Design and analysis of shoulder joint exoskeleton rehabilitation mechanism based on gear and rack transmission

Pang, Zaixiang; Wang, Tongyu; Yu, Junzhi; Wang, Zhanli; Liu, Shuai; Zhang, Xiyu

doi:10.1063/5.0051484

Due to insufficient muscle strength, a novel rehabilitation mechanism of the shoulder joint exoskeleton is proposed and analyzed based on gear and rack transmission, which solves the problem that stroke patients need repetitive exercise rehabilitation training. First, the range of motion angle of the shoulder joint is determined, and the mechanical structure of shoulder joint rehabilitation is designed. The arc rack is engaged with the output shaft gear of the reducer, and the servo motor is utilized as the driving force to realize the internal/external rotation movement of the shoulder joint. The motor connects the horizontal connecting rod and the rotating rod through the cross roller bearing to realize the abduction/adduction and flexion/extension movement of the shoulder joint. Second, the kinematics and dynamics of the shoulder joint exoskeleton are analyzed, investigated, and verified for the rehabilitation mechanism. The simulation analysis of the mechanism is completed in the virtual prototype. The changes in the joint angle and end trajectory and joint moment with time and angular velocity under the given working conditions are obtained, which directly reflect the movement of each joint of the rehabilitation mechanism. Finally, aiming at the nonlinear disturbance in the trajectory tracking control of the rehabilitation mechanism of the shoulder joint exoskeleton, the trajectory tracking approach of the mechanism is achieved aided with the closed-loop PD iterative learning control method. The results demonstrate that the actual trajectories are in good agreement with the desired trajectories, which can achieve effective tracking and improve the control quality of the system.

I. INTRODUCTION

Stroke has been one of the most threatening factors that endanger human health. Moreover, it shows a trend of rejuvenation. The disability rate is so high that the limb motor function of most patients is impaired. Shoulder subluxation (caused by hemiplegia) and shoulder joint motor dysfunction severely affect the patient’s quality of life, which harm the patients physiologically and psychologically.^1,2 To alter this situation, patients with shoulder joint motor dysfunction need appropriate small-scale rehabilitation training in the early and middle stages of rehabilitation training. The task of assisting the patients with shoulder joint motor dysfunction to regain a normal life effectively has become a social problem urgent to be solved counting on the scientific community of modern rehabilitation medicine.³

For the rehabilitation training of shoulder joints, rehabilitation physiotherapists generally perform traditional no-machine operation or “one-to-one” treatment with simple equipment. Generally speaking, there are many patients with shoulder joint rehabilitation. However, the technical level is uneven for rehabilitation trainers, which causes that the traditional rehabilitation training is inefficient and costly. At the same time, the working intensity for physiotherapists is extremely high sometimes, which is always along with the reduction in the accuracy of rehabilitation training.^4,5 With the continuous improvement of economic conditions and quality of life, patients with shoulder joint dysfunction have higher requirements for rehabilitation training. Traditional rehabilitation training hardly satisfies the actual demands of the patients.⁶ In order to solve this problem, the application of robot technology in medical rehabilitation training has provided new ideas for the study on new rehabilitation training devices and novel rehabilitation approaches. Meanwhile, it makes a possible method, which receives rehabilitation training more effectively for the patients.^7,8 At present, some shoulder joint rehabilitation training devices have been successfully applied to clinical treatments, such as shoulder joint rehabilitation therapy. Taking the structure of the rehabilitation device as the criterion of division, there are two types of rehabilitation devices in general. One is the terminal guidance device, and the other is the exoskeleton mechanism.^9,10 The terminal guidance device, which is characterized by the conciseness of the structure and the convenience of the manufacture, drives the shoulder joint to move by operating a handle to complete the rehabilitation training. However, there is only one structure that connects the device and the patient’s body so that it cannot realize the rehabilitation training for a specific joint.¹¹ In contrast, the exoskeleton mechanism for shoulder joint rehabilitation, which is composed of a joint rotation axis and the human joint axis, is designed based on the principle of ergonomics. The exoskeleton mechanism wears on the human arm, which drives the shoulder joint to perform rehabilitation motions, and contacts between the patient’s upper limbs and the robot. As a result, this method is equivalent to adding a structure surrounding the patient’s limb.¹²

Many scientific institutions and medical institutions at home and abroad have carried out to study shoulder rehabilitation robots one after another. A variety of shoulder rehabilitation robot structures have been proposed based on different mechanical structures and rehabilitation principles, for example, Intelli-Arm upper limb rehabilitation robot,¹³ CAREX-7 rehabilitation robot,¹⁴ MEDARM upper limb rehabilitation robot,¹⁵ ARMin 6-degrees of freedom (DOFs) rehabilitation robot,¹⁶ LIMPACT rehabilitation robot,¹⁷ and a variety of other robots for shoulder rehabilitation.^18,19 It can be seen that the exoskeleton rehabilitation robot adopts the method of introducing the active joint or the combination of the active joint and the passive joint to realize the human–machine motion compatibility design through the numerical analysis and theoretical proof. In the process of rehabilitation training, it is necessary to control the movement of the exoskeleton joint to realize the movement of the human shoulder joint. Due to the different movement tracks of the shoulder joint in patients with different physical signs, the applicability of the exoskeleton is limited to a certain extent.

Owing to the principle of the human upper limb anatomy, the designing requirements of the exoskeleton mechanism are developed for shoulder joint rehabilitation training by analyzing the physiological structure of the human upper limbs. Then, it designs the exoskeleton mechanism for shoulder joint rehabilitation training according to the former requirements and analyzes the basic machine. By simulation kinematics analysis of the exoskeleton mechanism, it obtains how the joint angle and the terminal trajectory of the exoskeleton mechanism change over time under a given working condition. Besides, by performing a dynamic simulation on the exoskeleton mechanism, it derives the joint torque’s relationship with the angular velocity and time, which intuitively presents the movement of each joint of the exoskeleton mechanism. The simulation results agree well with the actual motion of the exoskeleton mechanism. Then, the trajectory tracking of the exoskeleton mechanism is realized based on the closed-loop PD type iterative learning control method, which directs at the existence of nonlinear disturbance in the trajectory tracking control of the shoulder joint rehabilitation robot system. The simulation results indicate that the actual moving trajectory fits the expected trajectory well, which improves the control capability of the system. It also verifies the effectiveness of the structure of the exoskeleton mechanism for shoulder joint rehabilitation training.

