R}-2\underline{U}PS/RR$ ankle robotic system.

As for the control system, two servo linear actuators (linear actuator PC25PA042B03-0150XF1, Thomson, USA, along with DC Servo motor SMH40-1030D8ENL-2, Kinavo, China) with rated power 100 $W$ are employed as joints $\underline{P}_1$ and $\underline{P}_2$ to adjust the length of the kinematic chains with their stroke 150 $\textrm{mm}$ . An electric rotary table along with a DC Servo motor (SMH40-1030D10ENL-1, Kinavo, China) with rated power 100 $W$ is used to drive the horizontal motion, which is a series $R$ mechanism. The servo drives that we use for the three DC Servo motors are three DGFox DX060 drives (HDT, Italy), which have good interpolated position modes using the CANopen DS402 Protocol. A six-axis circular load cell (M3715C, SRI, China) is installed between the upper and lower parts of the moving platform using a data acquisition card (M8128B1, SRI, China), while three absolute rotary encoders (P3022-V1-CW360, China) are employed to measure the rotation angle of each DOF and to detect the moving platform’s trajectory in real time.

2.2. Inverse positions solution

The kinematic coordinate system of the $R-2UPS/RR$ ankle rehabilitation robot is as in Fig. 2. The $2UPS/RR$ is a 2-DOF parallel mechanism, which can realize the rotating movement of the moving platform around the rotating pair $R_2$ and $R_3$ by connecting with the rotating pair $R_1$ in series. The rotating pair $R_1$ is fixed on the frame of the mechanism, which can realize the rotation movement of the entire $2UPS/RR$ parallel mechanism around the rotating pair $R_1$ . Therefore, the mechanism has 3-DOFs and can achieve DO/PL, IN/EV, and AD/AB of the ankle joint.

Figure 2. Kinematic model and coordinate system of the mechanism. The blue dashed line box represents the reference directions of coordinates $\{ O \}$ and $\{ G2 \}$ .

The fixed coordinate system $\left \{ O \right \}$ of the mechanism is established at the intersection of the rotation axes of the rotating pairs $R_1$ and $R_2$ , and $x_0$ and $z_0$ coincide with the rotation axes of the rotating pairs $R_2$ and $R_1$ , respectively; the direction of $z_0$ is downward along the axis of rotation, and the direction of $y_0$ is determined by the right-hand coordinate criterion.

In the initial position, let the intersection of the rotation axes of the rotation pairs $R_1$ and $R_3$ be the point $K$ , and the distance from the point $K$ to the origin $O$ of the fixed coordinate system $\left \{ O \right \}$ at this moment is $p_0$ . The distance $p_0$ is the structure size of the organization and is a fixed value. However, the point $K$ is no longer located at the intersection of the rotation axes of the rotation pairs $R_1$ and $R_3$ since the rotation axes of the rotation pairs $R_1$ , $R_2$ , and $R_3$ of the mechanism do not converge at one point when the movable platform of the mechanism performs DO/PL movement. When the moving platform of the mechanism performs DO/PL and IN/EV, the motion trajectory of point $K$ is an arc with $O$ as the origin in the $y_0z_0$ plane and $p_0$ as the radius.

The moving coordinate system $\left \{ G_1 \right \}$ is established at the point $K$ , and the direction of $z_1$ is perpendicular to the moving platform downward. Initially, the directions of $x_1$ , $y_1$ , and $z_1$ coincide with the directions of $x_0$ , $y_0$ , and $z_0$ , respectively. Here, we constrain the rotation of the moving coordinate system $\left \{ G_1 \right \}$ around the $y_0$ and $z_0$ axes of the fixed system $\left \{ O \right \}$ , so that it can only rotate around the $x_0$ axis of the fixed system $\left \{ O \right \}$ . The moving coordinate system $\left \{ G_2 \right \}$ is also established at point $K$ , and $y_2$ coincides with the rotation axis of the rotating pair $R_3$ . Initially, the directions of $x_2$ , $y_2$ , and $z_2$ are consistent with the directions of $x_0$ , $y_0$ , and $z_0$ , respectively. Here, we constrain the rotation of the moving coordinate system $\left \{ G_2 \right \}$ around the $x_1$ and $z_1$ axes of the moving coordinate system $\left \{ G_1 \right \}$ , so that it can only realize the rotation around the $y_1$ axis of the $\left \{ G_1 \right \}$ .

