Design and Study of Particle Implantation Robot for Prostate Cancer Based on Imitating the Human Arm

Abstract

In the clinical treatment of prostate cancer, radioactive particle implantation surgery by a puncture robot is a safe and effective therapeutic approach. During surgery, patients are generally required to remain in the lithotomy position, within a limited and narrow working space for the complex procedure. However, the existing prostate particle implantation robots have poor motion flexibility, excessively large volume, complex inverse kinematics, and difficulty in control. Therefore, a prostate particle implantation robot based on imitating the human arm is proposed and designed in this paper. First, in accordance with the surgical requirements of prostate particle implantation and the bionics principle, the relevant physical parameters of the human arm are measured via X‐ray imaging, and the overall structure of the robot is designed. Kinematic analysis and the establishment of the spatial model for the main components of the robot—a robotic arm based on imitating the human arm (abbreviated as RAIHA robot)—are carried out to obtain the constraint conditions of each arm link in the working space and the constraint conditions between the joint angle of each arm and the driving force of the servo electric cylinder. Third, the objective optimization function for the RAIHA robot parameters is established based on the kinematic model and constraint conditions, and both the genetic algorithm (GA) and the NSGA‐Ⅱ‐based multiobjective optimization algorithm are used to optimize the RAIHA robot and its related structural parameters. Meanwhile, Matlab2017a/Simulink software is used to simulate the optimization results and verify their feasibility. Finally, to verify the effectiveness of the prostate particle implantation robot designed in this paper, a physical prototype is developed, and experiments are conducted to verify the rotational position of each arm joint, the influence of gravity torque on the angular position error of rotary joints, and the performance of terminal attitude control. The results show that the prostate particle implantation robot designed in this paper can not only well meet the task requirements of prostate implanting surgery in the human lithotomy position, but also overcome the influence of its own heavy moment, with the gravity torque self‐compensated by about 55.2%. Moreover, the robot exhibits excellent attitude adjustment performance and flexible movement, it ensures the stability of motion control for the terminal puncture needle. The robot structure designed in this paper can stably and accurately perform multiposture movements of the terminal puncture needle, reduce energy consumption, and meet the requirements of surgical tasks that demand a wide range of motion within the narrow yet effective maneuvering space at the human lithotomy site. The research work in this paper provides a theoretical reference for the design and development of puncture robots.

1. Introduction

Low dose rate (LDR) radioactive particle implantation for the treatment of early prostate cancer is a long term treatment method with the fewest side effects [1]. This method has been shown to be highly effective and safe, and it can serve as a viable alternative to surgical removal of the prostate. During the particle implantation surgery, doctors release particles in a targeted manner, the most common is iodine-I¹²⁵, it can destroy tumor cells within the prostate soft tissue while reducing the risk of unnecessary damage to nearby healthy tissue.

LDR radiation particle implantation therapy is typically an outpatient surgery, and it has several steps. First, before making a treatment plan, to determine the target point location of radioactive particles, the doctor places the anesthetized patient in the lithotomy position on the operating table and a puncture needle assisted system consisting mainly of a perforated mesh guide plate, a particle implant and an ultrasound probe is set up. Second, the doctor inserts an 18 G puncture needle through the perineum surface via the fixed mesh guide plate, directly to the prostate. Meanwhile, approximately 60–120 radioactive particles are released along a straight line under real-time ultrasound guidance. Finally, the puncture needle is removed, and the radioactive particles are left permanently in the prostate soft tissue to irradiate the tumor cells. Compared to other treatment such as external radiation therapy or prostatectomy, this internal radiation therapy provides a relatively short recovery time and minimizes toxicity to healthy tissue. However, the estimated accuracy of manual insertion of radioactive particles and nonprofessional prostate particle implantation robot has a large error [2]. The main reasons for this inaccuracy include: narrow operating space, poor flexibility of robot movement, and the doctor’s experience, among others. Therefore, it is highly necessary to develop a prostate particle implantation robot suitable for flexible movement in such a confined space, and this has also become a hotspot in current related research.

In recent years, several researchers have been developing robotic systems for prostate LDR radiation therapy. Yu et al. [3] and Podder et al. [4] designed single-channel and multichannel prostate particle implantation robots, both of which belong to rectangular coordinate structure. Salcudean et al. [5] designed a 4-degree-of-freedom prostate short-distance radiotherapy robot, which also belongs to the rectangular coordinate structure. The above robot systems can effectively reduce the movement and implant operation time of the target point. Vaida et al. [6] adopted a parallel structure, which not only has a guide plate, but also can freely adjust the insertion angle of the implanted needle. Guo et al. [7] designed a prostate puncture surgery robot with 5-degrees-of-freedom based on nuclear magnetic image navigation. The robot belongs to a parallel structure and can complete needle body posture adjustment within a small space. Zhang et al. [8–11] proposed and designed a cantilever prostate particle implantation robot that belongs to a series structure. A spring balancing mechanism is adopted in its rotating joint to achieve the self-weight balance of the upper and lower arms, reducing the interference of its own gravity torque. However, this robot can only realize the puncture motion in the horizontal direction, and can not complete the puncture task of changing the attitude angle. Li et al. [12] designed a pneumatic parallel prostate seed implantation robot. To realize the attitude adjustment of the puncture needle, the robot carried out the lifting parallel structure of the terminal particle implant in front and back shear fork, so as to realize the pitching attitude adjustment of the terminal puncture needle. However, the robot cannot adjust the roll and yaw attitude of the needle.

Through the analysis of existing prostate particle implantation robot structures, it can be seen that the main configurations are rectangular coordinate type, parallel type, and series type. However, there are distinct advantages and disadvantages among these structural categories. The motion flexibility of the prostate particle implantation robot with rectangular coordinate structure is poor, it cannot adjust the spatial attitude of the terminal needle body, and the volume is huge. The kinematics of parallel robots are complex, control is difficult, and the spatial attitude adjustment of the puncture needle is limited. However, if the attitude adjustment requirements of the surgical space are satisfied, the volume of the structure needs to be increased during the design process. The series structure enables large-range spatial motion and attitude changes; however, each rotating joint must implement complex mechanical gravity compensation to support the terminal-load (the weight of the particle implant device). Such compensation mechanisms are difficult to control accurately, resulting in increased structural volume. Therefore, although the above various structural types of prostate particle implantation robots can theoretically assist doctors to complete minimally invasive surgery for prostate particle implantation, but the main problems still exist are too large size and volume, poor movement flexibility, high control complexity, and few considerations on the prostate particle implantation robot’s ability to complete the prostate particle implantation task in the limited and narrow space of human lithotomy site and meet the stability of joint motion. Although many researchers adopt certain methods to overcome their own shortcomings in the above structural types (In addition to rectangular coordinate structure can not overcome the terminal attitude can not adjust the inherent shortcomings), new shortcomings will appear in the process (For example, in order to overcome the disadvantages of series and parallel structures, Vaida and Plitea [13] combined them and proposed a hybrid prostate puncture robot. Although the structure has their common structural advantages, such as good motion stability and high precision of puncture positioning; but it also leads to a particularly small range of space movement and terminal attitude change, and the fuselage is large, and the control operation is difficult). This is also the fundamental reason why relevant robots cannot operate in clinical trials at present. So, based on the research of the above scholars, this paper designs a particle implantation robot for prostate cancer base on imitation human arm. Its innovation lies in the fact that this robot adopts the series structure base on human arm, which not only enables its end to achieve a wide range of flexible movement in the limited and narrow space of the human lithotomy site, but also completes the “one-point-multiple-needle surgical task.” Moreover, the overall structure of this robot is compact, there is no need to carry out complex mechanical gravity torque compensation, and it meets the requirements of prostate particle implanting robot for safe operation and reduced injury.

