Configuration Design and Kinematic Performance Analysis of a Novel 4-DOF Parallel Ankle Rehabilitation Mechanism with Two Virtual Motion Centers

The ankle joint is a complex joint of the lower limbs, which is closely linked by ligaments and muscles, and plays a role in maintaining human body balance and providing power for walking in daily life [1]. The ankle joint is also one of the most vulnerable joints in the human body, there are two main causes for its injury: one is the joint sprain caused by external force in daily life, and the other is the ankle function injury caused by neurological diseases, such as stroke, spinal cord injury and other diseases [2, 3]. There are about 15 million people worldwide who suffer from stroke every year [4], including 795000 stroke patients in the United States every year. According to the prediction by the American Heart Association, the number of stroke patients will rise to 3.4 million by 2030 [5].

Traditional ankle rehabilitation therapy relies on manual assistance from doctors to help patients perform dorsiflexion/plantar flexion, inversion/eversion, and other movements. The goal is to gradually stimulate and repair the damaged central nervous system of the ankle. However, this rehabilitation process is time-consuming and repetitive, placing a significant burden on therapists [6]. Due to the imbalance in the doctor-patient ratio and the limitation of therapy time and resources, it is difficult for traditional ankle rehabilitation to ensure sufficient therapy frequency and intensity [7]. Therefore, the development of an ankle rehabilitation robot to replace traditional artificial rehabilitation therapy meets the market demand and is of great significance.

The series mechanism has limitations due to the motor's inability to be completely fixed, increased motion inertia, and difficulty in fitting within a smaller ankle space. In contrast, the parallel mechanism (PMs) has the advantages of smaller accumulative error, higher precision, larger loading capacity, easy closed-loop control, and stable motion [8]. Therefore, many scholars have extensively researched and designed ankle rehabilitation robots based on parallel mechanisms. Girone et al. [9] developed a 6-DOF ankle rehabilitation robot based on the Stewart platform, which can simulate the true motion of the ankle joint. However, it has drawbacks like redundant degrees of freedom, complex control, high cost, limited rotation space, and large space occupation. To address these issues, some scholars simplified the ankle joint as a 2-DOF universal (U) joint or a 3-DOF spherical (S) joint. They designed 2-DOF or 3-DOF parallel ankle rehabilitation robots with the unconstrained branch as the driving branch and the universal or spherical joint as the constraint branch. For example, Dai et al. [10] developed a high-performance 3-UPS/U ankle rehabilitation robot with two rotational degrees of freedom, which can achieve ankle dorsiflexion, plantar flexion, inversion, and eversion movement. Similarly, there are 3-SPS/S [11] and 3-RSS/S [12] ankle rehabilitation robots with three rotational DOFs. The ankle rehabilitation mechanism has the advantages of simple structure, good kinematic performance, small space occupation and relatively low cost. However, these rehabilitation robots have their limitations. The motion center of these robots does not align with the center of the human ankle joint, which can potentially cause secondary injuries. Additionally, the rotation space is constrained by the middle spherical joint’s rotation angle.

In order to solve the problem that the rotation center of the robot does not match the center of the human ankle joint, some scholars replace U or S joints with RR or RRR branches, such as Fang et al. [13] proposed a 3-RUS/RRR parallel mechanism and designed it as an ankle rehabilitation robot. Li et al. [14] proposed a 2-UPS/RRR ankle rehabilitation robot with three rotational DOFs, which has good kinematic performance and can complete active and passive rehabilitation therapy. Some scholars have designed a series of ankle rehabilitation mechanisms [15, 16] with the remote center of motion (RCM) [17], ensuring alignment between the human ankle center and the virtual rotation center of the mechanism. For example, Malosio et al. [18] designed an ankle rehabilitation robot based on the 3-RRR spherical robot (Agile eye). This robot features a relatively complex structure and an RCM, offering a favorable rehabilitation treatment effect on the ankle as the ankle center remains close to the motion center of the mechanism.

The ankle joint consists of the tibiotalar joint and subtalar joint, which have non-intersecting axes and a spatial interfacial relationship. However, the aforementioned scholars consider the ankle joint as an S joint or U joint with intersecting rotation axes, which deviates from the anatomical structure of the human ankle joint. This mismatch leads to a disparity between the actual motion of the rehabilitation robot and the ankle joint, resulting in poor coordination between human and machine motion. Moreover, the robot may exert non-DOF directional force on the human body, potentially causing secondary damage and negatively impacting rehabilitation outcomes and human comfort. Additionally, the differences in physiological structures among patients pose a challenge. The left and right feet have distinct anatomical characteristics, with opposite inclinations of the ankle axis. As a result, most ankle rehabilitation robots cannot accommodate the rehabilitation needs of both feet simultaneously. These factors emphasize the importance of developing ankle rehabilitation robots that accurately replicate the complex motion and spatial interfacial relationship of the human ankle joint while accommodating individual physiological differences between patients' left and right feet.