The main contributions of this paper are as follows:

A rehabilitation mechanism for the shoulder joint exoskeleton is proposed based on the human upper limb anatomy. The mechanism is engaged with the output shaft gear of the reducer through the arc rack, and the servo motor is used as the driving force to realize the internal/external rotation movement of the shoulder joint; the motor connects the horizontal connecting rod and the rotating rod through the cross roller bearing to realize the abduction/adduction and flexion/extension movement of the shoulder joint. By designing the number of teeth of the driven gear and the arc rack, the movement range of the patient’s shoulder joint rotation is limited to ensure the patient’s safety. The research results will play an important role in reappearing the motion of the human shoulder joint and promote the development of exoskeleton rehabilitation robot technology.
Through the passive adjustment module, the micro-motion of the rotation center of the human shoulder joint is realized, which has a certain activity margin relative to the exoskeleton shoulder joint, and the movement of the rehabilitation mechanism matches the movement of the human shoulder joint.
Based on the closed-loop PD iterative learning control method, the tracking trajectory is realized for the shoulder joint exoskeleton rehabilitation mechanism. The simulation results show that the actual trajectory is in good agreement with the desired trajectory, which improves the control ability and tracking function of the shoulder joint exoskeleton rehabilitation mechanism.

The rest of this paper is arranged as follows: In Sec. II, the shoulder joint is analyzed based on the human upper limb anatomy, and the free distribution of the shoulder joint and the freedom distribution of a robot are determined. In Sec. III, the modular design and modeling of the rehabilitation mechanism of the shoulder joint exoskeleton are completed. In Secs. IV and V, the kinematics and dynamics of the rehabilitation mechanism of the shoulder exoskeleton are analyzed, and the relationship between the joint angle and end trajectory with time and the joint moment with time and angular velocity under given working conditions is obtained. In Sec. VI, the simulation experiment is carried out, and the trajectory tracking of the shoulder joint exoskeleton rehabilitation mechanism is realized based on the closed-loop PD iterative learning control method. Finally, Sec. VII gives conclusions and future works.

II. STRUCTURAL DESIGN

A. Anatomy of the shoulder joint

Most of the motions in daily life are performed by the upper limbs, such as eating, picking things, cleaning the room, etc. The influence of upper limb impairment on the patients’ daily life is particularly obvious. Modern medical theory and clinical practice show that moderate and reasonable shoulder joint rehabilitation training for patients with shoulder joint impairment can promote the recovery of muscle function to a certain extent and help to regain some basic motor functions of the shoulder joint. When a patient with shoulder joint dysfunction is receiving rehabilitation training, the trainer should follow the scientific training plan to treat the impaired shoulder joint repeatedly by kinesitherapy.

The shoulder joint rehabilitation structure ought to coordinate with the human shoulder joint to complete the rehabilitation training of the impaired limbs. During the rehabilitation process, the exoskeleton mechanism needs to cooperate closely with the human body to achieve human–machine coordination and avoid aggravating the damage to the human shoulder joint during the rehabilitation training. Therefore, it is necessary to analyze the structure and movement characteristics of the human shoulder joint.

From the respective anatomy, the shoulder joint is one of the most flexible and complex structures of the loco-motor system, consisting of the glenoid bone and the glenoid of the scapula. It is a typical ball-and-socket joint, which rotates around three mutually perpendicular axes, respectively. The glenoid is shallow and covered with labrum along its edge, the joint capsule is thin and loose, and the long head of biceps brachii tendon passes through the capsule. The combination of coracohumeral ligament, coracoacromial ligament, and tendon is outside the joint capsule, which enhances its stability. The muscles of the human body have specific morphological structures when the body is doing the rehabilitation training; inside the muscles lie many vascular nerves. These skeletal muscles provide power for the shoulder joint and guarantee the stability of the shoulder joint.^20,21 Figure 1 shows the shoulder joint complex and the physiological structure of the joint. According to the structure characteristics, the shoulder joint can be considered as a combination of the inner joint of the shoulder strap joint and the outer joint of the scapular joint, which can be flexed/extended, circumflexed/extended, and abducted/adducted. The maximum angle of shoulder joint flexion/extension is 135°, and that of abduction/adduction is 135°. As for the internal/external rotation, it is 110°. This is a large motion range.²² Meanwhile, since these three moving pairs are at the beginning of the human upper limb kinematic chain and the operating space of the end effector (hand) is mostly affected by the motion of the joint who is at the beginning of the kinematic chain, fixing any one of the 3-degrees of freedom (DOFs) will result in failure to complete some regular movements for the hand. It is for this reason that the 3 DOFs of the shoulder joint are entirely reserved in the exoskeleton mechanism.^23,24 The shoulder joint’s and the robot’s motion ranges of each DOF are shown in Table I.

FIG. 1.

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Shoulder joint and the physiological structure diagram.

TABLE I.

The shoulder joint’s and the robot’s motion ranges of each DOF.

		Movement range of human	Movement range of robot
Parts	Degrees of freedom	(deg)	(deg)
Shoulder	Flexion/extension	0–90/0–45	0–90/0–10
Shoulder	Abduction/adduction	0–90/0–45	0–75/0–45
Shoulder	Internal/external rotation	0–80/0–30	0–80/0–30

		Movement range of human	Movement range of robot
Parts	Degrees of freedom	(deg)	(deg)
Shoulder	Flexion/extension	0–90/0–45	0–90/0–10
Shoulder	Abduction/adduction	0–90/0–45	0–75/0–45
Shoulder	Internal/external rotation	0–80/0–30	0–80/0–30

B. Design of rehabilitation mechanism for shoulder joint exoskeleton

The design of the exoskeleton mechanism for shoulder joint rehabilitation training should involve the consideration of rehabilitation medicine, mechanical structure principle, and the movement mechanism to ensure that the motion is consistent with the human shoulder joint motion without causing secondary injuries. The shoulder joint is the basis for the human upper limb to perform various basic activities. It is highly flexible but also highly vulnerable to be injured. The rehabilitation of motor functions is the prerequisite for the other upper limb joints to perform basic motor functions. The exoskeleton rehabilitation mechanism of shoulder joint rehabilitation training drives the joint to move around. The rigid structure is applied to assist the impaired limb to achieve effective rehabilitation motion. The design of the exoskeleton mechanism should ensure that the mechanism’s movement accords with laws of clinical human limb rehabilitation training, which prevents secondary injuries of the patient’s limb. Besides, the comfortableness of the mechanism should also be considered.

