To facilitate understanding, a simplified model of the forearm affected by DRF was established according to the anatomical features of human upper limb. As illustrated in Figure 2A, the proximal part of forearm that closes to the elbow has little impact on the distal fracture, thus this region is basically regarded to be fixed. M₁, M₂, M₃ and M₄ are the muscles and ligament tissues that connect different bones together. In the lateral view, notably, the distal region of radius is separated in a form of dorsal tilt, the same as the distinctive feature of CF.

Based on the presented diagram of DRF, the force received by fracture end is further analyzed. In free condition, as shown in Figure 2B, the clearance of two fracture surfaces is nearly eliminated under the combined action of M₃ and M₄ after the treatment of closed reduction. The contact force between two separated radial bones N→ can be divided into a normal force (Nn→) and a tangential force (Nt→) in reference to the fracture line, which are given by:

where α denotes the inclined angle of fracture line. Theoretically, Equation (3) is supposed to be satisfied if the fractured part of distal radius is able to maintain a position stability without any fixation.

where f represents the friction coefficient between two fracture surfaces. By substituting Equations (1, 2) into Equation (3), the stable condition can be further simplified as:

Nevertheless, a large number of environmental factors could potentially affect the maintenance of distal radius and the growth of bone tissues, it is clinically improbable for such a long-term recovery process without any external protection. An auxiliary protector to the wrist is conventionally effective on preventing the closed radial bone from an unexpected re-displacement, here the force caused by fixation and acted on fracture end is equivalent to G→, as shown in Figure 2C.

Subsequently, the resultant forces induced by both the internal tissues and external fixation in the normal and tangential direction can be expressed by Equations (5, 6), respectively.

Therefore, the fundamental requirement for fractured radius to reach a steady fixation state is:

and then the mathematical relationship between the inclined angle of fracture and the applied forces can be deduced as:

Here, the maximum inclined angle of fracture end is denoted with α_max, then the following relationship can be obtained:

Compared Equation (5) with Equation (1), it is evident that the action of external fixation increases the overall normal force of fracture end, which correspondingly enhances the friction between fracture surfaces. According to Equations (4, 9), the fracture end is believed to be more stable under the limitation of external fixation, and it is possible for the fractured part to preserve in a steady condition even if the tangent value of fracture angle is larger than the friction coefficient.

According to analysis, the application of external fixation is conducive to prevent the closed fracture radius from another displacement, which is essential for the regeneration of bone tissues. However, the immobilization is not persistently demanded, moderate exercises of wrist are strongly recommended to relieve the muscle stiffness if the recovery of DRF is progressed smoothly (Valdes, 2009; Ikpeze et al., 2016). The requirements on both fixation and motion provide a theoretical basis for the design of assistive device in early rehabilitation stage.

Clinically, a long-term limitation to the wrist joint will probably lead to partial loss on the fundamental motion functions for ADLs. Therefore, when the bone tissues are regenerated and the fracture region is basically closed, it is of great importance to further stimulate and enhance the muscle strength. In fact, the resistance training is a favorable practice for rehabilitation and recommended by increased researches. Resistance training is based on the principle that muscles will be naturally activated when they are required to overcome a resistance force, the muscles are getting stronger if patient do the resistant movements repeatedly and consistently. Based on the requirement on simplicity and repeatability, some passive elements are generally adopted for exercise, such as the springs and elastic bands.

The kinematic models of wrist movement under both the natural and resistant conditions are illustrated in Figure 3, where O is the origin of coordinate system, Q is the wrist joint, P is the centroid of hand, P₁ and P₂ are the fixed points of spring module in hand and forearm, l₁ and l₂ are the lengths of forearm and hand, l₃ is the distance between wrist joint and the fixed end of spring module in hand, l₄ is the distance between forearm and the fixed position of spring module in forearm, d is the distance between wrist joint and the hand centroid, k is stiffness coefficient of the spring module, and θ is the rotated angle of wrist joint. The mass of entire hand part is set to be m, and the coordinates of P, P₁, and P₂ are denoted with (x₁, y₁), (x₂, y₂), and (x₃, y₃), respectively. Accordingly, the specific parameters of involved points can be expressed as:

where Equation (10) describes the position characteristics of hand, and the corresponding velocity equation can be further deduced as:

Subsequently, the overall movement velocity v is calculated by Equation (14), and the kinetic energy E_k of hand can be obtained, as described in Equation (15).

In addition, the gravitational potential energy E_p1 is expressed as follows:

while the other one in this system, the elastic potential energy related to spring module, is given by:

where Δx is the length increment of spring and can be expressed as:

where x₂₀, y₂₀, x₃₀, and y₃₀ are the initial coordinate values under the condition of θ = 0.