The research process of this paper is as follows: First, the requirements of prostate particle implantation surgery are analyzed to determine its operable space range. Meanwhile, the bionics principle is used to measure and extract the human arm features, and the robotic arm based on imitating the human arm (RAIHA) robot, the terminal particle implantation device, and the space freedom frame are designed. Second, kinematics analysis is carried out and a space model is established for the main body of the robot (RAIHA robot), the inverse kinematics of its terminal, the motion trajectory equation, the constraints of the working space of each arm link, and the constraints between the joint angle of each arm and the driving force of the servo electric cylinder are obtained. Third, genetic algorithm (GA) and NSGA-Ⅱ multiobjective optimization algorithm are used to optimize the length of each arm and determine its structural design scheme. In addition, the optimization results of the RAIHA robot are simulated and verified to verify its feasibility. Lastly, by establishing the physical prototype of the robot, the verification experiment of the rotating position of each arm joint, the influence experiment of the rotating joint gravity torque and the control performance experiment of the terminal pose are carried out. The experimental results verify the effectiveness of the particle implantation robot for prostate cancer based on the imitating the human arm designed in this paper. The results of the research in this paper provide an important reference for the structural design and improvement of the prostate particle implantation robot.

2. Operable Space Analysis and Surgical Requirements

Operable space analysis is critical for the structural design of prostate particle implantation robots. The operable space will limit the size parameters and spatial layout of the robot structure. Since the actual operable space range varies among individuals, the analysis is based on mean value calculations to ensure universality. During subsequent structural design, the space occupied by the robot both at rest and in motion will have a margin relative to the operable space to ensure that there is no interference between the robot and the patient, and to ensure personal safety.

When a doctor uses the prostate particle implantation robot for radioactive particle implantation surgery, the patient generally assumes the lithotomy position [10], in Figure 1. During the operation, the angle between the spine and the thigh must be 100°–120°, both legs abducted 80°–110°, and the knee joint flexed 90°–130°.

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In this way, the entry point of the puncture needle can be fully visualized, thereby providing an optimal surgical field, enhancing the accuracy of particle implantation, and facilitating the doctors’ operation of the robot [14]. However, the lithotomy space is also very limited and narrow, and making it challenging for doctor to manually operate the robot for flexible attitude adjustments.

Meanwhile, the needle entry point of prostate particle implantation surgery is through the perineum region, combining this with the characteristics of human lithotomy position, the robot operable surgical space when the patient is in this position, as shown in Figure 2.

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Surgical safety, operational convenience and treatment efficacy are the fundamental prerequisites for robotic assisted operations. Therefore, by integrating the operable space analysis, the following surgical requirements should also be met during optimizing the structure of the analysis robot:

• 1.

  During surgery, it is necessary to ensure that the prostate particle implantation robot can enable the puncture needle to perform a wide range of spatial movements within a limited and narrow operational workspace (Figure 2, including: rough positioning of the needle entry mechanism, adjustment of the puncture angle, and attitude adjustment between the needle and the soft tissue target point), and ensure the stability of the movement of each joint of the robot without interference from gravity torque.

• 2.

  The robot’s components should be easily disassembled and assembled to facilitate preoperative sterilization, as well as the repair or replacement of parts.

• 3.

  The prostate brachytherapy robot is tasked with implanting radioactive particles into the patient’s prostate soft tissue, and thus requires a puncture mechanism fitted with a puncture needle.

• 4.

  The robot is required to be easy to operate and capable of achieving aseptic isolation during surgeries between the internal and external environments. Additionally, it should accurately simulate the doctor’s manual operations and inherit surgical expertise.

According to the surgical requirements for prostate brachytherapy, this paper designs and optimizes the overall structure of the prostate particle implantation robot.

3. Structure Design and Kinematics Analysis

The structural design of the prostate particle implantation robot follows the modular design concept [15], and the robot is divided into a RAIHA robot, a terminal particle implantation device and a two-degree-of-freedom spatial frame. The motion relationship between them is completely decoupled, which reduces the difficulty of control. Among them, in the design part of a RAIHA robot, a series structure is selected based on the analysis in the introduction. Meanwhile, to address the shortcomings of the series structure, the bionic principle is adopted in the structural design of the RAIHA robot [16].

3.1. Structure Design of a RAIHA Robot

3.1.1. Bionic Principle of a RAIHA Robot

After billions of years of evolution, organisms in nature have the ability to adapt to changes in the environment, thereby enabling them to survive and develop. Bionics focuses on natural organisms as the object of emulation and research, leveraging their morphological, functional, structural, and coloration characteristics to fulfill specific tasks [17–19]. Among advanced primates, the human arm has been perfectly evolved over a long period of production practice, and has a unique series structure that can overcome gravity torque interference to complete precise, smooth movement. Research on the structural characteristics of the human arm can provide a theoretical basis for the design of both the surgical robot suitable for prostate brachytherapy and related minimally invasive puncture robots.

This paper primarily imitates the structure and function of the human arm, combining with the characteristics of the doctor’s hand-held prostate particle implantation in Figure 3, the physical parameters of a large number of adult male and female arms are observed and measured, and then the corresponding physical information is obtained. However, complete structural bionics is not feasible at present, so this paper starts with the characteristics structure and key parameters of the human arm and designs a RAIHA robot.

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3.1.2. Configuration Analysis of a RAIHA Robot

The RAIHA robot designed in this paper adopts a serial structure. As analyzed in Chapter 1, although the serial structure offers advantages such as a large spatial movement range and flexible posture transformation, it also has inherent drawbacks including poor motion stability. Therefore, it is necessary to analyze the configuration of the robotic arm in the structure design, in Figure 4. Where, the blue identifier indicates the driver mode, the robotic body coordinate X indicates the vertical direction, Y indicates the side swing, and Z indicates the travel direction.

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As can be seen from Figure 3, the configuration of the doctor’s handheld particle implantation device belongs to the elbow structure, which mainly has three configurations as shown in Figure 4. These configurations can achieve a large range of flexible movement in space with fewer and shorter linkages. Configuration 1 is a series structure of the rotary motor directly drive connecting rod [20], the structure has the advantages of flexible motion and simple control, but the stability of motion and the bearing capacity of the terminal are poor. Configuration 2 is a series structure of two groups of parallelograms driven by a rotating motor [10], the structure can install the rotary motor of the upper arm joint and the lower arm joint on the upper end of the body, and the gravity of the motor driven by the lower arm joint can be transferred to the base. Compared with configuration 1, it reduces the moment of inertia around the joint of the upper arm, and the two sets of parallelogram structure improves the motion stability to a certain extent, but it also leads to poor motion flexibility and difficult control.