The moving coordinate system $\left \{ G_1 \right \}$ and the moving coordinate system $\left \{ G_2 \right \}$ are both located on the moving platform, with their origins coinciding. At the initial position, the $\left \{ G_1 \right \}$ and $\left \{ G_2 \right \}$ are both consistent with the direction of the fixed coordinate system $\left \{ O \right \}$ .

Here, we define $\alpha$ as the rotation angle of the moving system $\left \{ G_1 \right \}$ around the $x_0$ axis of the fixed system $\left \{ O \right \}$ ; define $\beta$ as the rotation angle of the moving system $\left \{ G_2 \right \}$ around the $y_1$ axis of the moving system G1; and define $\gamma$ as the rotation angle of the entire $2\underline{U}PS/RR$ parallel mechanism around the rotation pair $R_1$ . Then, the rotation transformation matrix of the moving coordinate system $\left \{ G_1 \right \}$ relative to the fixed coordinate system $\left \{ O \right \}$ can be expressed as in (1).

(1) \begin{equation}{}_{{G_1}}^OR ={R_{{z_0}}}\left ({\rm{0}} \right ) \cdot{R_{{y_0}}}\left ( 0 \right ) \cdot{R_{{x_0}}}\left ( \alpha \right ) \end{equation}

The position vector of the moving coordinate system $\left \{ G_1 \right \}$ relative to the fixed coordinate system $\left \{ O \right \}$ can be expressed as in (2).

(2) \begin{equation}{}_{{G_1}}^OP ={\left [{\begin{array}{*{20}{c}} 0&{-{p_0}\sin \alpha }&\quad {{p_0}\cos \alpha } \end{array}} \right ]^{\rm{T}}} \end{equation}

The rotation transformation matrix of the moving coordinate system $\left \{ G_2 \right \}$ relative to the moving coordinate system $\left \{ G_1 \right \}$ can be expressed as in (3).

(3) \begin{equation}{}_{{G_2}}^{{G_1}}R ={R_{{z_1}}}\left ({\rm{0}} \right ) \cdot{R_{{y_1}}}\left ( \beta \right ) \cdot{R_{{x_1}}}\left ( 0 \right ) \end{equation}

The position vector of the moving coordinate system $\left \{ G_2 \right \}$ relative to the moving coordinate system $\left \{ G_1 \right \}$ can be expressed as in (4).

(4) \begin{equation}{}_{{G_2}}^{{G_1}}P ={\left [{\begin{array}{c@{\quad}c@{\quad}c} 0&0&0 \end{array}} \right ]^{\rm{T}}} \end{equation}

Then, the coordinate transformation matrix of the moving coordinate system $\left \{ G_2 \right \}$ relative to the fixed coordinate system $\left \{ O \right \}$ is as in (5), where ${}_{{G_2}}^OR ={}_{{G_1}}^OR{}_{{G_2}}^{{G_1}}R$ , and ${}_{{G_2}}^OP{\rm{ = }}{}_{{G_1}}^OP +{}_{{G_2}}^{{G_1}}P$ .

(5) \begin{equation}{}_{{G_2}}^OT = \left [{\begin{array}{c@{\quad}c}{{}_{{G_2}}^OR}&{{}_{{G_2}}^OP}\\ \\[-8pt]0&1 \end{array}} \right ] \end{equation}

For the reason that the $S_1$ and $S_2$ are the spherical joint centers on the moving platform, and $U_1$ and $U_2$ are the universal joint centers on the fixed base, then we can get the (6) based on the closed vector loop chain in Fig. 3.

(6) \begin{equation} \left |{{l_i}} \right | ={S_i} -{U_i}\begin{array}{*{20}{c}}{}&{} \end{array}(i = 1,2) \end{equation}

where ${S_i}{\rm{ = }}{}_{{G_2}}^OR{S_i}^\prime +{}_{{G_2}}^OP$ , and ${S_i}^\prime$ is the position vector of the spherical joint center $S_i$ in the moving coordinate system $\left \{ G_2 \right \}$ . According to the designed mechanism, we can get (7).

