Four-degree-of-freedom rehabilitation training mechanism for upper limbs after breast cancer surgery

Abstract

The present application relates to postoperative rehabilitation training equipment technical field, disclose a kind of breast cancer postoperative upper limb four degrees of freedom rehabilitation training mechanism, comprising: installation support component, setting in training table;First shoulder joint drive module, setting in the installation support component, its output end connects shoulder joint connecting component;Second shoulder joint drive module, setting in the shoulder joint connecting component, its output end connects upper arm component;Elbow joint drive module, setting in the upper arm component, its output end connects forearm component;Hand drive module, setting in the forearm component, its output end connects hand training component.The present application is suitable for breast cancer patient postoperative rehabilitation upper limb training, can produce four training degrees of freedom, corresponding with the main rehabilitation action of human upper limb;Training mechanism can be adjusted length to adapt to different patients, improve the scope of application, still be provided with multiple limit grooves and limit rods, can limit the rotation range, improve safety.

Description

Technical Field

[0001] This invention relates to the field of postoperative rehabilitation training equipment technology, and in particular to a four-degree-of-freedom upper limb rehabilitation training mechanism after breast cancer surgery. Background Technology

[0002] After breast cancer surgery, sentinel lymph node biopsy, or axillary lymph node dissection, patients may experience limited shoulder joint movement, decreased upper limb muscle strength, pain, scar traction, soft tissue adhesions, and upper limb dysfunction, which can affect daily activities such as dressing, combing hair, picking up objects, and raising arms. Clinical rehabilitation typically involves exercises such as wall climbing, arm swinging, shoulder flexion and extension, adduction and abduction, elbow flexion and extension, and forearm or wrist rotation, tailored to the patient's recovery stage. These exercises are required to be slow, progressive, repetitive, and not exceed the patient's current range of motion. Traditional training relies primarily on patient-led practice or rehabilitation staff assistance. When patients practice independently, it is difficult to maintain consistent movement trajectories, repetitions, and speed. Rehabilitation staff assistance presents challenges due to high workload, insufficient repeatability, and difficulty in serving multiple patients simultaneously.

[0003] Existing upper limb rehabilitation robots are mostly geared towards multi-joint coordinated training after stroke or nerve injury. Some devices employ complex wearable exoskeletons, requiring repeated adjustments to multiple binding points. Furthermore, the matching process between the mechanism's rotation axis and the patient's physiological joint axis is cumbersome, making them unsuitable for post-mastectomy breast cancer patients whose primary needs are the recovery of shoulder joint range of motion and progressive upper limb elevation. In addition, existing adjustable arm length mechanisms often employ stepless sliding along long grooves. If insufficient locking is applied before training, the plates may shift position along the long groove.

[0004] In view of this, how to provide an upper limb training institution suitable for postoperative rehabilitation of breast cancer patients is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0005] The purpose of this invention is to provide an upper limb training device suitable for postoperative rehabilitation of breast cancer patients, in order to solve the problems existing in the prior art.

[0006] To achieve the above objectives, the present invention provides a four-degree-of-freedom upper limb rehabilitation training device after breast cancer surgery, comprising: Install the support components and set them on the training platform; A first shoulder joint drive module is mounted on the mounting support assembly, and its output end is connected to the shoulder joint connection assembly for driving the shoulder joint connection assembly to rotate in the plane containing the y-axis and z-axis; the shoulder joint connection assembly is capable of rotating along the plane containing the x-axis and y-axis. The second shoulder joint drive module is disposed on the shoulder joint connecting assembly, and its output end is connected to the upper arm assembly for driving the upper arm assembly to rotate in the plane containing the x-axis and z-axis. An elbow joint drive module is mounted on the upper arm assembly, and its output end is connected to the forearm assembly to drive the forearm assembly to rotate in the plane containing the x-axis and z-axis. A hand drive module is disposed on the forearm assembly, and its output end is connected to the hand training component for driving the hand training component to rotate relative to the forearm assembly.