The structure of the exoskeleton rehabilitation mechanism for shoulder joint rehabilitation training is designed based on the feature of shoulder joint movement and the characteristics of the patient’s body structure.^25,26 The exoskeleton mechanism consists of three main motion modules and one passive adjustment module through fixing on the lifting platform. The three main motion modules are exploited to realize the flexion/extension, internal/external rotation, and abduction/adduction movement of the shoulder joint, and the passive adjustment module realizes the regulation function of the joint locally.

In the 2-DOFs of flexion/extension and abduction/adduction movement, various parts of the mechanism are driven and connected by the harmonic reduction motor. However, the DOFs of inside/outside rotation take the upper arm as the axis cannot employ the structure above, which is because the structure needs to be worn, which restricts the application. To solve this problem, a circular arc slide is installed in the lifting platform. The circular arc slide meshes with the output shaft gear of the reducer through an arc-shaped rack on the circular arc slide rail via using the servo motor as the power source. The inside/outside rotation of the shoulder joint by the circular arc slide can be realized. Two sets of pulleys are installed on both sides of the arc rack to restrain the arc guide rail, and a passive gear is assembled between the driving gear and the arc rack, which reduces the output torque of the motor and the mass of the mechanism. Shoulders are assembled at both ends of the arc rack; therefore, the passive gear can only mesh with the arc rack within a certain angular range, which ensures the safety of the patient. On the other hand, the safety precaution is to limit the range of motion of the shoulder joint’s rotation by modifying the number of teeth of the passive gear and the arc rack. The exoskeleton mechanism of shoulder joint rehabilitation training is shown in Fig. 2. Figure 2(a) shows the three-dimensional model of the shoulder joint, and Fig. 2(b) shows the three-dimensional model of the rotational degree of freedom of the shoulder joint.

FIG. 2.

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Model of the shoulder joint. (a) The three-dimensional model of the shoulder joint. (b) Rotational degree of freedom.

The active module’s movement is realized by three motors. The motor with reducer on the J1 axis is connected to the horizontal connecting rod through cross roller bearings. The crossed roller bearing has to bear axial load and radial load at the same time, which can realize the abduction/adduction movement of the shoulder joint. The motor with a reducer on the J2 axis is connected with the rotating rod through crossed roller bearings. The crossed roller bearings mainly bear radial load, which can realize the flexion/extension movement of the shoulder joint. To realize the in/out rotation of the shoulder joint, the movement of the J3 axis is performed by a motor equipped with a reducer through a transmission mechanism, a pulley set, and an arc track, which is used to drive an arc rack.

The exoskeleton module of the rehabilitation robot can achieve 3-DOFs of the shoulder joint:

Flexion/extension movement of the shoulder joint. The horizontal link 3 rotates around the J₁ axis to the position and is shown in Fig. 3(a), and the upper arm rotates around the J₂ axis.
Adduction/abduction movement of the shoulder joint. The horizontal link 3 rotates around the J₁ axis to the position and is shown in Fig. 3(b), and the upper arm rotates around the J₂ axis.
In/out rotation of the shoulder joint. A part of the upper arm is surrounded by the semicircular ring and rotates around the J₃ axis.

The rehabilitation robot is a mechanical equipment based on the simplification of the skeletal structure of the human body. The shoulder joint is a complicated joint consisting of bones, muscles, and tissues. Therefore, the movement of the exoskeleton mechanism should fit the movement of the human shoulder joint as well as possible when designing an exoskeleton mechanism for shoulder joint rehabilitation training. When the shoulder joint is performing abduction/adduction movement, its rotating center does not remain static, that is to say, the rotating center of the glenohumeral joint deviates over time.^27,28 As a result, patients may feel uncomfortable to wear an exoskeleton during the rehabilitation training because the exoskeleton’s rotating center is fixed. The offset is directly related to the height and arm length of the patient. According to the theoretical and numerical analysis, the offset of the rotational center of the glenohumeral joint of the upper limb is usually less than 4 cm. In order to alleviate this pain to the most extent, the exoskeleton mechanism for the shoulder joint rehabilitation training exploits a regulation mechanism to passively modify the micro-motion of the rotation center of the human shoulder joint. As a result, the human shoulder gains a movement margin compared with the other exoskeleton mechanisms without hindering the movement of the 3-DOFs of the robot’s shoulder joint. Therefore, the allowable value in the passive regulation module designed is 5 cm, which can meet the use needs of patients in this paper. The passive adjustment module includes the following.

FIG. 3.

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Schematic diagram of shoulder closing movement: (a) flexion/extension and (b) adduction/abduction.

A passive regulation module was designed to ensure the reliability of the device in the rehabilitation process and the safety of patients. Both sides of the slide block are equipped with a group of springs, and the slide block is always maintained in the balance position under the action of the spring. When the human joint center is offset from the exoskeleton movement center, additional force will be generated, which will push the slider to move and eventually be converted into the elastic potential energy of the spring to be absorbed by the adjustment module to ensure the comfort and safety of the patient. Linear bearings are installed between the slider and the support, which can reduce the friction coefficient to 0.12 under the action of lubricating oil with the friction problem well solved.

The support is connected to the arc-shaped track that realizes the in/out rotation of the shoulder joint through screws. The slider can freely slide on the two tracks of the support, and each track has two stiffness coefficients. The same springs are distributed on both sides of the slider to ensure that the strap connected to the slider is at the initial position when the patient is wearing. The passive sliding movement along the J₃ axis direction can be achieved. A schematic diagram of the passive regulation module is shown in Fig. 4.

FIG. 4.

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Schematic diagram of the passive regulation module.