Configuration 3 is the series structure of cylinder drive, the structure not only has the advantages of configuration 1 and configuration 2, but also overcomes the disadvantages of poor motion stability in the series structure itself, which is caused by the rotation inertia of the joint and the gravity torque disturbance. In combination with Figure 3, the performance advantages of configuration 3 are further explained by taking the motion characteristic structure of the forearm joint when a person is holding the instrument as an example: in this movement, the upper arm bone, the lower arm bone, and the tendons of the upper arm form a triangular structure ΔD₀E₀F₀, which can effectively overcome the interference of gravity torque and keep the stability of torque transmission and displacement motion. In addition, since the motion structure between the shoulder arm and the upper arm and the forearm and the wrist is also triangular [21], this paper focuses on analyzing this configuration. Meanwhile, the characteristic structure of the prostate particle implant is observed when the doctor held the device, and the length of each arm segment and the angle of each joint of the human arm (male/female) are measured by the measuring tool.

3.1.3. Structure Analysis of a RAIHA Robot

Based on Figures 3 and 4c, this paper designs the structure that RAIHA robot, as shown in Figure 5. Each arm connecting rod is connected in series via rotating joints, with a total of 5° of freedom, including the shoulder joint, upper arm joint, forearm joint, wrist pitching joint, and wrist yaw joint. This configuration enables all-round flexible movement in three-dimensional (3D) space and realizes the simulation of doctors’ puncture surgical operations.

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The shoulder arm joint controls the RAIHA robot to perform yaw movements, while the upper arm joint and lower arm joint control the robot to perform forward-backward and lifting movements. The three joint arms constitute the main part of the RAIHA robot. The wrist joint controls the terminal particle implantation device to perform the pitch and yaw rotation. Each rotary joint is driven by a high-torque servo electric cylinder and installed in a manner similar to human tendons. This configuration can form a driving triangle with adjacent connecting rods, including the lateral swing triangle, upper arm triangle (ΔABC), lower arm triangle (ΔDEF), and wrist triangle. This driving structure can not only enable the RAIHA robot to have a wide range of spatial movement and multiattitude flexible motion, but also ensure the stability of the movement. In addition, each servo electric cylinder is equipped with pull rod displacement sensor and pull pressure sensor, which can detect changes in force and displacement at the terminal of the particle implantation device, as well as external disturbances to the robotic arm. This enables hybrid force-position control and ensures the accuracy and safety of particle implantation surgery.

As the robotic arm constitutes a key structural component of the prostate particle implantation robot system, significantly influencing its overall performance, this paper focuses on the analysis and research of a RAIHA robot.

3.1.4. Bionic Object and Mark Measurement Experiment

Since the bionic objects of the marker measurement experiment are the physical parameters of the arms of Asian adult males and females (adult doctors who actually operate), this paper randomly recruited a total of 20 clinicians from the urology department of Hailun People’s Hospital and Hospital of Traditional Chinese Medicine of Heilongjiang Province (10 men and 10 women) as subjects. Their heights range from 155 to 185 cm, and weight ranges from 45 to 95 kg.

The marker measurement experiment mainly collected the maximum and minimum values of the lengths of their shoulder, upper arm, and lower arm segments. Meanwhile, the average motion angles of the shoulder arm joint, upper arm joint, lower arm joint, and wrist joint of the subjects should also be measured. In addition, to accurately measure the length of each arm segment and the range of motion for each joint, the test method is to first using X-ray to acquire arm bone imaging [22], followed by applying a visual image recognition and processing algorithm to calibrate and measure human arm physical parameters under imaging [23], as shown in Figure 6. Where, L₀ in a is the length of the shoulder arm (the length of the shoulder joint to the upper arm joint), L₁ in b is the length of the upper arm (the length of the upper arm joint to the lower arm joint), and L₂ in c is the length of the lower arm (the length of the lower arm joint to the wrist joint).

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After the marker measurement experiment on the 20 subjects, the obtained measurement results are presented in Table 1 (values are rounded to integers). Among them, the rotation angle of the shoulder arm joint toward the body is 15°, and the rotation angle away from the body is 45°. These data results provide a reference for the further optimization of the RAIHA robot’s structure.

Table 1 Measurement results of physical parameters of human arm.

Physical parameters	Length (min)	Length (max)	Angle mean range
Shoulder arm (mm)	27	46	—
Upper arm (mm)	166	297	—
Lower arm (mm)	153	231	—
Shoulder arm joint (°)	—	—	60
Upper arm joint (°)	—	—	120
Lower arm joint (°)	—	—	86
Wrist pitch angle (°)	—	—	65
Wrist yaw angle (°)	—	—	62

3.2. Design of Terminal Particle Implantation Device

During surgery, the terminal particle implantation device is a mechanical device used to continuously implant radioactive particles into the target point of the lesion. According to the surgical characteristics of radioactive particle implantation, its structural design needs to meet the following four requirements:

• 1.

  The device is designed to achieve the positioning and rotation function of the inner/outer needle.

• 2.

  Featuring a continuous particle implantation function, the system shortens the operation time.

• 3.

  The terminal particle implantation device features a compact size and light weight, thereby reducing the movement inertia of the entire machine and improving its motion flexibility.

• 4.

  It can ensure the accuracy and safety of particle implantation surgery.

Accordingly, the terminal structure for particle implantation designed in this paper is illustrated in Figure 7.

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As shown in Figure 7, the device is installed on the terminal of a RAIHA robot, and it uses a bottom plate as the main frame to connect components. Among them, two stepping motors drive and control the translational movement of the inner and outer needles, thereby completing the puncturing and particle implantation tasks. A small servo motor is used to drive the rotational movement of the outer needle; in addition, the connecting part of the outer needle is equipped with a displacement sensor and a 6-DOF force sensor, which enables force-position hybrid control for the puncturing task and ensures both positional control accuracy and puncturing safety. To enable flexible movement of the needle in the terminal particle implantation, the parts are designed to be mounted on a base with dimensions (length × width: 300 mm × 60 mm).

The human arm has seven degrees of freedom. Among these, the upper arm (articulated with the shoulder) and the lower arm (articulated with the upper arm) both possess rotational freedom around their respective bones (axes); since these rotational movements serve the same function, one of the two degrees of freedom is considered redundant. Meanwhile, in the actual operation process, doctors mainly reduce puncture resistance and needle body deflection by rotating puncture, so as to improve puncture accuracy. Therefore, to simplify the structure and reduce control complexity, it is necessary to design a rotational puncture device for implementing the rotational puncture task in clinical procedures. The external needle rotation drive mechanism and internal needle rotation drive mechanism are illustrated in Figure 7.

3.3. Design of Space Double Degree of Freedom Robotic Frame and Overall Structure

As shown in Figure 8, the overall structure of a particle implantation robot for prostate cancer base on imitation human arm, has a total of 11° of freedom. In the Figure 8, the part marked by the red line is the space double degrees of freedom robotic frame. Among them, the vertical slide table and the vertical motor cylinder are responsible for the large-scale adjustment of the vertical direction, and the translation slide table can realize the large-scale spatial adjustment of the forward or left and right direction of a RAIHA robot through the transverse and longitudinal installation.

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Meanwhile, the robot’s overall structure is primarily constructed from aluminum material. ANSYS Workbench simulation software is utilized to optimize and analyze the entire robot structure, enabling it to meet both force requirements and overall lightweight demands while ensuring favorable dynamic driving performance.

3.4. Kinematic Analysis of RAIHA Robot

Since the free and flexible movement of the robotic arm is a prerequisite for the prostate particle implantation robot to achieve various posture adjustments and precise puncture within the confined space of the human lithotomy position, it is essential to determine the dimensions and connection positions of the robotic arm’s transmission components. Therefore, this paper conducts kinematic analysis and establishes a spatial model to provide a theoretical basis for subsequent solving processes.