[0007] Furthermore, the mounting support assembly includes: A vertical support frame, on which the first shoulder joint drive module is mounted; A detachable heightening component is detachably mounted on the training platform, and the vertical support frame is detachably mounted on the detachable heightening component.

[0008] Furthermore, the shoulder joint connection assembly includes: a first connecting plate, a second connecting plate, and a connector. The first connecting plate is fixedly connected to the second connecting plate via the connector and bolts and nuts. The first connecting plate is connected to the output end of the first shoulder joint drive module, and the second shoulder joint drive module is disposed on the second connecting plate.

[0009] Furthermore, the first connecting plate has a first arc-shaped limiting groove along its moving path, and the mounting support assembly has a first limiting rod corresponding to the first arc-shaped limiting groove. When the first connecting plate rotates to the first preset limit position, the first limiting rod connects with the end of the first arc-shaped limiting groove.

[0010] Furthermore, the upper arm assembly includes: a first upper arm plate and a second upper arm plate, the second upper arm plate having multiple sets of upper arm stop holes along its length direction, the first upper arm plate being bolted to a set of upper arm stop holes on the second upper arm plate; the second upper arm plate being connected to the output end of the second shoulder joint drive module, and the elbow joint drive module being disposed on the first upper arm plate.

[0011] Furthermore, the second upper arm plate has a second arc-shaped limiting groove along its moving path, and the second connecting plate is provided with a second limiting rod corresponding to the second arc-shaped limiting groove. When the second upper arm plate rotates to the second preset limit position, the second limiting rod connects with the end of the second arc-shaped limiting groove.

[0012] Furthermore, the forearm assembly includes: a first forearm plate and a second forearm plate, the second forearm plate having multiple sets of forearm stop holes along its length, the first forearm plate being bolted to a set of forearm stop holes on the second forearm plate; the second forearm plate being connected to the output end of the elbow joint drive module, and the hand drive module being disposed on the first forearm plate.

[0013] Furthermore, the second forearm plate has a third arc-shaped limiting groove along its moving path, and the first upper arm plate has a third limiting rod corresponding to the third arc-shaped limiting groove. When the second forearm plate rotates to the third preset limit position, the third limiting rod connects with the end of the third arc-shaped limiting groove.

[0014] Furthermore, the hand training component includes: A wrist connector is located at the end of the first forearm plate; A fixed support frame is mounted on the wrist connector; A rotating grip is provided, wherein the hand drive module is disposed on the fixed support frame, and its output end is connected to the rotating grip for driving the rotating grip to rotate relative to the first forearm plate. The rotating grip is located inside the fixed support frame.

[0015] Furthermore, it also includes: The host computer is electrically connected to the first shoulder joint drive module, the second shoulder joint drive module, the elbow joint drive module, and the hand drive module, respectively.