III. KINEMATICS AND DYNAMICS ANALYSIS

A. Kinematics analysis

The positive kinematics problem of the exoskeleton mechanism for shoulder joint rehabilitation training refers to deriving the posture and the position of the exoskeleton mechanism’s terminal by computing the geometric parameters and the joint angles of the exoskeleton mechanism in the base coordination system. The inverse kinematics problem is to make out the position and direction of the reference coordinate system by analysis of the geometric parameters and the posture of the connecting rod’s terminal, and then calculate the joint angle of the exoskeleton mechanism.^29,30

To obtain the relationship between the rotation and translation of the adjacent rods for the exoskeleton mechanism as well as the position and posture of its terminal, a reference coordinate system needs to be set up for each joint. As shown in Fig. 5, according to the prototype model, the Denavit–Hartenberg (D-H) coordinate system method is utilized to establish the kinematics model of the exoskeleton mechanism. The condition of each link can be determined by four parameters. Two of the parameters a_i and α_i are used to describe the geometric features of the connecting rod itself, which are determined by the distance and the angle between the two axes z_i−1 and z_i. The other two parameters, the offset distance d_i and the joint angle θ_i, present how the two links connect with each other. The specific values of the two parameters x_i−1 and x_i are ascertained by the distance and the angle between the two axes. The D-H model of the shoulder joint is shown in Table II.

FIG. 5.

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Coordinate system and motion parameters of rehabilitation robot joints.

TABLE II.

D-H table for the rehabilitation robot.

i	θ_i	α_i (deg)	d_i
1	θ₁	0	0
2	θ₂	90	L₁
3	θ₃	90	L₂

B. Positive kinematics analysis of the shoulder joint

Positive kinematics analysis is performed in the coordinate system of the shoulder joint. In other words, by coordinate transformation, a vector is transformed from the coordinate system {O_i−1} into the coordinate system {O_i},

\begin{align} {}_{i}^{i - 1}T & = Rot (Z, θ_{i}) \times Trans (0, 0, d_{i}) \times Trans (a_{i - 1}, 0, 0) \\ \times Rot (X, a_{i - 1}) \\ = [\begin{matrix} c θ_{i} & - s θ_{i} & 0 & a_{i - 1} \\ s θ_{i} c α_{i - 1} & c θ_{i} c α_{i - 1} & - s α_{i - 1} & - d_{i} s a_{i - 1} \\ s θ_{i} s α_{i - 1} & c θ_{i} s α_{i - 1} & c α_{i - 1} & d_{i} c α_{i - 1} \\ 0 & 0 & 0 & 1 \end{matrix}] . \end{align}

(1)

Equation (1) is a transformation matrix between two adjacent coordinate systems. Furthermore, the calculation formula of the transformation matrix from the foundation bed to the terminal of the shoulder joint can be derived as follows:

{}_{1}^{0}T = [\begin{matrix} c θ_{1} & - s θ_{1} & 0 & 0 \\ s θ_{1} & c θ_{1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],

(2)

{}_{2}^{1}T = [\begin{matrix} c θ_{2} & - s θ_{2} & 0 & 0 \\ 0 & 0 & - 1 & - L_{1} \\ s θ_{2} & c θ_{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],

(3)

{}_{3}^{2}T = [\begin{matrix} c θ_{3} & - s θ_{3} & 0 & 0 \\ 0 & 0 & - 1 & - L_{2} \\ s θ_{3} & c θ_{3} & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],

(4)

{}_{3}^{0}T =_{1}^{0} T_{2}^{1} T_{3}^{2} T = [\begin{matrix} r_{11} & r_{12} & r_{13} & P_{x} \\ r_{21} & r_{22} & r_{23} & P_{y} \\ r_{31} & r_{32} & r_{33} & P_{z} \\ 0 & 0 & 0 & 1 \end{matrix}] .

(5)

Equations (2)–(4) are vector mappings of the components of adjacent joints in the shoulder joint, and Eq. (5) is a vector mapping of the posture of the shoulder joint’s terminal relative to the foundation bed.

The details of Eq. (5) can be seen as follows:

\begin{matrix} r_{11} = \sin θ_{1} \sin θ_{3} + \cos θ_{1} \cos θ_{2} \cos θ_{3}, \\ r_{21} = \cos θ_{2} \cos θ_{3} \sin θ_{1} - \cos θ_{1} \sin θ_{3}, \\ r_{31} = \cos θ_{3} \sin θ_{2}, \\ r_{12} = \cos θ_{3} \sin θ_{1} - \cos θ_{1} \cos θ_{2} \sin θ_{3}, \\ r_{22} = - \cos θ_{1} \cos θ_{3} - \cos θ_{2} \sin θ_{1} \sin θ_{3}, \\ r_{32} = - \sin θ_{2} \sin θ_{3}, \\ r_{13} = \cos θ_{1} \sin θ_{2}, \\ r_{23} = \sin θ_{1} \sin θ_{2}, \\ r_{33} = - \cos θ_{2}, \\ p_{x} = L_{1} \sin θ_{1} + L_{2} \cos θ_{1} \sin θ_{2}, \\ p_{y} = L_{2} \sin θ_{1} \sin θ_{2} - L_{1} \cos θ_{1}, \\ p_{z} = - L_{2} \cos θ_{2}, \end{matrix}

(6)

$[\begin{matrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33} \end{matrix}]$ and ${[\begin{matrix} p_{x} & p_{y} & p_{z} \end{matrix}]}^{T}$ are the direction vector and the position vector of the shoulder joint, respectively.

This paper takes two special positions as examples to perform positive kinematics calculations.

Assuming θ₁ = θ₂ = θ₃ = 0° and inputting them into Eqs. (5) and (6) to obtain the shoulder posture vector matrix of the shoulder joint’s terminal can be seen as follows:
${}_{3}^{0}T =_{1}^{0} T_{2}^{1} T_{3}^{2} T = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & - L_{1} \\ 0 & 0 & - 1 & - L_{2} \\ 0 & 0 & 0 & 1 \end{matrix}] .$
(7)
Assuming θ₁ = θ₂ = 90°, θ₃ = 0°, similarly, the posture mapping vector matrix of the shoulder joint’s terminal relative to the foundation basement is obtained as follows:
${}_{3}^{0}T =_{1}^{0} T_{2}^{1} T_{3}^{2} T = [\begin{matrix} 0 & 1 & 0 & L_{1} \\ 0 & 0 & 1 & L_{2} \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] .$
(8)

C. Inverse kinematics analysis of the shoulder joint

Inverse kinematics is the premise and foundation of robot trajectory planning. Variables of different joints are solved according to the posture and position of the robot’s terminal. In this paper, the algebraic method is exploited to solve the inverse kinematics problems of the tandem structure. The homogeneous transformation matrix of the rehabilitation mechanism’s terminal can be computed as follows:

{}_{3}^{0}T =_{1}^{0} T_{2}^{1} T_{3}^{2} T = [\begin{matrix} r_{11} & r_{12} & r_{13} & P_{x} \\ r_{21} & r_{22} & r_{23} & P_{y} \\ r_{31} & r_{32} & r_{33} & P_{z} \\ 0 & 0 & 0 & 1 \end{matrix}] = {}_{1}^{0}T (θ_{1}) {}_{2}^{1}T (θ_{2}) {}_{3}^{2}T (θ_{3}) .