(7) \begin{equation}{\theta _1} = \gamma \end{equation}

By solving (6) and (7), the stroke of the two servo linear actuator $l_1$ , $l_2$ , and the rotation angle $\theta _1$ of the electric rotary table can be obtained.

Figure 3. The closed vector loop chain.

3.1. Calculation optimization of inverse solution of positions

Traditional inverse solution of positions requires matrix operations, which can consume a large number of CPU. To reduce the computational complexity, we use the linear least squares method to fit the multidimensional data of the inverse solution result. As the rotation angle $\theta _1$ of the electric rotary table is separate, we only need to fit the multidimensional data of the inverse solution of positions about the strokes of the two servo linear actuators $l_1$ , $l_2$ .

Define the general equation for the space curved surface as in (8). And, define the fitting equation as in (9).

(8) \begin{equation} l ={a_{00}} +{a_{10}}\theta _1 +{a_{01}}\theta _2 +{a_{20}}{{\theta _1}^2} +{a_{11}}{\theta _1}{\theta _2} +{a_{02}}{{\theta _2}^2} \end{equation}

(9) \begin{equation} \left \{ \begin{aligned} &f(\theta _1,\theta _2) ={a_{00}} +{a_{10}}\theta _1 +{a_{01}}\theta _2 +{a_{20}}{{\theta _1}^2} +{a_{11}}{\theta _1}{\theta _2} +{a_{02}}{{\theta _2}^2} \\ \\[-9pt]&f(\theta _1,\theta _2) = 0 \\ \end{aligned} \right. \end{equation}

According to the definition of the least square method, the total fitting error $V$ should be kept to a minimum.

(10) \begin{equation} \mathop V\limits _{\min } = f(\theta _1,\theta _2) - l \end{equation}

(10) is a multi-component function, and the necessary condition for the existence of its minimum value is that its corresponding partial derivative is zero as in (11). The equation provides a unique solution. Thus, the fitting function is determined.

(11) \begin{equation} \frac{{\partial V}}{{\partial{a_{00}}}}{\text{ = }}\frac{{\partial V}}{{\partial{a_{01}}}}{\text{ = }}\frac{{\partial V}}{{\partial{a_{10}}}}{\text{ = }}\frac{{\partial V}}{{\partial{a_{11}}}}{\text{ = }}\frac{{\partial V}}{{\partial{a_{02}}}}{\text{ = }}\frac{{\partial V}}{{\partial{a_{20}}}}{\text{ = }}0 \end{equation}

The surface fitting results of the two servo linear actuators based on their inverse solution of positions are as in Fig. 4, from which we can see that the result of the fitting is a saddle surface, which is smooth and the errors are within an acceptable range. The values of unknown parameters in the (9) are shown in Table I.

Table I. Parameter fitting value of the linear actuators.

Figure 4. Fitting results of the linear actuators’ inverse solution of positions.

3.2. Single-axis and multi-axis passive exercise

In order to carry out patient-passive rehabilitation training for the AJC in an all-round way, we carried out single-axis passive rehabilitation training and multi-axis rehabilitation training based on the inverse solution of positions, respectively. The single-axis passive exercise is the reciprocating rehabilitation training in one of the DO/PL, IN/EV, or AD/AB directions. The AD/AB movement is driven by the electric rotary table, while the DO/PL and IN/EV are driven through the two servo linear actuators. The multi-axis rehabilitation training is to better train the flexibility and ROM of the AJC, and based on previous studies [Reference Dong, Fan, Li, Zhou, Rong, Kong and Zhou20, Reference Washabaugh, Guo, Chang, Remy and Krishnan30, Reference Ren, Wu, Yang, Xu, Harvey and Zhang31], spatial trajectories they proposed contain straight lines and arcs with different slopes, which can test the performance of the proposed robot and control algorithm. For this purpose, we developed six kinds of spatial trajectories, that is, circle, triangle, spatial straight line, semicircle, semicircle ring, and rectangle, which are all realized based on the inverse solution of positions. The passive exercise strategy in single-axis and multi-axis is as in Fig. 5.