[0016] The present invention discloses the following technical effects: This invention is applicable to upper limb rehabilitation training for breast cancer patients after surgery. It can generate four degrees of freedom for training, corresponding to the main rehabilitation movements of the human upper limb. The second upper arm plate has multiple sets of upper arm stop holes along its length, and the second forearm plate has multiple sets of forearm stop holes along its length, which can adjust the length of the training mechanism to adapt to different patients and improve its applicability. The training mechanism is also equipped with multiple limit grooves and limit rods to limit the range of rotation and improve safety. The host computer can control the first shoulder joint drive module, the second shoulder joint drive module, the elbow joint drive module, and the hand drive module, thereby planning and controlling the patient's upper limb training movements and trajectories, which facilitates coordinated, smooth, and repeatable training. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is a schematic diagram of the structure of the present invention; Figure 2 This is a rear view of the structure of the present invention; Figure 3 This is a schematic diagram showing the interaction between the vertical support frame and the first connecting plate; Figure 4 This is a schematic diagram showing the fit between the upper arm assembly and the forearm assembly; The components include: 1. Installation support assembly; 101. Vertical support frame; 102. Detachable heightening component; 103. First limiting rod; 2. First shoulder joint drive module; 3. Shoulder joint connecting assembly; 301. First connecting plate; 3011. First arc-shaped limiting groove; 302. Second connecting plate; 3021. Second limiting rod; 303. Connector; 4. Second shoulder joint drive module; 5. Upper arm assembly; 501. First upper arm plate; 5011. Third limiting rod; 502. Second upper arm plate; 5021. Upper arm stop hole group; 5022. Second arc-shaped limiting groove; 6. Elbow joint drive module; 7. Forearm assembly; 701. First forearm plate; 702. Second forearm plate; 7021. Third arc-shaped limiting groove; 8. Hand drive module; 9. Wrist connector; 10. Fixed support frame; 11. Rotating grip. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0020] Those skilled in the art will understand that the term "comprising" as used in this application means the presence of the stated features, integers, steps, operations, elements, and / or components, but does not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof. It should be understood that when we say an element is "connected" or "coupled" to another element, it can be directly connected or coupled to the other element, or there may be intermediate elements present. Furthermore, "connected" or "coupled" as used herein can include wireless connections or wireless coupling. The term "and / or" as used herein includes all or any unit and all combinations of one or more associated listed items.

[0021] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0022] like Figures 1 to 4 As shown, this embodiment of the invention provides a four-degree-of-freedom upper limb rehabilitation training device after breast cancer surgery, comprising: Install support component 1 and set it on the training platform; The first shoulder joint drive module 2 is mounted on the mounting support assembly 1, and its output end is connected to the shoulder joint connecting assembly 3, which is used to drive the shoulder joint connecting assembly 3 to rotate in the plane containing the y-axis and z-axis; the shoulder joint connecting assembly 3 can rotate along the plane containing the x-axis and y-axis. The second shoulder joint drive module 4 is mounted on the shoulder joint connecting component 3, and its output end is connected to the upper arm component 5 to drive the upper arm component 5 to rotate in the plane containing the x-axis and z-axis. Elbow joint drive module 6 is mounted on upper arm assembly 5, and its output end is connected to forearm assembly 7 to drive forearm assembly 7 to rotate in the plane containing x-axis and z-axis. The hand drive module 8 is mounted on the forearm assembly 7, and its output is connected to the hand training component to drive the hand training component to rotate relative to the forearm assembly 7.

[0023] In this embodiment, the mounting support component 1 includes: A vertical support frame 101 is provided, and the first shoulder joint drive module 2 is mounted on the vertical support frame 101. The detachable heightening component 102 is detachably mounted on the training platform, and the vertical support frame 101 is detachably mounted on the detachable heightening component 102.

[0024] The function of the detachable height adjustment piece 102 is to adjust the height of the vertical support frame 101 to accommodate patients of different heights. When the detachable height adjustment piece 102 is not needed, the vertical support frame 101 can be directly connected to the training table.

[0025] In this embodiment, the shoulder joint connection assembly 3 includes: a first connecting plate 301, a second connecting plate 302, and a connector 303. The first connecting plate 301 is fixedly connected to the second connecting plate 302 through the connector 303 and bolts and nuts. The first connecting plate 301 is connected to the output end of the first shoulder joint drive module 2, and the second shoulder joint drive module 4 is disposed on the second connecting plate 302.

[0026] In this embodiment, the first shoulder joint drive module 2 is used to drive the shoulder joint connecting component 3 to rotate in the plane containing the y-axis and z-axis, corresponding to the adduction and abduction movements of the shoulder joint.

[0027] In this embodiment, the first connecting plate 301 has a first arc-shaped limiting groove 3011 along its moving path, and the mounting support assembly 1 is provided with a first limiting rod 103 corresponding to the first arc-shaped limiting groove 3011. When the first connecting plate 301 rotates to the first preset limit position, the first limiting rod 103 connects with the end of the first arc-shaped limiting groove 3011.