For any mechanical systems, conventionally, the Lagrangian mechanics can be described as the difference between kinetic and potential energies. According to previous discussion, the entire biomechanical system of wrist under a free condition involves only the kinetic energy and the gravitational potential energy, then the Lagrangian function of which is established as follows:

and then based on the principle of Lagrangian dynamics, the torque of wrist joint T₁ can be obtained as:

Furthermore, by taking the resistant effect of external device into consideration, the Lagrangian function is re-established as follows:

and the torque of wrist joint T₂ can be obtained as:

where T_e is calculated by:

Compared Equation (22) with Equation (20), it is evident that an extra torque T_e is generated under the application of spring resistance. To overcome a larger torque in the wrist joint, the muscles of forearm are required to consume more energies, which could ultimately contribute to increased strength and improved motion ability.

As presented in Figure 4, the designed AFD adopts a modularized structure, which is mainly comprised of three independent integrated parts. The two wearable modules of hand and forearm can be detached and substituted for personal customization, while the intermediate wrist module connects them together and is adjustable for basic wrist movement within the functional ROM.

The structural design and compositions of forearm and hand modules are shown in Figure 5. For the forearm module, the internal surfaces of ventral and dorsal components, which directly contacts with the human forearm, are shaped based on the acquired and processed 3D scanning contours of the right forearm of experimenter. This kind of reverse engineering was presented in many previous literature (Grski et al., 2019; Lin et al., 2016; Asanovic et al., 2019). The application of Voronoi holes not only improves the quality attribute and external appearance of AFD, but also contributes to diminish the possibility of dermatosis triggered by long-term wearing process to a certain extent. In addition, two platforms are designed for the installation and deployment of sensors, and the Velcro holes are used to fasten the ventral and dorsal components together with Velcro types. As to the hand module, the palmar part is equipped with the other installation platform of posture sensor, and the Velcro holes are configured for closed constraint between the palmar and dorsal components.

As shown in Figure 6, the wrist module is an intermediate part that integrates the hand and forearm modules into a coherent mechanical system, in which the two auxiliary rotation DOFs are achieved by the R/U and F/E adjusting mechanisms. The dorsal component of hand module is fixed with a trajectory slider of R/U mechanism, and the designed range of R/U movement is limited by an arc element. As for the F/E mechanism, it is configured to share an identical axis with the arc connector and the ventral component of forearm module. The rotary degree of F/E movement can be manually regulated under the loosen state, while the fixation of certain F/E posture is supposed to be configured as the tighten condition.

During the early treatment of rehabilitation, the fractured radius after closed reduction is strictly required to sustain a relatively static condition, thus the significance on fixation performance of the exoskeleton is greatly highlighted. In view of this, the design of AFD primarily adopts rigid structure and parts for ensuring the protection effect of wrist. However, due to the expansion of actual ROM and the impact of resistant force during strengthening exercise, the rigid components might give rise to an unpleasant experience for DRF patient. To solve this problem, a rigid-flexible coupled resistance training device (RTD) was proposed in this work, as shown in Figure 7.

Similar to the design of AFD, the main body of RTD includes a hand module and a forearm module. As to the hand module, notably, a flexible half-finger glove is added and stitched in the internal sides of palmar and dorsal components, which not only provides a rigid connection with the resistance module, but also increases the wearing comfortableness during exercise. More importantly, the previously displayed wrist module in AFD is canceled, instead a resistance module is attached to connect the two wearable modules. The resistance module is composed of four independent rope-spring kits, two of which are installed in the dorsal side of arm, while the others are distributed in the opposite side. One end of the rope-spring kit is fixed in the forearm module, and the other end is connected with the hand module, the direction of it is altered by the pulley mechanism, which also contributes to reduce the friction of rope during expansion and contraction.

Theoretically, which group of rope-spring kits is in application depends on the motion pattern of wrist. For example, the kit 1 and 2 receive pulling force when the DRF patient executes a flexion movement, while the kit 1 and 3 are tensioned if the radial deviation is performed. The resistant force is supposed to be increased along with the expanded motion degree of wrist, meanwhile the rope-spring kits distributed in the contrary side of wrist movement are bent and have no influence on the action. Therefore, by replacing the spring with different stiffness, the desired resistant force of RTD can be obtained.

As discussed before, the improvement of muscle strength can be realized by overcoming an additional torque provoked by the rope-spring kits. The adopted ropes are not easily deformed, hence the resistance force is mainly originated from the springs. As exhibited in Figure 7, the outlet position of rope in pulley mechanism is denoted as point A, and the fixed end of rope on the hand module is point B. In theory, the elongation of spring can be represented by the variation of distance between point A and point B. In this work, the workspace of hand module is analyzed using the modified Denavit-Hartenberg (D-H) method. The base frame {0} is attached in the forearm module, in which the Z₀ axis is coincide with the rotation axis of F/E movement. Here, the motion angles of wrist F/E and R/U are represented by θ₁ and θ₂, respectively. The associated D-H parameters for RTD are listed in Table 1.