(9)

Owing to the motion range of flexion/extension movements of the arm joint for an adult and considering the structure and moving characteristics of the exoskeleton mechanism for shoulder joint rehabilitation training, the angular range of each joint of the mechanism is appropriately defined as follows:

- 4 5^{◦} \leq θ_{1} \leq 9 0^{◦}; - 4 5^{◦} \leq θ_{2} \leq 13 5^{◦}; - 3 0^{◦} \leq θ_{3} \leq 3 0^{◦} .

(10)

Under the constraint of Eq. (10), combining with the equations in Eq. (6), the following equations are obtained:

\begin{gathered} \{\begin{cases} r_{13} = \cos θ_{1} \sin θ_{2}, \\ r_{23} = \sin θ_{1} \sin θ_{2}, \\ r_{33} = - \cos θ_{2}, \end{cases}) \\ t h e n θ_{1} = \arctan \frac{r_{32}}{r_{31}} . \end{gathered}

(11)

According to the system of equations in Eq. (6),

\begin{matrix} p_{x} = L_{1} \sin θ_{1} + L_{2} \cos θ_{1} \sin θ_{2}, \\ t h e n θ_{2} = \arcsin (\frac{p_{x} - L_{1} \sin θ_{1}}{L_{2} \cos θ_{1}}) . \end{matrix}

(12)

Finally, owing to the system of equations in Eq. (6),

\begin{gathered} \{\begin{cases} r_{31} = \cos θ_{3} \sin θ_{2}, \\ r_{12} = \cos θ_{3} \sin θ_{1} - \cos θ_{1} \cos θ_{2} \sin θ_{3}, \end{cases}) \\ t h e n θ_{3} = \arcsin (\frac{r_{31} \sin θ_{1} - o_{i} \sin θ_{2}}{\cos θ_{1} \cos θ_{1} \sin θ_{2}}) . \end{gathered}

(13)

D. Dynamics analysis

The dynamics analysis is to establish equations of the movements and to use the motion parameters of each joint to establish a dynamic equation to describe the relationship between the torque input of each joint and the motion output of the robot. In the process of rehabilitation treatment, doctors generally begin with passive low-speed movements and gradually translate to the method which applies an auxiliary force to a single joint. When the upper limb performs a compound movement, the positions of the limb’s terminal are only determined by the shoulder joint and the elbow joint and the wrist joint only serves to regulate the posture of the hand. Taking the clinical rehabilitation training method and the exoskeleton rehabilitation mechanism designed in this paper for reference, it can be derived that the abduction/adduction, flexion/extension movement of the shoulder joint, and the flexion/extension movement of the elbow joint play a major role in the process of realizing large-scale motion.

In this paper, based on the Lagrange equation, the kinetic equation is established using the variational method.³¹ The Lagrangian function L is defined as the difference between the kinetic energy of the system E_k and E_p,

L = E_{K} - E_{P} .

(14)

According to the Lagrange function L, the Lagrange equation of the system can be presented as

T_{i} = \frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{l}}) - \frac{\partial L}{\partial q_{i}} i = 1, 2, \dots, n .

(15)

The kinetic energy K of connecting rod i can be computed as follows:

K_{i} = \frac{1}{2} t r [\sum_{j = 1}^{i} \sum_{k = 1}^{i} \frac{\partial T_{i}}{\partial q_{j}} I_{i} {(\frac{\partial T_{i}}{\partial q_{k}})}^{T} {\dot{q}}_{j} {\dot{q}}_{k}],

(16)

where I_i is the moment of inertia at the center of mass.

Then, for a robot system with n links, the sum of kinetic energy is as follows:

K = \sum_{i = 1}^{3} K_{i} = \frac{1}{2} \sum_{i = 1}^{3} tr [\sum_{j = 1}^{i} \sum_{k = 1}^{i} \frac{\partial T_{i}}{\partial q_{j}} I_{i} {(\frac{\partial T_{i}}{\partial q_{k}})}^{T} {\dot{q}}_{j} {\dot{q}}_{k}] .

(17)

For a robot structure rotating around a fixed axis, the distribution of the mass of its rigid body is indicated by the inertial tensor. The inertia parameters of the robot include the inertia parameters of the mechanism and the transmission system. The inertia matrix is a 3 × 3 symmetric matrix, which represents the corresponding inertia product of the three rotational inertia and three axes, respectively,³²

I = (\begin{matrix} I_{x x} & - I_{x y} & - I_{x z} \\ - I_{y x} & I_{y y} & - I_{y z} \\ - I_{z x} & - I_{z y} & I_{z z} \end{matrix}) .

(18)

During the process of rehabilitation, the human–robot moves at the same time, so the quality of the patient’s upper limbs is an important factor affecting the dynamic characteristics. To verify the rationality of the designed mechanical structure, the body weight of the wearer is set as the ultimate bearing capacity of the exoskeleton robot. The relative mass of the upper arm is 2.60%, and the mass of the forearm and hand is 1.30% and 0.64%, respectively. According to the mass distribution data and the average weight of healthy men, the human upper limb model with mass parameters is imported into the ADAMS software to obtain the dynamic characteristics, which is exploited to simulate the influence of the patient’s upper limb in the process of rehabilitation. The principal moments of inertia of each joint of the man–machine system are obtained through the solving function of the ADAMS software, and the results can be seen in Table III.

TABLE III.

Main moments of inertia of each joint.

Joint	Centroid [x,y,z] (mm)	Mass (kg)
1	[162.083, −35.122, −180.471]	5.23
2	[190.703, 190.903, −308.783]	2.146
3	[45.363, 255.447, −314.387]	1.449

Joint	Centroid [x,y,z] (mm)	Mass (kg)
1	[162.083, −35.122, −180.471]	5.23
2	[190.703, 190.903, −308.783]	2.146
3	[45.363, 255.447, −314.387]	1.449

The gravity term coefficient D_i of each link can be represented as follows:

D_{i} = - \sum_{p = i}^{3} m_{i} g^{T} \frac{\partial T_{p}}{\partial T_{i}} r_{p} .