[0028] In this embodiment, the upper arm assembly 5 includes: a first upper arm plate 501 and a second upper arm plate 502. The second upper arm plate 502 is provided with a plurality of upper arm stop hole groups 5021 along its length direction. The first upper arm plate 501 is bolted to a group of upper arm stop hole groups 5021 on the second upper arm plate 502. The second upper arm plate 502 is connected to the output end of the second shoulder joint drive module 4, and the elbow joint drive module 6 is disposed on the first upper arm plate 501.

[0029] In this embodiment, the second shoulder joint drive module 4 drives the upper arm assembly 5 to rotate in the plane containing the x-axis and z-axis, corresponding to the flexion and extension movements of the shoulder joint.

[0030] In this embodiment, the second upper arm plate 502 is provided with a second arc-shaped limiting groove 5022 along its moving path, and the second connecting plate 302 is provided with a second limiting rod 3021 corresponding to the second arc-shaped limiting groove 5022. When the second upper arm plate 502 rotates to the second preset limit position, the second limiting rod 3021 connects with the end of the second arc-shaped limiting groove 5022.

[0031] In this embodiment, the forearm assembly 7 includes: a first forearm plate 701 and a second forearm plate 702. The second forearm plate 702 is provided with a plurality of forearm stop hole groups along its length direction. The first forearm plate 701 is bolted to a group of forearm stop holes on the second forearm plate 702. The second forearm plate 702 is connected to the output end of the elbow joint drive module 6, and the hand drive module 8 is disposed on the first forearm plate 701.

[0032] In this embodiment, the elbow joint drive module 6 drives the forearm assembly 7 to rotate in the plane containing the x-axis and z-axis, corresponding to the flexion and extension of the elbow joint.

[0033] In this embodiment, the second forearm plate 702 is provided with a third arc-shaped limiting groove 7021 along its moving path, and the first upper arm plate 501 is provided with a third limiting rod 5011 corresponding to the third arc-shaped limiting groove 7021. When the second forearm plate 702 rotates to the third preset limit position, the third limiting rod 5011 connects with the end of the third arc-shaped limiting groove 7021.

[0034] In this embodiment, the hand training component includes: Wrist connector 9 is located at the end of the first forearm plate 701; A fixed support frame 10 is mounted on the wrist connector 9; The rotating grip 11 and the hand drive module 8 are mounted on the fixed support frame 10. The output end of the rotating grip 11 is connected to the rotating grip 11 and is used to drive the rotating grip 11 to rotate relative to the first forearm plate 701. The rotating grip 11 is located inside the fixed support frame 10.

[0035] In this embodiment, the hand drive module 8 drives the rotating grip 11 to rotate around a rotation axis extending longitudinally along the patient's forearm; when the patient holds the rotating grip 11, the entire hand rotates with the rotating grip 11 to achieve forearm pronation and supination training.

[0036] In this embodiment, the plate structures such as the first forearm plate 701, the second forearm plate 702, the first upper arm plate 501, and the second upper arm plate 502 can all be provided with weight-reducing grooves. The weight-reducing grooves do not participate in the realization of the rotation action or the adjustment of the structural position.

[0037] In this embodiment, it also includes: The host computer is electrically connected to the first shoulder joint drive module 2, the second shoulder joint drive module 4, the elbow joint drive module 6, and the hand drive module 8, respectively. The first shoulder joint drive module 2, the second shoulder joint drive module 4, the elbow joint drive module 6, and the hand drive module 8 can be joint motors with reducers, absolute encoders, and power-off brakes. Each joint module has a built-in drive controller, which is connected to the host computer via a CAN bus. The host computer sends control commands, and the drive controller completes closed-loop position control of the corresponding joint module, feeding back the actual position, operating status, and fault status to the host computer.