In general, the homogeneous transformation matrix between two adjacent frames, namely {i−1} and {i}, is conventionally described as:

where Sθ_i, Cθ_i, Sα_i, and Cα_i represent sinθ_i, cosθ_i, sinα_i, and cosα_i, respectively. Therefore, by substituting the D-H parameters of RTD into Equation (24), the transformation matrix 01T and 12T can be obtained, then the transformation matrix between frame {0} and frame {2} is calculated by:

According to the discussion above, the point B of hand module in frame {0} is calculated by:

In terms of the obtained coordinate value of point B, the distance between A and B after a certain wrist movement is:

while the initial distance between A and B is:

then the actual extension value of spring can be expressed as follows:

For RTD, the resistance module includes four rope-spring kits, whose initial endpoint positions of A and B in base frame {0} are given by:

As mentioned before, the available range of F/E movement θ1∈[-75◦,70◦], and the range for R/U motion θ2∈[-20◦,35◦]. Based on previous analysis, the elastic deformations of four springs under different motion patterns can be calculated, the relationships between spring elongations and motion angles are exhibited in Figure 8.

In addition, the applied force of each group of two rope-spring kits can be calculated by:

where x is the spring elongation, k is the spring constant given by:

To investigate the influence of resistance levels on human muscle activity, two spring modules with different stiffnesses were employed in this work. According to Equations (32, 33) and the physical parameters of adopted extension springs listed in Table 2, the available resistant force of RTD for each motion patterns can be obtained. As presented in Table 3, the types of low and high resistance denote the conditions using springs with wire diameters of 0.8 and 1.0 mm, respectively.

As presented in Figure 9, the two exoskeleton devices are combined with respective data acquisition systems to evaluate their practical function. Notably, AFD and RTD are both equipped with two posture sensors, respectively distributing in the aforementioned installation platforms of hand and forearm modules. The required fixation for DRF could last a long time, thus a temperature module is attached in the forearm module of AFD, for monitoring the temperature variation of patient wearing the exoskeleton. The corresponding joint angle and temperature data are transmitted into a STM32 microcontroller, which is an information processing and exchanging center that communicated with computer based on a WiFi module. As to the RTD, the resistant effect is examined by an additional sEMG acquisition device, in which the electrodes (i.e., the red, green, and yellow ones) should be placed in the outer skin of forearm for bioelectric signal detection. The computer serves for data collection and visualization, the relevant joint angle and the cumulative exercise times will be calculated and displayed in real time, resulting to more intuitive observation for the patient and more quantitative supervision for the attending doctor.

In this work, the two groups of wrist movement were respectively performed for five consecutive times by experimenter, the posture data is recorded and the actual range of each action was calculated. As illustrated in Figure 11, the motion angle equals to zero when the wrist is at its neutral position, extension and ulnar deviation are defined as the positive direction, as opposite to the flexion and radial deviation. According to motion curves, it is apparent that the accessible ranges of AFD for flexion/extension and radial/ulnar deviation are all capable of covering the functional ROMs of wrist joint.

As for the elbow and shoulder joints, different thresholds were defined for the judgment on whether the basic range requirement of a motion behavior is satisfied or not. The employed thresholds for each motion patterns of elbow and shoulder are listed in Table 4. In this work, a continuous movement that successively exceeds the upper threshold and the lower threshold is regarded as one time of complete training action. To estimate the counting precision and sensitivity on training times, the experiments were deliberately configured by establishing conditions in which half of the total repetitive motions were correctly counted, while the other half were not. As illustrated in Figure 12, 10 times of reciprocating motion for each pattern have been conducted. According to the experimental results of each motion pattern, only the first five times of action can be effectively counted, while the last five are failed. The results are as expected, hereby the monitoring function of elbow and shoulder postures AFD is demonstrated to be reliable.

In contrast to previous studies that mainly focused on immobilizing fractured wrists using plaster splint or other fixation devices, the AFD introduces a novel, adjustable wrist exoskeleton mechanism. It provides stable protection for DRF while allowing limited joint movements to reduce muscle stiffness. Meanwhile, the AFD also offers effective motion data for rehabilitation exercises, contributing to better quantification of the active rehabilitation process in DRF patients.

The neuromuscular system of a human body includes plenty of muscles and nerves, every movement of the body is realized based on the communication from the brain to muscles. The muscular system is comprised of several specialized cells named muscle fibers, and the predominant function of them is for contractility. In general, the activities of muscle fiber can be detected and digitalized by the electromyography (EMG). As one of the EMG technology, surface electromyography (sEMG) has been extensively applied in wearable technologies (Xiao et al., 2024) for clinical diagnosis and functional supervision of muscle rehabilitation.