3.4.1. Establishment of Kinematic Equations

The establishment of a robotic arm kinematic model constitutes a conventional analytical method for realizing the mutual transformation between the rotational joints and the terminal position of each arm segment. Therefore, the D–H method and homogeneous transformation matrices are employed to describe the pose relationships between adjacent links [24].

Since the main body of the RAIHA robot consists of the shoulder arm, the upper arm, and the lower arm, which drive the wrist joint movement, the force received by the puncture needle is also transmitted to each joint of the main body through the wrist joint. While the wrist joint is only responsible for driving the terminal particle implantation to control and adjust the pitch and yaw angle according to the measurement results shown in Table 1. Therefore, this paper only analyzes the kinematics of the main part (3°-of-freedom).

The D–H connecting rod coordinate system is established as shown in Figure 9. Among them, the base coordinate system O₀X₀Y₀Z₀ is established on the shoulder arm joint, the shoulder arm joint coordinate system O₁X₁Y₁Z₁ is based on the base coordinates, coordinate systems of the upper arm, lower arm, and wrist (O₂X₂Y₂Z₂, O₃X₃Y₃Z₃, and O₄X₄Y₄Z₄) are built on the end of the previous link, respectively. Meanwhile, combined with the initial angle position of the human hand-held terminal particle implantation device as ∠O₁O₂O₃ = −150°, ∠O₂O₃O₄ = 90°, and ∠O₃O₄O₅ = 150°.

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In Figure 8, the Z_i axis of each arm joint coordinate system is along the direction of the joint axis, and the X_i axis is along the direction of the connecting rod, where i = 1 – 4, and the wrist coordinates are parallel to the forearm coordinates. Combined with the bionic structure of the human arm, the connecting rod transformation parameters of the RAIHA robot are shown in Table 2. In addition, the variation range of connecting rod length and joint angle of each arm should be optimized based on the working space and the measurement results of physical parameters of the human arm in Table 1.

Table 2 D–H coordinates parameters of RAIHA robot.

Connecting rod, i	Connecting rod length, ai − 1 (mm)	Joint angle, αi − 1 (°)	Joint distance, di (mm)	Range of joint angles, θi (°)
1	0	0	0	θ1
2	a1	−90	0	θ2
3	a2	0	0	θ3
4	a3	0	0	0

According to the kinematic coordinates established in Figure 6 and the parameters defined in Table 2, the general formula for the transformation matrix of the wrist and each link coordinate system relative to the base coordinate system can be derived using the Paul [25] transformation method. 1 $\begin{array}{}\begin{array}{c}{}^{i-1}{T}_i=\left(\text{Rot}\left(z,\theta \right)\times \text{Trans}\left(00,,d\right)\times \text{Trans}\left(a,00,\right)\times \text{Rot}\left(x,\alpha \right)\right)\\ =\left[\begin{array}{cccc}\cos {\theta }_i& -\sin {\theta }_i& 0& a_{i-1}\\ \sin {\theta }_i\cos {\alpha }_{i-1}& \cos {\theta }_i\cos {\alpha }_{i-1}& -\sin {\alpha }_{i-1}& -\sin {\alpha }_{i-1}{d}_i\\ \sin {\theta }_i\sin {\alpha }_{i-1}& \cos {\theta }_i\sin {\alpha }_{i-1}& \cos {\alpha }_{i-1}& \cos {\alpha }_{i-1}{d}_i\\ 0& 0& 0& 1\end{array}\right]\end{array},\end{array}$

Where ^i − 1T_i is the coordinate position transformation matrix of connecting rod a_i relative to connecting rod a_i − 1, where i = 1–4.

As shown in Table 2, the D–H parameters of each connecting rod i are brought into the Equation (1), and the coordinate transformation matrix of each connecting rod a_i relative to a_i − 1 is obtained, respectively. Then, the transformation matrix of each arm connecting rod is multiplied right in turn to obtain the positive solution of the kinematics Equation (2) of the RAIHA robot. 2 $\begin{array}{}{}^0{T}_4=\left[\begin{array}{cccc}{a}_1{c}_{23}& -a_1{s}_{23}& -s_1& c_1\left(a_1+a_2{c}_2+a_3{c}_{23}\right)\\ s_1{c}_{23}& -s_1{c}_{23}& c_1& s_1\left(a_1+a_2{c}_2+a_3{c}_{23}\right)\\ -s_{23}& -c_{23}& 0& -a_2{s}_2-a_3{s}_{23}\\ 0& 0& 0& 1\end{array}\right],\end{array}$

Where ⁰T₄ is the transformation matrix of the wrist position relative to the base coordinates; s₁ is sinθ₁; s₂ is sinθ₂; c₁ is cosθ₁; c₂ is cosθ₂; s₂₃ is sin(θ₂ + θ₃); c₂₃ is cos(θ₂ + θ₃); a_i is the length of the connecting rod, i = 1–3.

Then, the inverse kinematics analysis of Equation (2) is carried out to solve the transformation relationship between the motion trajectory of the wrist position and each arm joint. Because the main part of the RAIHA robot has 3° of freedom, there are many solutions in solving the kinematics equation, so the inverse transformation method of the homogeneous matrix is used to solve the inverse kinematics of the main body [26].

According to the forward kinematics solution Equation (2), the homogeneous matrix T_e of the position of the wrist of the RAIHA robot is set as: 3 $\begin{array}{}{T}_e=\left[\begin{array}{cccc}{n}_x& o_x& a_x& p_x\\ n_y& o_y& a_y& p_y\\ n_z& o_z& a_z& p_z\\ 0& 0& 0& 1\end{array}\right],\end{array}$

Where n_x, n_y, and n_z are normal vector elements whose wrist position is relative to the base coordinate system; o_x, o_y, and o_z are the direction vector elements of the wrist position relative to the base coordinate system; a_x, a_y, and a_z are the approximate vector elements of the wrist position relative to the base coordinate system; p_x, p_y, and p_z are the origin vector elements of the wrist position relative to the base coordinate system.

An Equation (4) is established according to the corresponding equality between the forward solution matrix (Equation (2)) and the wrist position matrix (Equation (3)), and the transformation equations of joint angles θ₁, θ₂, and θ₃ of the shoulder arm, upper arm, and lower arm are solved, as shown in Equation (5). 4 $\begin{array}{}{}^0{T}_4=T_e,\end{array}$ 5 $\begin{array}{}\left\{\begin{array}{c}{\theta }_1=\text{arctan}\left(\frac{p_y}{p_x}\right)\hfill \\ {\theta }_2=\text{arctan}\left(\frac{k_2}{k_1}\right)-\text{arctan}\left(\frac{k_3}{\sqrt{k_1^2+k_2^2-k_3^2}}\right)\hfill \\ {\theta }_3=-\text{arctan}\left(\frac{s_3}{c_3}\right)\hfill \end{array}\right.,\end{array}$

Where k₁ = p_ys₁ + p_xc₁ − a₁; k₂ = −p_z; k₃ = a₃s₃; $s_3=\sqrt{1-c_3^2}$; $c_3=\left[k_1^2+k_2^2-a_2^2-a_3^2\right]/2a_2{a}_3$

3.4.2. Analysis of Motion Trajectory Planning

The motion trajectory of the main part of the RAIHA robot refers to the motion position of the wrist relative to the trajectory curve passed by the base coordinate system in one motion cycle, which contains the position, velocity, and acceleration information of the robotic arm during the motion process. Therefore, the planning methods of different motion trajectories have important effects on its motion performance [27, 28].

