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In order to develop a uterine manipulator for laparoscopic surgery, it is necessary to comprehend the surgical procedure and female reproductive system. In laparoscopic surgery such as uterine cancer surgery, the surgeon checks the internal conditions of the patient through a camera. During the entire surgical procedure, the uterus is continuously manipulated by a separate assistant with a uterine manipulator. The manipulator is inserted through the patient’s vagina up to the cervix, the cup seats on the cervix, and the tip extends through the cup into the uterus. To operate via the vagina, the manipulator design size should accommodate the dimensions of a human vagina. Accordingly, the average vagina length is 62.7 mm, within a large range (40.8-95 mm), and the narrowest diameter is 26.15 mm^[12]. The uterus has a mass of 270 g on average, and the manipulation angle range should be ± 45° during surgery as shown in Fig. 1. Since previously developed uterine manipulators require communication between the surgeon and staff, the control process is inefficient and, and it requires more of the surgeon’s valuable time to coordinate. Moreover, the conventional uterine manipulators have a rigid structure and may cause damage to the human body. In order to overcome the aforementioned limitations, a robotic uterine manipulator could be designed to satisfy the following requirements.

Fig. 1 
Uterine manipulator used in laparoscopic surgery with the manipulation angle range

• 1. Have a flexible structure with smooth bending movement to prevent injury.
• 2. The manipulator should be able to control the uterus position at an angle of ± 45° in 2-DOF
• 3. To lift and control the uterus, the tip of the manipulator must generate sufficient force.

The proposed robotic uterine manipulator consists of a series of PAMs that create bending movement and a linear servo motor which controls fore/aft translation. The configuration is shown in Fig. 2. The actuator consists of a series of three segments of McKibben PAMs separated by two 4 mm-thick disks, spaced equally. Each segment consists of three PAMs placed radially at 120° intervals around the circumference of the disks. The diameter of the disks and end effector is 25 mm, which is smaller than the vagina. Each PAM is located 8 mm from the center of each disk and the ends pass through an 8 mm diameter hole in the disks. The cup and tip are attached to the end of the actuator, which includes an end-effector, which fixes the manipulator to the uterus. Since translational movement is necessary during surgery, a linear servo motor (L-12R, Fergelli) is installed in the manipulator to allow 100 mm fore/aft translational movement of the uterus.

Fig. 2 
Configuration of the proposed robotic manipulator

The McKibben PAM is a popular artificial muscle device due to its superior properties, and a schematic diagram of the McKibben PAM is shown in Fig. 3. The McKibben PAM is composed of an inflatable rubber tube inside a braided mesh. They are attached to pneumatic fittings (LCN-0640-M5, PISCO) at each end. One end is capped and the other is connected to the air supply. When the rubber tube is pressurized with air, it expands radially and contracts longitudinally until the braided angle of the mesh reaches its designed limit. The PAM contraction occurs when the braided angle α is less than 54.7°. The contraction ratio of every McKibben PAM is different but does not exceed 35%^[13]. In this study, braided mesh with a diameter of 8 mm are used, and their specifications are shown in Table 1. With a simple PAM experiment, a contraction ratio of 25% is measured with 0.6 MPa of internal air pressure.

Fig. 3 
A schematic of the McKibben PAM and characteristics used in proposed robotic uterine manipulator

where l₀ denotes initial PAM length and l denotes contracted length.

The objective of mapping between actuator space and configuration space is to derive the relationship between PAM lengths (l₁,l₂,l₃) and arc parameters (R,θ_n,ϕ) of the constant curvature model.

The bending actuator is divided by disks spaced equally apart as shown in Fig. 5. To express the bending angle, the thickness of the disk is ignored. Instead, the calculated length of the PAM is reduced by the total thickness of the disks. The radius of curvature of a segment(R) is derived as follows^[14].

Fig. 5 
(a) Configuration of an actuator segment. (b) Diagram of the section to determine the bending angle θ

where d is the radial distance between the PAM and center of the disk, and l_i is the length of the ith PAM in a segment. Also, the rotation angle of the actuator(ϕ) is derived as

The bending angle of a segment(θ_n) is expressed by the curvature radius of each segment(R) and the length of an imaginary centerline of the segment(l_c), which is expressed as

where

To obtain relationship between configuration space and task space, a homogeneous transformation matrix is used. It is assumed the + z₀ axis of the base coordinate to be tangent to the base of the segment.