(19)

The position of the centroid and the mass of the robot system in a specific posture are obtained through simulation software, as shown in Table IV.

TABLE IV.

Location and mass of center of gravity.

Joint	Centroid [x,y,z] (mm)	Mass (kg)
1	[162.083, −35.122, −180.471]	5.23
2	[190.703, 190.903, −308.783]	2.146
3	[45.363, 255.447, −314.387]	1.449

Joint	Centroid [x,y,z] (mm)	Mass (kg)
1	[162.083, −35.122, −180.471]	5.23
2	[190.703, 190.903, −308.783]	2.146
3	[45.363, 255.447, −314.387]	1.449

The total kinetic energy of the structural connecting rod component of the 3-DOF shoulder joint rehabilitation robot can be solved as follows:

K_{b} = \sum_{i = 1}^{3} K_{i} = \frac{1}{2} \sum_{i = 1}^{3} tr [\sum_{j = 1}^{i} \sum_{k = 1}^{i} \frac{\partial T_{i}}{\partial q_{j}} I_{i} {(\frac{\partial T_{i}}{\partial q_{k}})}^{T} {\dot{q}}_{j} {\dot{q}}_{k}] .

(20)

When solving the total kinetic energy of the robot, the kinetic energy of the transmission device, such as the motor and the reducer, should also be considered,

K_{a} = \frac{1}{2} \sum_{i = 1}^{n} I_{a i} {\dot{q}}_{i}^{2} .

(21)

The total kinetic energy of the robot can be obtained from Eqs. (20) and (21),

\begin{align} K = K_{b} + K_{a} & = \frac{1}{2} \sum_{i = 1}^{3} t r [\sum_{j = 1}^{i} \sum_{k = 1}^{i} \frac{\partial T_{i}}{\partial q_{j}} I_{i} {(\frac{\partial T_{i}}{\partial q_{k}})}^{T} {\dot{q}}_{j} {\dot{q}}_{k}] + \frac{1}{2} \sum_{i = 1}^{n} I_{a i} {\dot{q}}_{i}^{2} . \end{align}

(22)

Assuming that the homogeneous coordinate of the bar i’s centroid in coordination system i is r_i, then its expression in coordination system O is $r_{i}^{c} = T_{i} r_{i}$ ⁠, so the total potential energy of the robot can be obtained as

P = \sum_{i = 1}^{3} P_{i} = - \sum_{i = 1}^{3} m_{i} g^{T} T_{i} r_{i} .

(23)

Substituting Eqs. (22) and (23) into L = K − P, the Lagrange function of the robot is obtained as

\begin{align} L = K - P & = \frac{1}{2} \sum_{i = 1}^{3} t r [\sum_{j = 1}^{i} \sum_{k = 1}^{i} \frac{\partial T_{i}}{\partial q_{j}} l_{i} {(\frac{\partial T_{i}}{\partial q_{k}})}^{T} {\dot{q}}_{j} {\dot{q}}_{k}] + \frac{1}{2} \sum_{i = 1}^{n} I_{a i} {\dot{q}}_{i}^{2} \\ + \sum_{i = 1}^{3} m_{i} g^{T} T_{i} r_{i} . \end{align}

(24)

Equation (24) is substituted into the Lagrange equation to obtain the dynamic equation of the robot as follows:

T_{i} = \sum_{j = 1}^{3} D_{i j} {\ddot{q}}_{i} + I_{a i} {\ddot{q}}_{i} + \sum_{j = 1}^{3} \sum_{k = 1}^{3} D_{i j k} {\dot{q}}_{j} {\dot{q}}_{k} + D_{i} i = 1, 2, \dots, n .

(25)

According to Eq. (25), the simplified dynamic equations of each joint of the robot are as follows:

\begin{align} T_{1} & = D_{11} {\ddot{q}}_{1} + D_{12} {\ddot{q}}_{2} + D_{13} {\ddot{q}}_{3} + D_{111} {\dot{q}}_{1}^{2} + D_{122} {\dot{q}}_{2}^{2} + D_{133} {\dot{q}}_{3}^{2} \\ + 2 D_{12} {\dot{q}}_{1} {\dot{q}}_{2} + 2 D_{113} {\dot{q}}_{1} {\dot{q}}_{3} + 2 D_{123} {\dot{q}}_{2} {\dot{q}}_{3} + l_{a 1} {\ddot{q}}_{1} + D_{1}, \end{align}

\begin{align} T_{2} & = D_{21} {\ddot{q}}_{1} + D_{22} {\ddot{q}}_{2} + D_{23} {\ddot{q}}_{3} + D_{211} {\dot{q}}_{1}^{2} + D_{222} {\dot{q}}_{2}^{2} + D_{233} {\dot{q}}_{3}^{2} \\ + 2 D_{212} {\dot{q}}_{1} {\dot{q}}_{2} + 2 D_{213} {\dot{q}}_{1} {\dot{q}}_{3} + 2 D_{223} {\dot{q}}_{2} {\dot{q}}_{3} + I_{a 2} {\ddot{q}}_{2} + D_{2}, \end{align}

\begin{align} T_{3} & = D_{31} {\ddot{q}}_{1} + D_{32} {\ddot{q}}_{2} + D_{33} {\ddot{q}}_{3} + D_{311} {\dot{q}}_{1}^{2} + D_{322} {\dot{q}}_{2}^{2} + D_{333} {\dot{q}}_{3}^{2} \\ + 2 D_{312} {\dot{q}}_{1} {\dot{q}}_{2} + 2 D_{313} {\dot{q}}_{1} {\dot{q}}_{3} + 2 D_{323} {\dot{q}}_{2} {\dot{q}}_{3} + I_{a_{3}} {\ddot{q}}_{3} + D_{3} . \end{align}

IV. SIMULATION ANALYSIS

A. Simulation analysis of kinematics

To verify the correctness of the kinematics of the rehabilitation robot, it is necessary to perform the movement of a set of motion angles with each joint of the robot. A motion is designed, which simulates the motion of eating food in the virtual prototype in ADAMS. The STEP function of each joint is defined as follows:

Joint 1: STEP(time, 0, 0, 12, −pi/4) + STEP(time, 10, 0, 30, pi/4).
Joint 2: STEP(time, 0, 0, 12, pi/9) + STEP(time, 12, 0, 30, −pi/9).
Joint 3: STEP(time, 0, 0, 12, −pi/6) + STEP(time, 12, 0, 30, pi/6).