To address the aforementioned challenges, Liu et al. [19, 20] proposed a series of generalized spherical parallel ankle rehabilitation mechanisms based on the ankle bone structure. These mechanisms feature two rotational center points corresponding to the tibiotalar joint and subtalar joint centers. The distance between the two center points corresponds to the size of the talus of the ankle joint. The mobile platform of the mechanism allows rotation around these two center points individually or simultaneously, replicating the compound motion of the ankle joint more accurately and exhibiting promising rehabilitation potential. However, these mechanisms utilize multiple arc-shaped connecting rods, making the structure complex and requiring higher processing and installation accuracy. Moreover, during compound motion of the ankle joint, the rotation axis of the mechanism changes with its pose variations. Consequently, it becomes challenging to ensure precise alignment between the mechanism's motion axis and the ankle joint's two rotation axes. While the mechanism aligns the rotation centers with the joint centers of the ankle, this limitation may still pose a risk of secondary damage to the ankle joint.

Therefore, this paper proposes a novel parallel ankle rehabilitation robot with three rotational and one translational (3R1T) DOF, taking into account the physiological and anatomical model of the human ankle joint. It features two virtual motion center (VMC) points and two rotation planes, enabling continuous rotation along specific lines without unintended motion. The VMC points correspond to the tibiotalar joint center and subtalar joint center. The distance between these points can be adjusted to accommodate different ankle heights. The robot also considers variations in deviation angles and inclination angles among patients. It offers benefits such as improved human-machine motion compatibility, suitability for diverse patients, easy wearing, and high safety.

In summary, the main work and innovation of this paper are as follows. Firstly, based on the motion properties and anatomical model of the human ankle joint, the mapping relationship between the ankle joint and the rehabilitation mechanism is determined. A novel equivalent model of the series ankle joint is proposed, which can provide guidance for the configuration design of the ankle rehabilitation robot. Secondly, a novel parallel mechanism with three rotational and one translational (3R1T) DOF is proposed. The mechanism has unique motion properties and has two virtual center points, which can realize fixed-point rotation around two virtual centers. This mechanism can be well applied in the field of ankle rehabilitation. Thirdly, according to the kinematic properties of the mechanism, it can be equivalent to a series branch chain. Combined with DH method, the inverse kinematics model of the rehabilitation mechanism can be easily established. Then, the inverse kinematics model, kinematic performance, and workspace of the mechanism are analyzed. Finally, based on the OpenSim biomechanical software, the human musculoskeletal model is coupled with the rehabilitation robot to establish a human-machine rehabilitation simulation model, and the evaluation index is defined to evaluate the rehabilitation effect.

The paper is organized as follows: In Section 2, considering the physiological anatomy model of the human ankle joint, a novel ankle rehabilitation robot is designed, and its mobility is analyzed based on screw theory. In Section 3, based on the intersection theory of the motion plane, the motion property and the full-cycle mobility of the mechanism are further distinguished, and the continuous axis of the mechanism is distinguished based on the constraint power and its derivative. In Section 4, according to the motion property of the mechanism, it can be equivalent to a series of branched chains, and the inverse kinematics model of the mechanism is established. In Section 5, given the size of the prototype, the kinematic performance of the rehabilitation mechanisms is discussed based on the motion/force transmission index and constraint index. Then, considering the singular configuration, driving stroke, and interference conditions of the mechanism, the workspace of the rehabilitation robot is analyzed. In Section 6, the human-machine rehabilitation simulation model is established to verify and evaluate the rehabilitation performance of the mechanism. Section 7 is the conclusion.

The coordinate system is established as shown in Figure 13. The origin O of the global coordinate system O-XYZ is the center of the fixed platform A₁A₃A₄. The Y-axis is along the axis direction of the revolute joint R₁₁, and the Z-axis is along the normal direction of the base platform. Similarly, the mobile coordinate system P-xyz is established, and P is the center of the mobile platform B₂B₃B₄. The following parameters are defined: the circumcircle radius of triangle A₁A₃A₄ is R₁, the circumcircle radius of triangle B₂B₃B₄ is r₁, the length of platform slide radius OA₂ is R, the length of mobile platform slide radius PB₁ is r, the length of connection center line AB is d₁, and the length of connection line BP is d₂. The inclination angle of the axis of the revolute joint R₂₂ is θ₁, and the inclination angle of the axis of the revolute joint R₁₄ is θ₂.