Combined with the requirements of prostate particle implantation surgery, when planning the motion trajectory, the motion trajectory of the wrist should be followed relative to the base coordinate system in the direction of lifting arm height X₀, swinging Y₀, and advancing distance Z₀ to avoid sudden changes in speed and acceleration, so as to ensure the safety of puncture and reduce the damage to soft tissue. In this paper, quintic polynomial is used to plan its motion trajectory. Meanwhile, according to the boundary conditions: (1) The velocity and acceleration at the beginning and end of the puncture movement are required to be zero to prevent mutations. (2) The wrist end should meet the coordination of movement in each posture direction when moving. So, the trajectory equation of the origin vector of the wrist position matrix with respect to the base coordinates is as follows: 6 $\begin{array}{}\left\{\begin{array}{c}{p}_x=p_i-v_0{t}_x+\frac{10\left(H+0.5v_0\right)}{{\left(0.5T\right)}^3}{t}_x^3-\frac{15\left(H+0.5v_0\right)}{{\left(0.5T\right)}^3}{t}_x^4+\frac{6\left(H+0.5v_0\right)}{{\left(0.5T\right)}^3}{t}_x^5\\ p_y=p_j-v_1{t}_y+\frac{10\left(S_1+0.5v_1\right)}{{\left(0.5T\right)}^3}{t}_y^3-\frac{15\left(S_1+0.5v_1\right)}{{\left(0.5T\right)}^3}{t}_y^4+\frac{6\left(S_1+0.5v_1\right)}{{\left(0.5T\right)}^3}{t}_y^5\\ p_z=p_k-v_2{t}_z+\frac{10\left(S_2+0.5v_2\right)}{{\left(0.5T\right)}^3}{t}_z^3-\frac{15\left(S_2+0.5v_2\right)}{{\left(0.5T\right)}^3}{t}_z^4+\frac{6\left(S_2+0.5v_2\right)}{{\left(0.5T\right)}^3}{t}_z^5\end{array}\right..\end{array}$

In the formula, p_i, p_j, and p_k are the starting position of wrist motion relative to the axis direction of base coordinates X₀, Y₀, and Z₀; v₀ is the initial velocity of the RAIHA robot at the initial moment; v₁ is the initial velocity of the lateral movement of the robotic arm base on imitation human arm at the initial moment; v₂ is the initial velocity of the forward and backward motion of the RAIHA robot at the initial moment; t_x, t_y, and t_z are, respectively, the time when the wrist position is in different motion modes relative to the base coordinates; T is the motion cycle; S₁, S₂, and H are the displacement of the wrist position with respect to the base coordinate system Y₀, Z₀, and X₀ during one motion cycle.

According to the position of the needle tip shown in Figure 9, combined with Equation (6), the motion trajectory equation of the puncture needle tip is obtained, as shown in Equation (7): 7 $\begin{array}{}\left\{\begin{array}{c}{\beta }_x=\left(p_x+l_0+l_1\right)\sin {\theta }_4\\ {\beta }_y=\left(p_y+l_0+l_1\right)\sin {\theta }_5\\ {\beta }_z=\left(p_z+l_0+l_1\right)\cos {\theta }_5\end{array}\right.,\end{array}$

Where β_x, β_y, and β_z are the transformation coordinates of the needle tip relative to the base coordinate system; θ₄ is the wrist pitch angle; θ₅ is the wrist roll angle; l₀ is the fixed distance between the wrist roll joint and the wrist joint; l₁ is the variable distance from the tip of the needle to the wrist roll joint.

As shown in Figure 10, the displacement trajectory curves in each direction of the terminal particle implantation planned by using quintic polynomials Formula (7) in one movement period are shown. Among them, the motion period is temporarily set to 2 s (the motion period can be changed if the speed is adjusted), and the motion displacement S in each direction of the space is temporarily set to 100 mm.

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As can be seen from Figure 10a,b, at the beginning and end of the movement, the motion velocity of the terminal position is relatively gentle, which improves the motion stability and prevents the servo electric cylinder (Servo motor) from being blocked due to high output torque in the process of high-frequency start and stop. Meanwhile, the further analysis in Figure 10c shows that the acceleration of this motion trajectory does not have step phenomenon, so it can effectively overcome the inertial force generated by its own motion acceleration step (equivalent to exerting a large impact force on the terminal particle implantation when it starts to move). Therefore, the stability of the center of gravity and the movement coordination of the robotic arm when it starts to move are ensured.

As shown in Figure 11, it is the space motion trajectory diagram of the tip of the terminal puncture needle base on quintic polynomial are constructed according to Equations (6) and (7). Among them, Figure 11a,b,c show that the tip of the puncture needle driven by the robotic arm presents displacement directions of 30°, 45°, and 60° in the horizontal and vertical directions, respectively, and the spatial displacement distances are 50, 70.7, and 86.6 mm, respectively. As shown in Figure 11, the accuracy and effectiveness of Equations (6) and (7) in the planned motion trajectory can be fully verified, and the displacement motion of the wrist position and posture can be realized.

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4. Design Parameters Optimization of RAIHA Robot

Aiming at the narrow surgical space of human lithotomy, it is necessary to optimize the design parameters of the RAIHA robot in order to enable the robot to perform the particle implantation operation safely and reliably. Since establishment of the corresponding optimization objective function to obtain the optimal solution is a common method for parameter optimization of robotics arm [29]. Accordingly, this paper derives the mathematical model, identifies the constraint conditions and optimization objectives, and thereby establishes the parameter optimization objective function for the RAIHA robot. Meanwhile, intelligent algorithms are employed to optimize the design parameters. Finally, simulations are performed on the optimized design parameters to verify their effectiveness.

4.1. Analysis of Spatial Motion Model

Since the design of each arm segment length of the RAIHA robot must be determined based on the physical parameters measured in Table 1, the robot’s reasonable dimensions are optimized in accordance with the requirements of prostate particle implantation surgery. Therefore, the motion space of the RAIHA robot is modeled by geometric method (graphic method), as shown in Figure 12.

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Where, a₁, a₂, and a₃ are the initial positions of the robot arm; a_2-1 and a_2-2 are the limit positions of the upper arm; a_3-1, a_3-2, a_3-3 and a_3-4 are the limit positions of the lower arm; φ₀, φ₁, and φ₂ are the swing arcs of the upper and lower arms (minimum and maximum angles); φ₃ and φ₄ are the swing arcs of the joint of the upper arm and lower arm at the maximum limit angle position; τ₁ is the distance between the wrist terminal and the needle entry point of the human body (the initial distance between the needle entry point and the wrist, its value is equal to the length of the terminal particle implantation is 300 mm); τ₂ is the space distance of the robotic arm in the Z direction (Figure 2, its value ranges from 550–650 mm); τ₃ and τ₄ are half of the space of motion of the mechanical arm in the Y direction (Figure 1, the distance between the knees is 225–260 mm and the distance between the feet is 275–300 mm).

In addition, combined with the structural characteristics of the human arm and the operable surgical space of the human lithotomy position, the rotation range of the shoulder and arm joint needs to be set in advance: ε is −60°−120° (initial position is −90°; where, ε₁ is 45° and ε₂ is 15°); The rotation range of the upper arm joint is −150°−30° (Initial position is −60°); lower arm joint rotation range γ is 60°–150° (initial position is 90°); among them, ε₁, ε₂, δ, and γ are defined in terms of spatial cartesian coordinates.