The curve of the movement angle of each joint over time is as follows when the operating time is set to 30 s and the step number is set to 300. To analyze the movement of the terminal of the robot, the x axis represents the motion angle of each joint, and the y axis represents the displacement of the terminal. The motion simulation curves of the three joints are shown in Fig. 6. Figure 6(a) shows the curve of the movement angle of each joint changing with time; Figs. 6(b)–6(d) represent the relationship between the movement angle and end displacement of joints 1–3.

FIG. 6.

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Motion simulation curves of three joints: (a) three joints, (b) joint 1, (c) joint 2, and (d) joint 3.

As shown in Fig. 6, the displacement of the exoskeleton mechanism for shoulder joint rehabilitation training and the motion angle of each joint change smoothly without any leaps, which fits the prediction of positive kinematics. The simulation results can verify the correctness of the positive kinematics of the exoskeleton mechanism for shoulder joint rehabilitation training.

B. Simulation analysis of inverse kinematics

The inverse kinematics simulation of the exoskeleton rehabilitation mechanism for shoulder joint rehabilitation training uses MATLAB to carry out terminal motion trajectory planning. Then, the parameters’ solution of the joint is imported into ADAMS for simulation.

In the actual terminal trajectory planning, the initial position and the target position need to be given according to the specific work task. Under the constraints of the moving range of the joints, the trajectory is derived by interpolation. This paper designs a trajectory whose starting point and ending point are (0.00, 297.00, −300.00) and (−210.01, 148.51, −89.98), respectively, in which 46 points are inserted. Figure 7 shows the trajectory of the terminal of the exoskeleton mechanism for shoulder joint rehabilitation training.

FIG. 7.

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End motion trajectory.

The joint motion parameters can be obtained when determining the terminal trajectory of the rehabilitation robot at the same time. Then, the motion trajectory curve of each joint is derived by simulation. The curves are shown in Fig. 8, in which the abscissa represents time and the ordinate represents the joint variables of the three joints. It can be seen that the curves are smooth and there are no leaps, which meets the standard for the control performance. It verifies the correctness of the inverse kinematics analysis of the rehabilitation robot.

FIG. 8.

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Joint motion simulation trajectory.

C. Dynamic analysis

In this paper, the data processing and analysis of the target curve are performed with the application of the post-processing analysis module ADAMS/Postprocessor of ADAMS, which gives the dynamic relationship among the moments, the angular velocities, and the joint angles of the different joints of the rehabilitation robot. The simulation is performed after the operating time is set to 30 s and the step number is set to 300. Figures 9–11 show the change in the driving torque and angular acceleration of each joint.

FIG. 9.

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Variation curve of driving torque and angular velocity of joint 1.

FIG. 10.

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Variation curve of driving torque and angular velocity of joint 2.

FIG. 11.

View large Download slide

Variation curve of driving torque and angular velocity of joint 3.

By the observation of the simulation results and the analysis of the terminal trajectory, it can be concluded that as the posture of the rehabilitation mechanism changes, the torque of each joint changes smoothly without any leaps. This ensures the stability of the mechanism’s structure. It can also be concluded from the simulation diagram that the torque and angular acceleration of joint 2 are significantly greater than those of the other two joints. This is because the rotation movement of joint 2 and the gravity share the same direction, so joint 2 needs to overcome the gravity while the other two joints do not. Using the post-processing calculation of ADAMS, the maximum value of moment of joint 2 is 8.06 Nm. The rated torque of the motor selected for this joint is 1.5 Nm, and the reduction ratio of the reducer is 50. Therefore, the rated torque provided can fully meet the requirements. The dynamic simulation verifies the reliability of the virtual prototype and the correctness of the dynamic model, which can provide a theoretical basis for structural analysis and optimization of the robot and for later research on the control system.

V. SIMULATION EXPERIMENT

The control system of the rehabilitation robot is a nonlinear system, which is highly complicated and strongly coupled. There are always uncertainties, such as errors, various disturbances, and some unknown parameters in the possession of robot modeling. The controlling quality and performance of the shoulder rehabilitation robot are influenced by the environmental conditions and its complexity. The main task of the shoulder rehabilitation robot proposed in this paper is to drive the patient’s arm to perform various rehabilitation motions and each motion should be performed repeatedly, which requires high repetitive positioning accuracy.³³

In this section, the trajectory tracking of the shoulder rehabilitation robot is realized based on the closed-loop PD iterative learning control method. Iterative learning control can be described as follows: in a limited time interval $[0, T]$ ⁠, and the expected trajectory y_d(t) $(t \in [0 T])$ of the shoulder rehabilitation robot is given, the control input is found, which is the joint moment u_k(t) $(t \in [0, T])$ of the robot, so that the obtained trajectory y_k(t) tracks the expected trajectory y_d(t) as much as possible in the time interval $t \in [0, T]$ ⁠. To increase the stability of the system and the speed of the control process, the closed-loop PD iterative learning control method is used in this paper. Moreover, the control law and control block diagram are shown in Fig. 12.

FIG. 12.

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The block diagram of closed-loop PD type iterative learning control.

Closed-loop PD type control law: $T_{k + 1} (t) = T_{k} (t) + K_{p} (q_{d} (t) - q_{k + 1} (t)) + K_{d} ({q^{'}}_{d} (t) - {q^{'}}_{k + 1} (t))$ ⁠, where K is the gain indice, and K_p = 100 and K_d = 500.

The process of the iterative learning control algorithm can be summarized as follows:

Set k = 0, give the expected position $q_{d} (t)$ and initial control input.
The control input $T_{k} (t)$ is applied to the controlled object to start the repeated operation. At the same time, the sample of $q_{k} (t)$ is taken and restored.
After the repeated operation, the error output $e_{k} (t) = q_{d} (t) - q_{k} (t)$ is calculated, then the new control input $T_{k + 1} (t)$ is calculated and stored by the learning law.
Check the stopping condition of iteration. If the condition is satisfied, stop the operation; otherwise, set K = K + 1 to continue the execution until the stopping condition is met.