Then in the global coordinate system O-XYZ, the coordinates of points A and B can be expressed as:where \({}^{O}{\varvec{R}}_{P}\) is the attitude matrix of the mobile platform, \({\varvec{B}}_{{}}^{P} = [0,0,r \cdot \tan (\theta_{2} )]^{{\text{T}}}\), \({\varvec{P}} = [x,y,z]^{{\text{T}}}\).

The attitude of the mobile platform can be determined by the length d₁ of the connecting line AB, the continuous rotation \(\sigma_{1}\) around the k₁ axis, the continuous rotation \(\sigma_{2}\) around the connecting line AB, and the continuous rotation \(\sigma_{3}\) around the k₂ axis. The k₁ axis is an axis passing point A on the continuous rotation plane C₁ and is parallel to the constraint force line vector \(\user2{\$ }_{2}^{r}\) along the direction of the Y-axis. The k₂ axis is an axis passing point B on the continuous rotation plane C₂ and is parallel to the constraint force line vector \(\user2{\$ }_{2}^{r}\). Under the initial assembly configuration, the angle between planes C₁ and C₂ is \(\uppi /2 - \sigma_{0}\), because the human body generally tilts 23° inward from the subtalar joint, so \(\sigma_{0}\) can be set to 23°.

In summary, the motion of the mobile platform can be equivalent to a 4-DOF RPRR series manipulator, rotating around the k₁ axis, moving around the connection line AB, rotating around the connection line AB, and rotating around the k₂ axis. Therefore, the homogeneous transformation matrix is as follows:

The position coordinates of point P and the attitude matrix \({}^{O}{\varvec{R}}_{P}\) of the mobile platform can be obtained by Eq. (11), \({\varvec{P}} = {\varvec{A}} + {}^{A}{\varvec{P}}\). Since points B₂, B₃ and B₄ are respectively fixed on the mobile platform, the coordinates of points B₂, B₃, and B₄ can be obtained from the following:

Since point B₁ is always in the XZ plane and slides relative to the mobile platform, the coordinates of point B₁ can be defined as \([B_{1x} ,0,B_{1z} ]\). Since the distance from point B₁ to point P and point B is always constant, there are the following constraints:

Similarly, since the A₂ point is always in the XY plane and sliding relative to the base platform, the coordinates of point A₂ can be defined as \([A_{2x} ,A_{2y} ,0]\). Because the distance between point A₂ and point O is constant, and the line A₂B₂ is always perpendicular to the axis of R₂₅, s₂₅ is defined as the direction vector of R₂₅, there are the following constraints:

Through Eqs. (13) and (14), the coordinates of points B₁ and A₂ in the global coordinate system can be solved respectively. Finally, the length of each branch prismatic joint is obtained:

A rehabilitation robot serves as a crucial tool in assisting patients with safe and comfortable rehabilitation therapy. Therefore, it is essential to evaluate the effectiveness of rehabilitation robots. Due to the complexities of the human bone structure and joints, as well as limitations in measuring instruments, the use of biomechanical analysis software has become a prominent trend for evaluating the effects of rehabilitation robots in the rehabilitation process. One widely utilized software in this domain is OpenSim [35]. OpenSim enables the simulation of musculoskeletal dynamics and neuromuscular control, providing an accurate biomechanical model for muscle dynamics and joint kinematics modeling. It accurately reproduces human and animal movements, making it applicable in various fields, including biomechanics research, surgical process simulation, and medical rehabilitation device development.

Rehabilitation therapy modes can be broadly categorized into three categories: passive rehabilitation, collaborative rehabilitation, and active rehabilitation [36]. The choice of rehabilitation mode depends on the extent of the patient’s injury. In the early stage of rehabilitation, when the injured limbs may not respond, the passive rehabilitation mode is adopted [37]. During the middle stage of rehabilitation, the patient and the rehabilitation robot move together [38]. In the later stage of rehabilitation, when the patient has gained a certain level of muscle strength, impedance active rehabilitation therapy with a certain intensity can be conducted [36]. In this paper, we evaluate and analyze the effect of an ankle rehabilitation robot on passive rehabilitation of the human foot using OpenSim biomechanical software. The ankle rehabilitation robot model, established in SolidWorks software, is imported into OpenSim software. By setting the physiological parameters of the human body through OpenSim XML language, we combine the human neuromuscular skeleton model with the rehabilitation robot to create a human-machine coupling model for rehabilitation simulation, as shown in Figure 19.