Figure 5. Patient-passive strategy in single-axis and multi-axis.

The linear position interpolation of the single servo drives is shown in Fig 6. During the set time period $ \Delta T$ , the driver follows point-to-point linear interpolation. Passive rehabilitation training is carried out by using the linear position interpolation mode of the DGFox DX060 servo drives, and the interpolation cycle is $8\,\textrm{ms}$ . The upper computer sends control commands to the lower computer, which receives the control commands and controls the movement of the servo motors based on the CANopen DS402 protocol.

Figure 6. Linear position interpolation.

5.1. Experiments of the patient-passive exercise

For the reason that the AD/AB movement is independent and decoupled from motion in the other two directions, driven by the electric rotary table, and also the patient-passive exercises in DO/PL and IN/EV are just realized through different combination drive forms of the two servo linear actuators, we only tested the single-axis patient-passive exercise in DO/PL direction, while for the multi-axis patient-passive exercise, the trajectories of circle, triangle, spatial straight line, semicircle, semicircle ring, and rectangle were all tested with healthy subjects. It should be noted that in order to facilitate the drawing of the spatial curve, we only generate the predetermined trajectories through the combined motion around the $x-axis$ and the $z-axis$ . And it is important to note that, for the toe of the AJC, the trajectory of the movement around the $x-axis$ and the movement around the $z-axis$ in space is on a spherical surface, for the reason that the $x-axis$ and the $z-axis$ intersect at one point.

5.1.1. Passive exercise in DO/PL direction

The trajectories of the rehabilitation training are predefined, and reciprocating movement is performed within the AJC’s ROM. Experimental result of patient-passive exercise in DO/PL from one subject is as in Fig. 8, where the sampling frequency is $10\,\textrm{Hz}$ , the desired value is the predefined trajectory while the actual value is the actual trajectory. It should be noted that, unlike most passive rehabilitation training that uses sine or cosine functions as the predefined trajectories, we use a passive training trajectory with constant velocity characteristics to improve the effectiveness of rehabilitation training; the speed remains constant in most of the time, while the speed becomes $10$ % of the original value when there is $2^{\circ }$ left before the movement reaches the threshold.

Figure 8. Patient-passive exercise in DO/PL direction.

From the results we can see that the actual trajectories of the patient-passive exercise in DO/PL direction are almost identical to the predefined trajectories, showing good trajectory tracking performance.

5.1.2. Circle trajectory

The circle trajectory is through the combined motion around the $x-axis$ and the $z-axis$ , with the DO/PL and AD/AB all range from $ -{18^{\circ } } \sim{18^{\circ }}$ . The result of one subject is as in Fig. 9, where the left figure is the plane representation of the experimental result while the right one is its spatial representation on the spherical surface with the unit of the coordinate system radian. The desired value is the predefined trajectory while the actual value is the actual trajectory.

Figure 9. Circle trajectory tracking performance.

From Fig. 9 we can see that the robot can drive the AJC along with the predefined trajectory with very high precision, showing very good trajectory tracking performance.

5.1.3. Triangle trajectory

The triangle trajectory is also through the combined motion around the $x-axis$ and the $z-axis$ , with the DO/PL ranges from $ -{18^{\circ } } \sim{18^{\circ }}$ while the AD/AB ranges from $ -{20^{\circ } } \sim{20^{\circ }}$ . Result of one subject including both the plane representation and spatial representation on the spherical surface is shown as in Fig. 10.

Figure 10. Triangle trajectory tracking performance.

5.1.4. Spatial straight line trajectory

The spatial straight line trajectory is also through the combined motion around the $x-axis$ and the $z-axis$ , with the DO/PL ranges from $ -{18^{\circ } } \sim{18^{\circ }}$ while the AD/AB ranges from $ -{20^{\circ } } \sim{20^{\circ }}$ . Result of one subject including both the plane representation and spatial representation on the spherical surface is shown as in Fig. 11.

Figure 11. Spatial straight line trajectory tracking performance.