When the rotation angle of the segment(ϕ) is 0, the bending direction of the segment defines the + x₀ axis, and + y₀ axis is defined by right-hand rule, as shown in Fig. 6. In this situation, the actuator lies in the x-z plane, and bends with a constant radius of curvature(R) at a bending angle of a segment(θ_n). The position of the end point of a segment(P) is expressed as:

Fig. 6 
Diagram of a segment of constant curvature model in (a) 2-D space and (b) 3-D space

If we rotate the segment at an angle of ϕ about + z₀ axis, the homogenous transformation matrix can be derived as follows^[15].

where R_z and R_y represent rotation matrix about +z axis and +y axis respectively. To consider the thickness of the disks, translational matrix A is considered.

where t in matrix A denotes the disk thickness of 4 mm. Post-multiplying T by A can represent (i-1)th segment coordinate system, expressed as:

where c and s represent cos and sin respectively. Since the actuator is composed of three segments in series, the position of the end of the actuator and orientation can be computed as:

and position of end of the actuator is

where the workspace of the 2-DOF bending actuator can be derived and is shown in Fig. 7. The initial lengths of the PAM in a segment(l₁,l₂,l₃) are 27.33 mm, because of thickness (t) of two disks. At a 25 % contraction ratio with 0.6 MPa of internal air pressure, the maximum decreased length is 20.5 mm. The workspace is computed with decreasing the length of two PAMs. Decreasing all three PAMs prevents it from bending. It shows that the workspace is symmetric about the y-z plane and has order 3 rotational symmetry when viewed from the x-y plane, which is the front of the manipulator. The workspace of the manipulator would vary with a different sized end-effector, cup and tip.

Fig. 7 
Workspace of the manipulator. The length of the PAMs are decreased at a 0.2 mm decrement. (a) 3-D space (b) side view (c) top view (d) front view

To evaluate accuracy of the proposed kinematic model, computed workspace data from section 3.2 is compared with experimental position data of the actuator end and is shown in Fig. 10. The computed workspace is drawn with points where x coordinates are set at 0, and to compare, a video tracking method is utilized. The upper bending motion can be generated by pressurizing PAM 1 in Fig. 8(b), and downward movement is generated by pressurizing PAM 2 and PAM 3 at the same time. It shows larger positional errors as it moves away from the center, and the maximum deviation is measured at 17.65 mm. In addition, pressurizing two PAMs, which generates downward bending motion, generates more errors because of interference between two contracted PAMs. It affects the reduction of the contraction ratio^[18] and it is directly related to the workspace of the manipulator. It can be said that the error comes from a combination of modeling assumptions, the low stiffness of the actuator, and measurement errors.

Fig. 10 
The experimental result of video tracked position data of end of the actuator compared with computed workspace

The bending angle experiment was performed only in +y axis by pressurizing only PAM 1. The experimental configuration for the bending angle measurement and definition of bending angle of the manipulatorθ is shown in Fig. 11. The bending angle was measured with a 6-DOF gyro-accelerometer sensor (GY-521, HiLetgo) mounted on the top of the end-effector. The internal pressure was set from 0 to 0.6 MPa and the experiment was conducted 10 times.

Fig. 11 
The measurement of the bending angle θ with a sensor attached on the end effector

The experimental results of bending angle measurement is shown in Fig. 12. The maximum bending angle of the manipulator was 85.3 ± 2.02° with 0.6 MPa of internal pressure.

Fig. 12 
The bending angle of the manipulator due to the internal pressure

To measure the force generated from the manipulator, a blocked force test is conducted, and the test configuration is shown in Fig. 13. The manipulator is mounted on the platform and the load cell (CB1-K5, DACELL), which contacts the end of the actuator, is installed on a rotatable stage. Tip force is measured only in the +y direction to confirm that the manipulator generates enough force to lift and control the uterus. Since the mean weight of the uterus is 270 g, the actuator must generate at least 3 N. The rotatable stage fixed the load cell at 0° and the internal pressure of the PAM is increased from 0 to 0.6 MPa.

Fig. 13 
Configuration of tip force measurement. The stage for the load cell is set to 0°. (a) manipulator (b) rotatable stage (c) load cell

The experimental results are shown in Fig. 14. These results confirm that the manipulator generates maximum 5.45 ± 0.02 N on the tip with 0.6 MPa of internal pressure.

Fig. 14 
Tip force due to the pressurization of the PAM from 0 to 0.6 MPa

To measure the tip force for different bending angles of the manipulator, the load cell is rotated to 0° on the stage and the actuator engaged the load cell as shown in Fig. 15. The maximum applied pressure is 0.6 MPa.

Fig. 15 
Tip force measurement for different angles. The load cell rotates on the stage as pressure is increased from 0 to 0.6 MPa to measure the maximum force at the tip

The experimental results of tip force measurements for different actuator bending angles is shown in Fig. 16. The actuator generated 3.31 ± 0.05 N of tip force at 50° of bending angle, and a smaller bending angle generated a higher tip force.

Fig. 16 
Tip Force for different bending angles with a maximum pressure of 0.6 MPa