When wearing a rehabilitation robot on the human body, flexible materials such as elastic straps are usually used. On the one hand, they can constrain the movement of the human body and exert the functions of the rehabilitation robot, and on the other hand, they can protect the human body to a certain extent from harm, because the rehabilitation robot and the human body are actually not perfectly compatible. The rehabilitation mechanism’s mobile platform is connected to the human foot through a BushingForce element. BushingForce is a 6-DOF spring damping force with three translations and three rotations, which can be free or constrained by flexible forces. It can establish moving springs/damping or rotating springs/damping in six directions, which can effectively simulate the flexible binding effect between human feet and the platform.

Based on the human-machine coupling model, the passive rehabilitation simulation is carried out for the early rehabilitation stage of patients with serious ankle injuries. At this time, the robot performs active motion according to the rehabilitation needs, drives the ankle joint to perform dorsiflexion/plantar flexion, inversion/eversion, and other actions, and the human ankle muscle follows the rehabilitation robot to passively stretch. The given motion of the tibiotalar joint and the subtalar joint is as follows:

Foot movement is mainly controlled by the muscles on the lower leg. During the dorsiflexion / plantar flexion, and inversion/eversion movement of the foot, the muscles that play a major role include extensor digitorum longus, flexor digitorum longus, tibialis anterior, tibialis posterior, extensor hallucis longus, flexor hallucis longus, peroneus brevis, gastrocnemius, soleus, and so on. Therefore, the above muscles are selected as the analysis objects. Muscle activation, muscle passive force, muscle fiber length, and human-machine interaction force were selected as output parameters to evaluate the rehabilitation effect of the rehabilitation robot. During the rehabilitation process, the muscle force of the patient should not increase sharply to avoid foot discomfort, and the human-machine binding force curve and muscle fiber length curve should not produce violent fluctuations. When performing rehabilitation therapy according to the rehabilitation trajectory given by Eq. (23), the human-machine interaction force is the flexible binding force curve between the human foot and the rehabilitation robot, as shown in Figure 20; the activation of each muscle is shown in Figure 21; the length of each muscle fiber is shown in Figure 22; the passive tension of each muscle is shown in Figure 23.

It can be seen from Figure 20 that the maximum human-machine flexible binding force is about 36 N, and the fluctuation range is small, which will not make the foot feel uncomfortable or hurt. Figure 21 shows that the muscle activation degree is 0, which means that the muscle is only passive stretching, and does not produce active force, in line with the patient's initial passive rehabilitation; it can be seen from Figure 22 that under the rehabilitation trajectory shown in Eq. (23), the transition of the muscle fiber length curve of the ankle joint is smooth, without sharp elongation, and will not cause discomfort to the patients. Under this rehabilitation trajectory, the variation trends of muscle length curves in Figure 22(a) are consistent, and they are synergistic muscles. Similarly, the muscles in Figure 22(b) are also synergistic muscles. The variation trends of muscle length curves in Figure 22 are opposite, and they are antagonistic muscles. It can be seen from Figure 23 that the four muscles, i.e., tibialis posterior (tib_ post _ r), peroneus longus (per _ long _ r), soleus (soleus_ r), and peroneus brevis (per _ brev _ r), are subjected to the maximum force, while the other muscles are subjected to small tension. Among them, the maximum tension of the muscle is not more than 300 N, and the muscle curve is excessively smooth without rapid growth. In the rehabilitation process, it will not cause secondary damage to the ankle joint and will not make the human body uncomfortable.

In this study, we determined the equivalent model of an ankle rehabilitation robot based on the anatomical structure and motion properties of the human ankle. We established a mapping relationship between the human ankle and the configuration design of the rehabilitation mechanism.

Although the proposed rehabilitation mechanism can theoretically simulate the rehabilitation movement of ankle joints in different patients, there are practical considerations to address. These include addressing processing and assembly errors of the robot, as well as mitigating wearing errors of both the robot and the patient. Future work should focus on adding flexible joints and components to enhance the compliance, safety, and comfort of the robot system. Additionally, studying the human-machine interaction force between the robot and the patient is essential. Furthermore, it is worth noting that the prototype proposed in this study adopts an arc-shaped sliding rail design, which may pose challenges such as machining difficulties, high accuracy requirements, and increased cost. As a potential solution, turntable bearings can be considered as a replacement for the arc-shaped sliding rails during the later stages of prototype manufacturing.