To perform flexible puncture effectively and safely, the puncture needle can be fully covered in the needle insertion area as shown in the red dotted box in Figure 12b, combined with the operable space for prostate particle implantation and the height and size of the terminal particle implantation as shown in Figures 12 and 2, it is required that the difference H = R_max − R_min between the arc length radius of φ₂ (or φ₄) and φ₁ (or φ₃) should be (70 and 210 mm). Meanwhile, to ensure the safety of lateral swing, 2τ₃ ≤ 400 mm, 2τ₄ ≤ 500 mm. Therefore, the constraints of the working space of each arm link are Equation (8). 8 $\begin{array}{}\left\{\begin{array}{c}70210<H=R_{\max }-R_{\min }\le \hfill \\ {\tau }_1\le S<{\tau }_2-{\tau }_1\hfill \\ 0400\le \left(a_1+R_{\min }\right)\left(\sin {\epsilon }_1+\sin {\epsilon }_2\right)<\\ 0500\le \left(a_1+R_{\max }\right)\left(\sin {\epsilon }_1+\sin {\epsilon }_2\right)<\\ \epsilon ={\epsilon }_1+{\epsilon }_2\hfill \end{array}\right.,\end{array}$ where $R_{\max }=\sqrt{a_2^2+a_3^2-2a_2{a}_3\cos {\gamma }_{\max }}$;

$R_{\min }=\sqrt{a_2^2+a_3^2-2a_2{a}_3\cos {\gamma }_{\min }}$.

4.2. Analysis of Force Model

Since the RAIHA robot uses a servo electric cylinder to imitate the tendons of the human arm to drive, it can be known by combining the cosine theorem that the change of the rotation angle of each joint can be controlled by controlling the displacement of the servo electric cylinder. Meanwhile, the influence of force (power consumption) should also be considered in the design of the humanoid arm, which can not only ensure the compact and lightweight structure, but also meet the rotation angle requirements of each joint. Therefore, the position of the servo electric cylinder head installed in each connecting rod arm has a very important impact on the joint angle and driving force. So this paper analyzes the stress condition of the robotic arm, and its statics model is shown in Figure 13.

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Among them, views a and b are XZ and XY planar graphs of the statics model of a RAIHA robot; l₂, l₃, and l₄ are the distance between the installation position (K, F, and H) of the servo electric cylinder head and the connecting rod of the shoulder arm, the upper arm and the lower arm, respectively, and the shoulder joint, the upper arm joint and the lower arm joint; m is the center of mass of the load-bearing part of the upper arm joint, which is composed of the upper arm EG, the lower arm GI, the terminal particle implantation IJ and JE (auxiliary lines of the regular quadrilateral EGIJ formed at an ideal joint angle). n is the center of mass of the load-bearing part of the lower arm joint, which is a triangle GIJ composed of the lower arm GI, the terminal particle implantation IJ and JG (diagonal of the quadrilateral EGIJ); G₁, G₂, and G₃ are the gravity forces on the shoulder joint, the upper arm joint, and the lower arm joint. In addition, s₁ is the installation distance between the servo electric cylinder and the side swing joint, s₂ is the installation distance parallel between the servo electric cylinder and the lower arm, and s₃ is the installation distance parallel between the servo electric cylinder and the connecting rod of the upper arm; u₁, u₂, and u₃ are the total initial length of the servo electric cylinder.

First, this paper carries out force analysis on the XY plane shown in Figure 13b, and sets the deflection degree ε of the RAIHA robot and the servo electric cylinder is contracted. Without considering the friction torque of the joint, the combined moment M_D of the shoulder joint can be expressed as the Equation (9). 9 $\begin{array}{}{M}_D=G_1{p}_1\text{\hspace{0.17em}sgn}\left(\epsilon \right)+l_2{f}_K\cos \left|\epsilon \right|-l_2{f}_K{\sin }^2\left|\epsilon \right|,\end{array}$ where sgn(ε) = 1 for ε < −90°, sgn(ε) = 0 for ε = −90° (the robotic arm is perpendicular to the upper spider DD₁), sgn(ε) = −1 for ε > −90°; p₁ is the force arm of gravity G₁ of robotic arm on shoulder joint; f_K is the pull force of the shoulder servo electric cylinder.

When ε = −90°, Equation (9) can be expressed as: 10 $\begin{array}{}\epsilon =\text{arccos}\frac{M_D}{l_2{f}_K}.\end{array}$

Then, the force analysis of the joint torque of the upper arm in the XZ plane shown in view a is carried out in this paper, the deflection degree of the upper arm δ is set and the servo electric cylinder is contracted. Without considering the friction torque of the joint, the joint torque M_E of the upper arm can be expressed as Equation (11). 11 $\begin{array}{}\left\{\begin{array}{c}{M}_E=-G_2{p}_2+l_3{f}_F\cos \delta \hfill \\ \delta =\text{arccos}\frac{M_E+G_2{p}_2}{l_3{f}_F}+\frac{\pi }{6}\hfill \end{array}\right.,\end{array}$ where p₂ is the force arm of the robotic arm relative to the center of mass m on the joint of the upper arm; f_F is the pull force of the upper arm servo electric cylinder.

Finally, the force analysis of the lower arm joint torque in the XZ plane is carried out, and the deflection degree of the lower arm is set to be γ and the servo electric cylinder is contract. Without considering the friction torque of the joint, the combined torque M_G of the lower arm joint can be expressed as Equation (12). 12 $\begin{array}{}\left\{\begin{array}{c}{M}_G=-G_3{p}_3+l_4{f}_H\cos \gamma \hfill \\ \gamma =\text{arccos}\frac{M_G+G_3{p}_3}{l_4{f}_H}+\frac{\pi }{3}\end{array}\right.,\end{array}$ where p₃ is the force arm of center of mass n against the lower arm joint; f_H is the pull force of the lower arm servo electric cylinder.

According to Equations (11) and (12), the joints of the upper and lower arms bear the gravity torque brought by the wrist. If the robot arm of a human arm moves at a constant speed (considering the joint torque is zero), the gravity torque at the joint is converted into the pulling force of the servo electric cylinder. Although the center of gravity and rotation angle will change constantly, the tensile force is mainly affected by its force arm (l₂, l₃). Therefore, the structure of the prostate particle implantation robot designed in this paper can compensate for the gravitational torque of each rotating joint to a certain extent.

In addition, according to the analysis of Equations (10), (11) and (12), if the angle of each joint is kept constant, the force arm (the distance between the installation position of the servo electric cylinder’s head and each joint’s axis) is inversely proportional to the driving force of the servo electric cylinder. Meanwhile, if the driving force is kept constant, the torque arm is proportional to the rotation angle of each arm joint. If the force arm is kept constant, the driving force is proportional to the angle of rotation of each joint. This indicates that the position of the servo cylinder head installed on each link arm affects the driving force and rotation angle of the link arms, but to meet the requirement that the robotic arm can perform flexible displacement movements within the human lithotomy position space, it is necessary to analyze the installation position of the servo cylinder head on each arm rationally, so as to ensure that the joints of each arm are within the working range of rotation, and reduce the driving force of the servo cylinder (the smaller the driving force, the smaller the size of the servo cylinder, and the more compact the overall structure).

