An ankle rehabilitation robot based on 3-RRS spherical parallel mechanism

This article presents the design modeling of a novel 3-RRS spherical parallel mechanism for ankle rehabilitation applications. The kinematics of the 3-RRS spherical parallel mechanism is established. The degree of freedom of 3-RRS spherical parallel mechanism is calculated using screw theory. The inverse kinematics of 3-RRS spherical parallel mechanism is solved. Eight groups of inverse solutions of 3-RRS spherical parallel mechanism are obtained. A method for forward position analysis is developed with variation and iteration approaches, which is suitable for motor position control. The ankle rehabilitation robot can be widely used in clinical treatment and can also be used at home, hotels, and fitness centers for ankle muscle relaxation.


Introduction
Ankle sprain is one of the common sport injuries, particularly in complex movement such as in the ball games, where there are sudden changes in direction, or urgent start and stop bring in the problems. 1 If the injuries are not treated timely or thoroughly, it can lead to ankle ligament laxity and joint instability and cause repeated injuries. In the worst cases, it can lead to more serious problems such as ankle dysfunctions.
The development of robot technology makes possible to perform ankle rehabilitation effectively with a robotic device. An ankle rehabilitation robot can not only reduce the labor intensity of the doctor but also can quantitatively evaluate patient rehabilitation condition. The robots make them possible for regularly, moderately rehabilitation exercises.
Some ankle rehabilitation robots in experimental study or the market products have been reported in the literature. A 2-RRR/UPRR (R-revolute joint, U-universal joint, P-prismatic joint) robot mechanism for ankle rehabilitation is presented by Bian et al., 2 where the mechanism can realize three rotations around the remote center and satisfy the requirements of ankle rehabilitation motions. Wang et al. 3 presented a novel parallel ankle rehabilitation robot and achieved the kinematic solution and simulation analysis. A parallel robot for ankle rehabilitation was reported by Shah and Basah. 4 Hou et al. 5 proposed a concept of ankle rehabilitation mechanism, which can train the movements of flexion, inversion, and eversion of ankle. A decoupled parallel mechanism designed for ankle rehabilitation is presented by Zeng et al.; 6 the proposed mechanism can realize independent rotational rehabilitation under the traction simultaneously. Other parallel ankle rehabilitation robots can be found in Jamwal et al., 7 Dai, et al., 8 Girone, et al., 9 Zhao, et al., 10 among others.
It can be noted that the existing ankle rehabilitation robots have some limitations. First, most robots have only 1 or 2 degrees of freedom (DOFs) and cannot be used for comprehensive rehabilitations in all degrees of angle motions. Second, for most robots, their rotation centers can easily shift away from the rotation center of human ankle. Such a shift can cause uncomfortable feeling in the patients, or in the worst case cause secondary injury of ankle joint. An ankle rehabilitation robot with simple structure, which can fully meet the ankle structure characteristics and realize ankle rehabilitation motion, is needed.
In this work, we propose a spherical parallel manipulator for ankle rehabilitations. The robot is based on a 3-revolute-revolute-spherical joint's (RRS) spherical parallel mechanism (SPM), which generates rotation analogy to human ankles.

The motion of human ankle
Prior to the design of the SPM, we looked at the biomechanics of human ankle motion to determine robot specifications properly.

Movement of human ankle joint
As shown in Figure 1, ankle joint has three kinds of movements: dorsiflexion and plantar flexion, inversion and eversion, adduction and abduction.
The ranges of motion of ankle joint are shown in Table 1.

The motion curve of foot relative to ankle
Based on the anthropometric measurements, 14 the distance between the moving center of the ankle and the bottom of the foot is generally 60-100 mm long. The size of the foot is 210-270 mm long for 96% female over the age of 14 and 240-285 mm long for 96% male over the age of 15.
In Figure 2, the distance between the moving center O of the ankle and the bottom of the foot B takes as 80 mm. The distance between the moving center O of the ankle and the front of foot A takes a value of 240 mm. Taking this foot size as an example, we analyze the motion curve of the foot relative to ankle.
The moving center O of the ankle is as the coordinate origin. The motion curve of the front of foot A relative to moving center O of the ankle based on the coordinate system in Figure 1 is shown in Figure 3.