Here, the contraction activities of extensor carpi ulnaris (ECU) and flexor carpi ulnaris (FCU) during F/E movement, flexor carpi radialis (FCR) and flexor carpi ulnaris (FCU) during R/U movement, were systematically investigated. ECU is a forearm muscle extending from the lateral epicondyle of the humerus to the posterior border of the ulna, it plays a crucial role in wrist extension. FCU is a flexor muscle that originates from the medial epicondyle of the humerus to the posterior border of the ulna, it is mainly associated with wrist flexion and ulnar deviation. FCR is emanated from the medial epicondyle of the humerus and extends distally to the base of the second metacarpal bone, it contributes to radial deviation of the wrist, facilitating movement toward the thumb side. The selected three muscles are distributed in a superficial layer of forearm, and the bioelectric signals of this superficial muscle are more effectively and evidently to be detected on the skin. The sEMG signals of muscles under different experimental conditions were monitored in real time based on a sEMG acquisition kit presented in Figure 9.

For each experimental subject, the F/E and R/U movement of wrist under the condition without any resistance, the condition with low resistance and the condition with high resistance were required to be independently performed for five consecutive times. Compared with the early stage, the ROM for resistance training is limited to 55° for wrist flexion and 35° for wrist flexion, while radial deviation is limited to 15° and ulnar deviation to 25°. However, the directly detected bioelectric data may contain a significant amounts of noise and artifacts, which can negatively affect signal interpretation (Yin et al., 2024). Algorithmic intervention served as a popular strategy to reduce such distortions and enhance signal clarity. Here, the sEMG signals were further filtered by the wavelet transform algorithm, with the basis function of Daubechies (db4), the decomposition level of 2, and a sampling rate of 1,000 Hz. Wavelet transform offers an effective denoising method for electromyographic signals due to its time-frequency localization and multi-scale resolution, it removes noise while preserving signal details, enhancing signal-to-noise ratio and facilitating accurate analysis of EMG activity patterns. The eventually acquired sEMG signal data of one of the subjects are presented in Figure 13.

The integrated EMG (iEMG), a frequently-used feature extraction methods of sEMG signals, was employed as an evaluation index of the muscular contraction level. As described in Equation (34), iEMG is the mathematical integration of rectified EMG signals over a specified period of time, where T denotes the length of integration time window, and Data[t] is the EMG signal.

The calculated iEMG values of different muscles under three experimental conditions are shown in Figure 14. The results demonstrate an apparent growth of muscle activation with the increase of external resistance applied in each motion pattern for all subjects. As depicted in Figure 15, compared to the experimental condition without any constraint, the iEMG values exhibit distinct patterns under varying resistance levels. Specifically, during the F/E movement, the iEMG values increase by approximately 35% and 22% for FCU and ECU under low resistance, respectively, and by approximately 91% and 86% under high resistance, respectively. Similarly, during the R/U movement, the iEMG values increase by approximately 12% and 31% for FCR and FCU under low resistance, respectively, and by approximately 37% and 82% under high resistance, respectively.

According to the experimental results of RTD, the application of an resistant force is essential for the patients with diverse muscular strength. Compared to previous studies that mainly focused on the passive motion of wrist joint under the assistance of exoskeleton robots, the RTD proposed in this study aims to enhance the active rehabilitation ability of DRF patients, which is of great significance for helping them regain normal function of wrist movement. Notably, the springs employed on the RTD can be freely replaced with ones of different stiffnesses according to the baseline muscle strength of patients. For patients with weak muscle strength (e.g., old patients or the patients in the early stage of rehabilitation), it is recommend that the RTD be equipped with low resistance. To continue gaining benefits from strength enhancing activities, the resistance force is required to be progressively increased.

In this section, the application performance of AFD and RTD for wrist rehabilitation were experimentally evaluated. The research focus lies in validating the joint monitoring function and exploring the resistance training influence of exoskeleton devices. Therefore, although the data were obtained from healthy human subjects, the summarized principles related to joint motion and resistance training can be generalized to patients suffering from DRFs with varying characteristics. However, despite the validated advantages, there are still some limitations to the current exoskeleton devices. Due to the physiological differences between healthy individuals and fracture patients, further integration of the exoskeleton devices with force/torque sensors is conducive to gain a deeper understanding of the wrist dynamics in DRF patients. Moreover, the exoskeleton device can be further integrated with additional mechanisms to automatically impose physical intervention that limit joint movement when the monitored force/torque data reach safety thresholds defined by doctors based on clinical experience during joint motion. Meanwhile, there are also some potential issues that could affect the application of exoskeleton devices. In clinical practice, patients may experience issues such as poor comfort and compliance when wearing exoskeleton devices for extended periods. During the long-term application of the exoskeleton devices, it is necessary to develop more systematic safety protocol and monitoring plan in conjunction with the diagnoses of doctor.