Therefore, combined with the above analysis, this paper defines the constraint condition between the angle of each arm joint and the driving force of the servo electric cylinder as Equation (13). 13 $\begin{array}{}\left\{\begin{array}{c}{f}_{\min }=\min \left[f_k+f_F+f_H\right]\\ {\theta }_{\max }=\left[\epsilon +\delta +\gamma \right]\hfill \end{array}\right.,\end{array}$

Where: f_min is the minimum value of the sum of the driving forces of each driving element; θ_max is the maximum value of the sum of the angles of each joint.

4.3. Establishment of Optimization Objective Function

• 1.

  Optimization objective function of connecting rod length:

To make the structure of the RAIHA robot more compact and more flexible and efficient when performing puncture surgery tasks, the structural size of the RAIHA robot should be as small as possible, that is, the “sum of the length of each connecting rod” of the robotic arm should be as small as possible.

Therefore, the first optimization objective based on “compact structure” criterion is: 14 $\begin{array}{}\min \left[a_1+a_2+a_3\right].\end{array}$

Therefore, the combination of Equations (8) and (14) can establish the first optimization objective function.

• 2.

  Optimization objective function of driving force and joint angle:

To enable the prostate particle implantation robot to meet the target workspace operation task, it is required that the control space of the RAIHA robot should be as large as possible, so as to realize the multiattitude needle adjustment at the terminal, and effectively complete the task of “one point and more needles.” Meanwhile, when the servo electric cylinder continues to output the driving force, an excessively small driving force can not maintain the normal movement of the prostate particles implantation robot, resulting in the failure to complete the given puncture particle implantation task. On the other hand, an excessive driving force may cause the servo motor of the servo electric cylinder to exceed its rated power, causing servo motor cylinder damage. Therefore, the driving force value should be as small as possible on the basis of meeting the maximum rotation angle of each arm joint and normal driving. In this way, the robot system can not only successfully complete the surgical task of prostate particle implantation, but also ensure its dynamic performance, extend its service life and reduce energy loss. So, the second optimization objective is min [f(l_j)], where f(l_j) represents the optimization objective function individual. Combining the second optimization objective, cosine theorem and Equation (13), the second optimization objective function is obtained, as shown in Equation (15). 15 $\begin{array}{}\left\{\begin{array}{c}{f}_1=f_{\min }=\sum _{j=24\sim }^{i=13\sim }\frac{G_i{p}_i}{l_j\cos {\theta }_i}\hfill \\ f_2=\frac{1}{{\theta }_{\max }}=\sum _{j=24\sim }^{i=13\sim }\frac{1}{\text{arccos}\frac{s_i^2+l_j^2-u_i^2}{2s_i{l}_j}}\hfill \\ \min \left[f\left(l_j\right)\right]=\left[f_1\left(l_j\right),f_2\left(l_j\right)\right]\hfill \end{array}\right.,\end{array}$ where f₁ is an optimization objective function related to driving force; f₂ is an optimization objective function related to joint angle.

According to Equation (15), it can be seen that the structural design parameters l, s, and u of the RAIHA robot have a very important impact on the motion performance of the prostate particle implantation robot and on meeting the requirements of the surgical space task. Therefore, it is necessary to optimize it in this paper.

4.4. Length Optimization of Each Arm Based on GA

4.4.1. Population Range Selection of GA

Since the establishment of the space motion model of the robotic arm is the basis for optimizing the length of each arm, this paper combined the inverse kinematics solution Equation (5), the terminal motion trajectory Equations (6) and (7) and the first optimization objective function, and used Matlab2017 software to construct the space motion model according to the operable space shown in Figure 1.

Among them, according to the analysis in section 4.1, the rotation angles of the shoulder arm joint (ε₁, ε₂), the upper arm joint δ, and the lower arm joint (γ_min, γ_max) are selected, as shown in Equation (16). 16 $\begin{array}{}\left\{\begin{array}{c}{\epsilon }_1\in \left(04,\pi /\right)\hfill \\ {\epsilon }_1\in \left(012,\pi /\right)\hfill \\ \delta \in \left(023,\pi /\right)\hfill \\ {\gamma }_{\min }=2356\pi /,{\gamma }_{\max }=\pi /\hfill \end{array}\right..\end{array}$

Meanwhile according to the measurement results of human arm physical parameters in Table 1 in Section 3.1.4, it can be seen that the maximum and minimum length of the shoulder arm, upper arm and lower arm are taken as the population range, and three groups of optimization solutions are carried out for the defined population respectively. As shown in Equation (17). 17 $\begin{array}{}\left\{\begin{array}{c}{a}_1\in \left(2746,\right)\hfill \\ a_2\in \left(166297,\right)\hfill \\ a_3\in \left(153231,\right)\hfill \end{array}\right..\end{array}$

4.4.2. Simulation Solution of Optimal Parameters

The parameter optimization of the RAIHA robot belongs to the extreme value problem of a single objective function, which can be solved by using the GA toolbox in Matlab2017. First, the optimization objective function (Equation (14)) of robotic arm is compiled into the fitness function of GA in Matlab2017, and the workspace motion model is compiled into the constraint function form of GA. Then, we set the ranges of the parameters to be optimized, along with the initial population size to 100, mutation probability to 0.04, crossover probability to 0.98, and evolutionary generation to 500. Finally, the GA is used to solve the optimal parameters (where the motion period T = 2 s).

The variation trend of fitness function value with evolutionary algebra in the evolution process of GA is shown in Figure 14.

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As can be seen from Figure 14, when the GA has evolved to about 160 generations, its fitness function (objective function) is in complete convergence. At the end of the calculation, the optimal solution of the parameters value of the RAIHA robot is rounded, and the optimized results are as follows: the length of the shoulder arm, the upper arm, and the lower arm are 35, 217, and 190 mm.

4.5. Optimization Analysis of Structural Design Parameters

Through the analysis in Section 4.2, it can be known that the selection problem between the rotation angles of each arm joint (ε, δ, and γ) and the driving force received by the servo electric cylinder belongs to a multiobjective problem (that is, the two optimized objectives are in a conflicting relationship with each other). The essence of the multiobjective optimization problem lies in that in most cases, it is impossible to optimize multiple (two or more) objectives, and only by coordinating, weighing and compromising among the objectives can all the optimized objective values be made as optimal as possible. Therefore, the optimal solution obtained is no longer a single solution that meets the minimum (or maximum) of all target fitness functions at the same time, but an optimal solution determined according to the Pareto governing principle [30, 31]. Among them, Pareto governing principle as: Set the multiobjective optimization problem as minf(x) = [f₁(x), f₂(x),…, f_m(x)], when the feasible solution x₁ and x₂ are fully satisfied Equation (18), that is, x₁ is not worse than x₂ on all objectives, and is strictly better than x₂ on at least one objective, x₁ is said to be a nondominant solution, x₂ is a dominant solution, and x₁ is Pareto dominant. 18 $\begin{array}{}\left\{\begin{array}{c}{f}_i\left(x_1\right)\le f_i\left(x_2\right),\forall i\in \left\{12,,\cdots ,m\right\}\hfill \\ f_j\left(x_1\right)<f_j\left(x_2\right),\mathrm{\exists }j\in \left\{12,,\cdots ,m\right\}\hfill \end{array}\right..\end{array}$

Pareto optimal solution set in multiobjective optimization problem is composed of multiple nondominant solutions, and the set of corresponding objective vectors is called Pareto frontier. In the process of searching for the optimal solution, it is necessary to make these solution sets as close as possible to the Pareto frontier. Therefore, in this paper, NSGA-Ⅱ algorithm and the second optimization objective function (Equation (15)) are used to optimize the driving structure parameters l, s, and u of the RAIHA robot.