5.1.5. Semicircle trajectory

The semicircle trajectory is also through the combined motion around the $x-axis$ and the $z-axis$ , with the DO/PL ranges from ${0^{\circ } } \sim{18^{\circ }}$ while the AD/AB ranges from $ -{18^{\circ } } \sim{18^{\circ }}$ . The radius of the semicircle is $18^{\circ }$ . Result of one subject including both the plane and spatial representation is shown as in Fig. 12.

Figure 12. Semicircle trajectory tracking performance.

5.1.6. Semicircle ring trajectory

The semicircle ring trajectory is also through the combined motion around the $x-axis$ and the $z-axis$ , with the DO/PL ranges from ${0^{\circ } } \sim{18^{\circ }}$ while the AD/AB ranges from $ -{18^{\circ } } \sim{18^{\circ }}$ . The radius of the inner ring and the outer ring is $9^{\circ }$ and $18^{\circ }$ , respectively. Result of one subject including both the plane representation and spatial representation on the spherical surface is shown as in Fig. 13.

Figure 13. Semicircle ring trajectory tracking performance.

5.1.7. Rectangle trajectory

The rectangle trajectory is also through the combined motion around the $x-axis$ and the $z-axis$ , with the DO/PL ranges from $ -{18^{\circ } } \sim{18^{\circ }}$ while the AD/AB ranges from $ -{20^{\circ } } \sim{20^{\circ }}$ . Result of one subject including both the plane representation and spatial representation on the spherical surface is shown as in Fig. 14.

Figure 14. Rectangle trajectory tracking performance.

5.1.8. Results of the patient-passive exercise experiments

From Figs. 8, 9, 10, 11, 12, 13, and 14 we can see that the ankle rehabilitation robot basically moves according to the predetermined trajectories in space on a spherical surface. To further quantitatively evaluate the performance of the developed patient-passive exercise, we calculated the mean deviation (MD) and root-mean-square deviation (RMSD) of the trajectory tracking results between the predicted trajectory and the actual trajectory in all the experiments, where $MD = \frac{{\sum \limits _{{\rm{i}} = 1}^n{\left |{{\theta _p} -{\theta _a}} \right |} }}{n}$ , $RMSD{\rm{ = }}\sqrt{\frac{{\sum \limits _{{\rm{i}} = 1}^n{{{\left ({{\theta _p} -{\theta _a}} \right )}^2}} }}{n}}$ , ${\theta _a}$ and ${\theta _p}$ represent the actual and desired trajectories, respectively. The results is as in Table II, from which we can see that although the errors in the separate $x-axis$ and $z-axis$ directions are visually large, the error after space vector synthesis is not large compared with our previous work [Reference Li, Fan, Dong and Rong22], reflecting better trajectory tracking performance.

Table II. Evaluation of results in patient-passive exercise experiments.

MD, mean deviation; RMSD, root mean square deviation.

5.2. Experiments of the patient-active exercise

For the reason that the amount of IN/EV movement is very small, and it is rarely used in normal times and can basically be ignored, during the patient-active exercise, we also try to use only the combined motion around the $x-axis$ and the $z-axis$ , to give a better display on a spherical surface. During the experiments, we constrained the movement in the direction of IN/EV, with the experimental results of two patient-active exercises are as in Figs. 15 and 16, respectively, where the predicted value is deduced based on the patient-active exercise strategy in (13) while the actual value is the actual trajectory of the robot.

Figure 15. #1 experiment of patient-active exercise.

Figure 16. #2 experiment of patient-active exercise.

From Fig. 15 and 16, we can see that the actual trajectories can basically move according to the deduced trajectories obtained from (13) in space on a spherical surface. To quantitatively evaluate the performance of the developed patient-active exercise, we also calculated the mean deviation (MD) and root-mean-square deviation (RMSD) of the trajectory tracking results between the predicted trajectory and the actual trajectory in all the experiments, with the results shown in Table III.

Table III. Evaluation of results in patient-active exercise experiments.

From Table III, we can see that the error after space vector synthesis is also very small. What needs special explanation is that the position reaching the limit $ \pm{20 ^{\circ } }$ shown in Fig. 16 is the effect of the software limit we set, which is to protect the AJC’s safety during rehabilitation training.