[0038] The specific control method of the host computer is as follows: The host computer sets the minimum and maximum software position limits to the drive controllers of each drive module via the CAN bus. The drive controllers execute closed-loop position control and software position limits based on feedback from the absolute encoder. Mechanical and software limits work together for the three degrees of freedom of elbow flexion and extension, shoulder flexion and extension, and shoulder adduction and abduction; the wrist rotational degrees of freedom (forward and backward rotation) are limited to software position limits only.

[0039] Each drive module's drive controller monitors conditions such as overcurrent, overvoltage, undervoltage, overtemperature, encoder malfunction, communication malfunction, and position tracking error. When a fault is detected, the drive controller stops the drive output of the corresponding drive module and puts the power-off brake into a locked state; when the power is cut off, the power-off brake automatically enters a locked state to suppress free rotation of the output terminal.

[0040] To improve safety, the host computer interface is equipped with a stop training command, and combined with software position limits, drive module fault monitoring, power failure brake and three mechanical limits to form a layered protection system.

[0041] In single-joint training mode, the host computer selects one of the four drive modules to perform reciprocating motion, while the other joints maintain the target position; in wrist pronation and supination training mode, only the hand drive module 8 is controlled to reciprocate within the set angle range.

[0042] In the arm swing training mode, the host computer coordinates and controls the first shoulder joint drive module 2, the second shoulder joint drive module 4, and the elbow joint drive module 6 according to the preset shoulder and elbow joint timing sequence, so that the patient's upper limb can complete the swing training.

[0043] In the simulated wall climbing training mode, the host computer synchronously generates the target position sequence of shoulder joint flexion and extension, shoulder joint adduction and abduction, and elbow joint flexion and extension according to the preset path, so that the patient's hand gradually rises or falls along the simulated wall.

[0044] To reduce the impact during startup and shutdown, the host computer can use a fifth-order polynomial to generate the position trajectory. The position, velocity, and acceleration boundary conditions at the start and end of the training are set, the fifth-order polynomial coefficients are obtained, and then the target position at each moment is discretized according to the control cycle and sent to the corresponding joint module via the CAN bus.

[0045] Before training begins, the host computer reads the actual positions of the four absolute encoders, checks communication and fault status, and loads corresponding software position limits based on the affected side, training movements, and the patient's tolerable range of activity. During training, the target position is continuously compared with the actual position, and training is stopped if any abnormality is detected.

[0046] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," "axial," "radial," and "circumferential" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing this invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0047] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0048] In this invention, unless otherwise explicitly specified and limited, the terms "installation," "connection," "linking," and "fixing," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components, unless otherwise explicitly limited. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0049] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A four-degree-of-freedom upper limb rehabilitation training device after breast cancer surgery, characterized in that, include: Install the support assembly (1) and set it on the training platform; The first shoulder joint drive module (2) is disposed on the mounting support assembly (1), and its output end is connected to the shoulder joint connecting assembly (3) for driving the shoulder joint connecting assembly (3) to rotate in the plane containing the y-axis and z-axis; the shoulder joint connecting assembly (3) can rotate along the plane containing the x-axis and y-axis. The second shoulder joint drive module (4) is disposed on the shoulder joint connecting assembly (3), and its output end is connected to the upper arm assembly (5) for driving the upper arm assembly (5) to rotate in the plane containing the x-axis and z-axis. An elbow joint drive module (6) is mounted on the upper arm assembly (5), and its output end is connected to the forearm assembly (7) to drive the forearm assembly (7) to rotate in the plane containing the x-axis and z-axis. A hand drive module (8) is disposed on the forearm assembly (7), and its output end is connected to the hand training component for driving the hand training component to rotate relative to the forearm assembly (7).