Because each group of rehabilitation motions needs to be repeatedly performed for at least ten times, the definition of the number of iterations is 10 in this paper. The following diagrams are the charts of the training effects and the errors of the three joints. As shown in Figs. 13–15, the red track is the expected track, and the blue track is the training track of each time. The red, blue, and green curves in Fig. 16 are the training error curves of joint 1, joint 2, and joint 3, respectively.

FIG. 13.

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Verification of the trajectory of joint-1.

FIG. 14.

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Verification of the trajectory of joint-2.

FIG. 15.

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Verification of the trajectory of joint-3.

FIG. 16.

View large Download slide

Three joint trajectory tracking error during iteration.

As can be seen in Fig. 16, the error decreases gradually, which means that the time of iteration increases, and improves the actual trajectory’s accuracy continuously. Gradually, the actual trajectory becomes closer and closer to the expected trajectory. Finally, the error output of the system converges to zero. Therefore, the system can achieve complete tracking and meet the requirements of rehabilitation training.

The errors of each joint after each iterative learning are shown in Table V. It can be seen that after the sixth learning, the errors of all three joints are less than 0.1. Iterative learning control can not only ensure that the system converges to a small error range, but also achieve complete tracking and satisfy with the rehabilitation requirements. It infers that the control method is effective and superior for the upper limb rehabilitation training.

TABLE V.

Iterative learning times and joint error.

Number of studies joint error	1	2	3	4	5	6	7	8	9	10
Joint 1	1.195	1.149	0.727	0.436	0.108	0.062	0.051	0.040	0.028	0.021
Joint 2	0.301	0.282	0.089	0.131	0.086	0.016	0.015	0.008	0.006	0.005
Joint 3	0.603	0.465	0.178	0.262	0.171	0.032	0.029	0.028	0.017	0.010

Number of studies joint error	1	2	3	4	5	6	7	8	9	10
Joint 1	1.195	1.149	0.727	0.436	0.108	0.062	0.051	0.040	0.028	0.021
Joint 2	0.301	0.282	0.089	0.131	0.086	0.016	0.015	0.008	0.006	0.005
Joint 3	0.603	0.465	0.178	0.262	0.171	0.032	0.029	0.028	0.017	0.010

VI. CONCLUSIONS

An exoskeleton rehabilitation mechanism for shoulder joint rehabilitation is proposed based on the anatomy of human upper limbs for the mechanism of movement and the range of motion. The mechanism is meshed with the gear of the reducer output shaft through the arc rack on the arc slide rail and is transmitted to the arc rack by the servo motor as the driving force to realize the internal/external rotation of the shoulder joint. A harmonic motor is used to drive and connect the flexion/extension and abduction/adduction of the shoulder joint. The variation curve of the joint angle and end trajectory with time and the relationship between the joint torque and time and angular velocity under given working conditions are obtained through the kinematics and dynamics simulation analysis of the rehabilitation mechanism. It can directly show the movement of each joint in the exoskeleton rehabilitation mechanism of shoulder joint rehabilitation movement. The rationality of the design of the exoskeleton rehabilitation mechanism is verified for shoulder joint rehabilitation. The trajectory tracking of the shoulder rehabilitation robot is realized based on the closed-loop PD type iterative learning control method. The simulation results show that the actual running trajectory can track the desired trajectory, improve the control quality of the system, and make the shoulder rehabilitation robot system obtain better tracking performance.

In future work, the structure will be optimized to make the rehabilitation mechanism of the shoulder joint exoskeleton more compact and anthropomorphic. In addition, in order to improve the control accuracy and interactive ability of the system, the human–machine interactive control approaches of the shoulder joint exoskeleton rehabilitation mechanism will be studied in depth. The design method in this paper will contribute to the further research of the rack and pinion mechanism. Moreover, the development of shoulder exoskeleton rehabilitation institutions will also promote the development of rehabilitation medical equipment in upper limb rehabilitation training.

ACKNOWLEDGMENTS

This work was supported, in part, by the National Natural Science Foundation of China under Grant Nos. 51875047 and 61873304, in part, by the Jilin Province Education Department Project under Grant No. JJKH20200658KJ, and, in part, by the Foundation of Jilin Province Science and Technology under Grant No. 20170307012YY.

The authors declare no conflict of interest.

The authors declare no potential conflict of interest with respect to the research, authorship, and/or publication of this article.

AUTHORS’ CONTRIBUTIONS

In this work, Z.P. and T.W. conceived and designed the experiments; J.Y. gave some constructive suggestions; X.Z. performed the experiments; Z.P. and S.L. analyzed the data; Z. W. guided the writing of the article and made some modifications; and Z.P. wrote the paper.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Author(s)

Design and analysis of shoulder joint exoskeleton rehabilitation mechanism based on gear and rack transmission

I. INTRODUCTION

II. STRUCTURAL DESIGN

A. Anatomy of the shoulder joint

B. Design of rehabilitation mechanism for shoulder joint exoskeleton

III. KINEMATICS AND DYNAMICS ANALYSIS

A. Kinematics analysis

B. Positive kinematics analysis of the shoulder joint

C. Inverse kinematics analysis of the shoulder joint

D. Dynamics analysis

IV. SIMULATION ANALYSIS

A. Simulation analysis of kinematics

B. Simulation analysis of inverse kinematics

C. Dynamic analysis

V. SIMULATION EXPERIMENT

VI. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHORS’ CONTRIBUTIONS

DATA AVAILABILITY

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Design and analysis of shoulder joint exoskeleton rehabilitation mechanism based on gear and rack transmission

I. INTRODUCTION

II. STRUCTURAL DESIGN

A. Anatomy of the shoulder joint

B. Design of rehabilitation mechanism for shoulder joint exoskeleton

III. KINEMATICS AND DYNAMICS ANALYSIS

A. Kinematics analysis

B. Positive kinematics analysis of the shoulder joint

C. Inverse kinematics analysis of the shoulder joint

D. Dynamics analysis

IV. SIMULATION ANALYSIS

A. Simulation analysis of kinematics

B. Simulation analysis of inverse kinematics

C. Dynamic analysis

V. SIMULATION EXPERIMENT

VI. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHORS’ CONTRIBUTIONS

DATA AVAILABILITY

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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