The 3-RRS SPM for ankle rehabilitation
The conceptual design of the 3-RRS rehabilitation robot is shown in Figure 4, which includes the base, the moving platform, support, motor, crank, coupler, foot support, expansion device, and so on. Three motors drive the cranks to generate the rotations of the moving platform around the center of the sphere.
The adjustable length of the expansion device is 50 mm, which can be applied to most patients. Compared with the existing ankle rehabilitation robots, the ankle rehabilitation robot based on 3-RRS SPM mainly shows some unique features as follows:  1. The mechanism can implement 3-DOF rotation to be completely compatible for human ankle motion. 2. The mechanism is adjustable and controllable by adjusting the telescopic rod in the expansion device to adapt to the different sizes of feet. 3. The location of the spherical center O of 3-RRS SPM remains during movement. On the other hand, the center position can be adjusted according to the distance between ankle movement center and the bottom of foot adjusting the telescopic rod of the mechanism, to make the ankle movement center coincide with the spherical center. 4. The mechanism has a large range of motion in its three axes. This allows the magnitude of rotation adjustable to satisfy the requirements of different rehabilitations.

Kinematics of 3-RRS SPM
Kinematic parameters and coordinate systems of 3-RRS SPM Figure 5 shows the kinematics model of 3-RRS SPM. It is composed of the base, the crank, the coupler, and the moving platform. It has three limbs with identical structures.
In each limb, there are two revolute pairs R and one spherical pair S. The point O represents spherical center. The point P represents the geometrical center of the moving platform. The point G represents the geometrical center of the base. A i , B i , and C i represent the center of the kinematic pair in limb i (i = 1, 2, 3), respectively.
The structure parameters of 3-RRS SPM are shown in Table 2. The cone angles u 1 and u 2 of the moving platform and the base are 54.74°. Three axes OA 1 , OA 2 , and OA 3 of the revolute joints connecting the cranks and the base are perpendicular to each other in space and intersect at the spherical center. Three lines OC 1 , OC 2 , and OC 3 between the centers C i (i = 1, 2, 3) of the spherical pairs and the spherical center O are perpendicular to each other in space. The fixed coordinate system O-XYZ is established at the base. The coordinate origin O is located at the spherical center. The OX, OY, and OZ axes coincide with OA 1 , OA 2 , and OA 3 , respectively.  The moving coordinate system m-xyz is attached to the moving platform. The coordinate origin m is also located at the spherical center O. The mz, mx, and my axes coincide with OC 1 , OC 2 , and OC 3 , respectively.
The kinematic modeling of 3-RRS SPMs is welldocumented based on constraint equations of spherical linkages. 15,16 In this work, we develop their model with a different approach.
The DOF calculation of 3-RRS SPM Figure 6 shows one limb of the 3-RRS SPM. One spherical pair can be seen as three revolute pairs that are concurrent but not coplanar. The axes of six revolute pairs in three limbs intersect at the spherical center O of 3-RRS SPM. Based on the principle that the inverse proportion of screw is independent to the choice of coordinate system, taking limb 1 as an example, the kinematic screw system of limb 1 in the m-xyz system is In limb 1, 6 S 11 , 6 S 12 , and 6 S 13 in different planes intersect one point in space. 6 S 14 , 6 S 15 , and 6 S 13 are perpendicular to each other in space. Thus, 6 S 11 , 6 S 12 , 6 S 13 , 6 S 14 , and 6 S 15 are linearly independent. The reciprocal basis screw in limb 1 is as follows According to the relationship between force screw and kinematics screw, limb 1 is subjected to the action of three forces from the axes X, Y, and Z, not subjected to the action of the couple.
From the structure in Figure 5, the spherical pair S in each limb degenerates into the revolute pair R. Thus, there are two passive DOFs in each limb. The DOF of 3-RRS SPM is calculated as follows using the modified Gru¨bler-Kutzbach equation where d is the order of the mechanism, n is the link numbers including the frame, g is the number of kinematic pairs, f i is the DOF of ith kinematic pair, y is the number of redundant constraints, and z is the number of passive DOFs.