2. The four-degree-of-freedom upper limb rehabilitation training device according to claim 1, characterized in that, The mounting support assembly (1) includes: A vertical support frame (101) is provided on which the first shoulder joint drive module (2) is mounted; A detachable heightening component (102) is detachably mounted on the training platform, and the vertical support frame (101) is detachably mounted on the detachable heightening component (102).

3. The four-degree-of-freedom upper limb rehabilitation training device according to claim 1, characterized in that, The shoulder joint connection assembly (3) includes: a first connecting plate (301), a second connecting plate (302) and a connector (303). The first connecting plate (301) is fixedly connected to the second connecting plate (302) through the connector (303) and bolts and nuts. The first connecting plate (301) is connected to the output end of the first shoulder joint drive module (2), and the second shoulder joint drive module (4) is disposed on the second connecting plate (302).

4. The four-degree-of-freedom upper limb rehabilitation training device according to claim 3, characterized in that, The first connecting plate (301) has a first arc-shaped limiting groove (3011) along its moving path. The mounting support assembly (1) is provided with a first limiting rod (103) corresponding to the first arc-shaped limiting groove (3011). When the first connecting plate (301) rotates to the first preset limit position, the first limiting rod (103) connects with the end of the first arc-shaped limiting groove (3011).

5. A four-degree-of-freedom upper limb rehabilitation training device according to claim 3, characterized in that, The upper arm assembly (5) includes: a first upper arm plate (501) and a second upper arm plate (502). The second upper arm plate (502) is provided with a plurality of upper arm stop hole groups (5021) along its length direction. The first upper arm plate (501) is bolted to a set of upper arm stop hole groups (5021) on the second upper arm plate (502). The second upper arm plate (502) is connected to the output end of the second shoulder joint drive module (4). The elbow joint drive module (6) is disposed on the first upper arm plate (501).

6. A four-degree-of-freedom upper limb rehabilitation training device according to claim 5, characterized in that, The second upper arm plate (502) has a second arc-shaped limiting groove (5022) along its moving path. The second connecting plate (302) has a second limiting rod (3021) corresponding to the second arc-shaped limiting groove (5022). When the second upper arm plate (502) rotates to the second preset limit position, the second limiting rod (3021) connects with the end of the second arc-shaped limiting groove (5022).

7. A four-degree-of-freedom upper limb rehabilitation training device according to claim 5, characterized in that, The forearm assembly (7) includes: a first forearm plate (701) and a second forearm plate (702). The second forearm plate (702) has multiple sets of forearm stop holes along its length. The first forearm plate (701) is bolted to a set of forearm stop holes on the second forearm plate (702). The second forearm plate (702) is connected to the output end of the elbow joint drive module (6). The hand drive module (8) is disposed on the first forearm plate (701).

8. A four-degree-of-freedom upper limb rehabilitation training device according to claim 7, characterized in that, The second forearm plate (702) has a third arc-shaped limiting groove (7021) along its moving path. The first upper arm plate (501) is provided with a third limiting rod (5011) corresponding to the third arc-shaped limiting groove (7021). When the second forearm plate (702) rotates to the third preset limit position, the third limiting rod (5011) is connected to the end of the third arc-shaped limiting groove (7021).

9. A four-degree-of-freedom upper limb rehabilitation training device according to claim 7, characterized in that, The hand training component includes: A wrist connector (9) is disposed at the end of the first forearm plate (701); A fixed support frame (10) is disposed on the wrist connector (9); Rotating grip (11), the hand drive module (8) is disposed on the fixed support frame (10), and its output end is connected to the rotating grip (11) for driving the rotating grip (11) to rotate relative to the first forearm plate (701). The rotating grip (11) is located inside the fixed support frame (10).

10. A four-degree-of-freedom upper limb rehabilitation training device according to any one of claims 1-9, characterized in that, Also includes: The host computer is electrically connected to the first shoulder joint drive module (2), the second shoulder joint drive module (4), the elbow joint drive module (6), and the hand drive module (8), respectively.

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