The forward kinematic analysis of 3-RRS SPM
The forward position solution of 3-RRS SPM is to solve the orientation of the moving platform (a, b, g) for the given three driving angles (u 1 , u 2 , u 3 ). As shown in Figure 5, the position of each kinematic pair axis is represented by unit vector u i , v i , w i in limb i (i = 1, 2, 3), respectively. The direction of each kinematic pair axis is pointing from the spherical center O to the corresponding kinematic pair center.
In the 3-RRS SPM, the unit vectors u 1 , u 2 , u 3 coincide with OX, OY, and OZ in the base coordinate system O-XYZ, respectively. The unit vectors w 1 , w 2 , w 3 coincide with mx, my, and mz in the moving coordinate system m-xyz, respectively. v 1 , v 2 , v 3 coincide with OB 1 , OB 2 , and OB 3 , respectively.
When Z-Y-X Euler angles (a, b, g) are used to represent the orientation of the moving coordinate system m-xyz relative to the base coordinate system O-XYZ, the transformation matrix R m b of m-xyz system relative  to O-XYZ system, based on the exponential product and the Rodrigues' formula, can be expressed as Equation (4)  Here and hereafter c = cos and s = sin. When 3-RRS SPM is in the initial position, the coordinates of the unit vectors u Suppose the center of three spherical pairs C i (i = 1, 2, 3) connected the coupler and the moving platforms are all located on the unit sphere, the coordinate of C i (i = 1, 2, 3) in the O-XYZ system, based on the exponential product and the Rodrigues' formula, can be expressed as where h i (i = 1, 2, 3) is the angle between links A i B i and B i C i , called passive angle In the 3-RRS SPM, the following equation always holds Differentiating and simplifying equation (7) yield From equation (6), the following expression can be obtained Let the distance between point C 1 and point C 2 be l 12 , then For the other two limbs C 1 C 3 and C 2 C 3 , there exists the same relationship From equations (10) and (11), the following expression can be obtained , Cone angle of the moving platform :C i OP u 2 Cone angles of the base :A i OG r Radius of the curved link A i B i r b Radius of the curved link B i C i r c Distance OC i between the spherical center O and the spherical joint center C i h 1 Thickness of the crank and the coupler h 2 Height of the small boss on the crank and the coupler l s Distance between the spherical joint center and the coupler boss; distance between the spherical joint center and the moving platform The matrix J is the Jacobian matrix of 3-RRS SPM. The Jacobian matrix J expresses the mapping relationship between the displacement of the moving platform and the driving angles of 3-RRS SPM when the passive angles h i (i = 1, 2, 3) are used as driving angles.
From equation (12), dh i = J À1 dl, the iteration equation of h = ½ h 1 h 2 h 3 can be obtained where k is the iteration time.
After the passive angles h i (i = 1, 2, 3) are obtained using Newton iteration method, the rotation matrix R m b can be obtained using Rodrigues' exponential product equation. Furthermore, the coordinates p O of center P of the moving platform in the O-XYZ system are obtained as follows The inverse kinematic analysis of 3-RRS SPM The inverse position solution is to solve three driving angles (u 1 , u 2 , u 3 ) for a given orientation (a, b, g) of the moving platform. The coordinates of the unit vector w i (i = 1, 2, 3) in the m-xyz system are w 1m w 2m w 3m The vector w im (i = 1, 2, 3) can be transformed to the base system O-XYZ using transformation matrix R m When the angles a, b, and g are known in the transformation matrix R m b , the unit vectors v i and vectors w i are connected by the coupler in each limb. The dimensional angle of link B i C i is d 2 . This relationship can be used to establish the constraint equation as follows that is The constraint equations of 3-RRS SPM are as follows cd 2 = cd 1 cacb À sd 1 sbsu 1 + sd 1 sacbcu 1 cd 2 = (sasbsg + cacg)cd 1 + (casbsg À sacg)sd 1 su 2 + cbsgsd 1 cu 2 cd 2 = cd 1 cbcg + (sasbcg À casg)sd 1 su 3 + (casbcg + sasg)sd 1 cu 3 Equation (19) yields where , Equation (20) is a first-order equation about sin u i and cos u i .
Substituting it into equation (20) yields Thus Therefore, the inverse position solution of 3-RRS SPM is obtained as follows u i has two solutions for limb i. This shows that the inverse position solutions have eight sets of solutions for a given position and orientation of 3-RRS SPM.

Case study
The structure parameters of 3-RRS SPM are shown in Table 3. Figure 7 is the simulation model of 3-RRS SPM.
The moving platform and the base of 3-RRS SPM are parallel to each other when 3-RRS SPM is located in the initial position. When the driving angles u 1 = À308, u 2 = À458, and u 3 = À458, the forward kinematics leads to h = ½ 0:1 0:2 0:4 , the transformation matrix R m b is further obtained The values of the driving angles can be obtained as shown in Table 4 using inverse position solution equations and substituting the structure parameters in Table  3 and the value of transformation matrix R m b in equation (24). To sum up, the driving angles (u 1 , u 2 , u 3 ) have eight kinds of combination for a given position and orientation of the moving platform.
Observing Table 4, the second sets of the inverse position solutions, that is, u 1 = À30:028, u 2 = À44:998, and u 3 = À44:998, are the same as the driving angles u 1 = À308, u 2 = À458, and u 3 = À458 within the range of acceptable errors. This verifies the inverse position solutions of 3-RRS SPM.

Conclusion
This article proposes a new ankle rehabilitation robot based on 3-RRS SPM, which accords with the topology structure of ankle. 3-RRS SPM can move along a spherical trajectory in the space, and therefore, it can realize the motion of ankle.
The kinematics of 3-RRS SPM is analyzed. Eight sets of inverse kinematics are obtained. The case verifies the inverse position solutions of 3-RRS SPM.
The ankle rehabilitation robot has a high potential for rehabilitation applications; it can be widely used in clinical treatment and can also be used at home, hotels, and fitness centers for ankle muscle relaxation.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.