4.5.1. Multiobjective Optimization Algorithm Based on NSGA-Ⅱ

NSGA-Ⅱ is based on GA and operates on multipopulation individuals, so the algorithm has implicit parallelism. The sensitivity of the algorithm to the initial value is reduced because the initial value of the multipopulation individuals is different. In evolutionary operation, constraints can be conveniently added to the population to avoid infeasible solutions and improve the efficiency of the algorithm. In each generation of evolution, NSGA-II first performs evolutionary operations (e.g., selection, crossover, and mutation) on population P to generate population Q.

Then the two populations are merged, noninferior sorting and crowding distance sorting are performed to form a new population P_t, and iterative calculation is repeated until the end. The algorithm is iterated by partial order selection based on Pareto rank and crowding distance. According to this ranking order, the selection operation is carried out in the evolution process, thus giving the Pareto optimal individual at the front of the list a greater chance of being inherited by the next generation population. Figure 15 shows the Flowchart of NSGA-Ⅱ.

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4.5.2. Multiobjective Optimization Simulation Solution

According to the geometric structure relationship of the humanoid arm as shown in Figure 13a mult-objective simulation model is established by combining Equations (10), (11), and (12). Building on the kinematics model, the multiobjective optimization algorithm based on NSGA-Ⅱ is developed in Matlab2017 software, and the multiobjective simulation model is solved through the optimization objective function of Equation (15), so as to obtain the Pareto optimal frontier. On this basis, the nondominant solution set is analyzed and the optimal solution is selected.

Simulation experiment parameters are set as follows:

• 1.

  Parameters of RAIHA robot.

• ①

  The shoulder arm length is set to 35 mm, the lower arm length is set to 190 mm, and the upper arm length is set to 217 mm. The shape and volume of each arm are obtained from data obtained through strength analysis and structural optimization using ANSYS software.

• ②

  Material density ρ_aluminum = 2.7 g/cm³.

• ③

  Acceleration of gravity g = 9.8 m/s².

• ④

  The maximum driving force of the servo electric cylinder is 400 N.

• ⑤

  The friction torque of joint motion is not considered, and the torque of the whole system is balanced (the resultant torque is 0 N).

• 2.

  Motion parameters.

• ①

  Initial joint angle: shoulder arm joints is −90°, upper arm joints is −60°, and lower arm joints is −135°.

• ②

  Servo electric cylinder head and each arm connection position range: l₁∈ [0, 35], l₂∈ [0, 217], l₃∈ [0, 190].

• ③

  The pre-set installation distances between the servo electric cylinder and the shoulder-arm joint, the servo electric cylinder and the upper arm parallel, and the servo electric cylinder and the forearm link parallel are: s₁∈ [0, 500], s₂∈ [0, 500], and s₃∈ [0, 500].

• ④

  The initial length range of preset servo electric cylinder: u₁∈ [0, 500], u₂∈ [0, 500], u₃∈ [0, 500].

• ⑤

  Motion period T = 2 s.

• 3.

  NSGA-Ⅱ algorithm parameters

• ①

  Initial population number: Pop = 200.

• ②

  Maximum genetic algebra: Gen = 200.

According to the simulation optimization calculation, the Pareto frontier distribution and planning points are obtained, as shown in Figure 16.

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As can be seen from Figure 16, since point A and point E are close to the two extremes of Pareto frontier, respectively, the sum of joint angles of the RAIHA robot in this simulation motion is smaller when it is close to point A in the Figure 16. When approaching point E, the driving torque of the servo electric cylinder is larger. To specifically analyze the influence of different optimization schemes in Pareto front on the space motion and driving force of the RAIHA robot, three points B, C, and D in the figure are selected as three optimization plans. Among them, point B is the minimum total driving force of the large arm servo electric cylinder when it meets the normal motion, point C is the tentative optimal value of the rotation angle of each arm joint, and point D is the maximum total driving force of the small arm servo electric cylinder when it reaches the rated power.

Table 3 lists the optimization parameters and target values for plan B, C, and D.

Table 3 Optimization scheme and parameter target value.

Parameter and target value	Plan B	Plan C	Plan D
l	l2 (mm)	36.3	25.3	22.4
l3 (mm)	89.2	49.7	30.6
l4 (mm)	54.3	31.8	17.2
s	s1 (mm)	276.4	225.2	189.6
s2 (mm)	317.1	283.6	245.2
s3 (mm)	198.2	185.1	151.6
u	u1 (mm)	244.4	262.3	285.3
u2 (mm)	286.6	307.9	345.7
u3 (mm)	233.8	258.7	275.5
θmax	ε1 (°)	8.8	15.9	22.3
ε1 (°)	31.2	45.7	68.8
δ (°)	96.6	122.3	156.3
γ (°)	65.5	89.5	112.5
fmin	fK (N)	20.3	45.6	135.8
fF (N)	94.2	167.2	557.4
fH (N)	41.7	108.8	408.5

As can be seen from Table 3, if plan D is adopted, the force on the upper arm servo electric cylinder and the small arm servo electric cylinder of the RAIHA robot will exceed the maximum driving force. If two plans B and C are adopted, the force condition of the servo electric cylinder of the RAIHA robot meets the requirements of its maximum driving force.

4.6. Optimization Results Simulation Experiment

In order to verify the feasibility of the optimization results and select the best optimization plan, this paper combined the length of the connecting rod of each arm of the RAIHA robot, the parameter values shown in Table 3 and the double-free frame with preset space to build the model of the prostate particle implantation robot. In addition, the working space of the wrist of the RAIHA robot is simulated by SimMechanics mode in Matlab2017/Simulink. Figure 17 shows the model.

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Figure 18 shows the simulation experiment results of three optimization plans B, C, and D for the working space of the wrist of the RAIHA robot. If the structural plan B is adopted for design, the motion range of the wrist motion space in the XY and XZ working planes is too small, resulting in the inability of the terminal particle implantation to carry out large range of flexible movement in the limited and narrow space of the human lithotomy site.

[IMAGE OMITTED. SEE PDF]

If scheme D is used for structural design, the motion range of the wrist end in the XY and XZ planes is too large and exceeds the motion constraints range of the human lithotomy position, this can easily pose risks to the patient and cause damage to the robot. When the structural is designed by plan C, the motion range of the wrist on the XZ working plane is: Z direction (−51.6 to 152.3 mm), meet the constraint range of Z direction (200 to 250 mm); (−138.6 to −351.8 mm) in the X direction. Operating range in the XY working plane: the distance of χ₁χ₃ in the Y direction of the knee is (−167.7 to 213.6 mm); The distance of the ankle in Y direction χ₂χ₃ is (−288.5 to 213.6 mm).

In addition, in the case of setting the angular motion of each arm joint, taking plan C as the simulation experiment object, Monte Carlo method is used to conduct posture simulation analysis of the wrist end (blue dot) and puncture needle tip (red dot) of the RAIHA robot in the XY working plane, as shown in Figure 19. Among them, the black dotted line is the location of the human prostate region; The green dotted line is the dividing line of the human prostate urethra; The black point is the entry point. The number of spatial distribution points is 10000 points. As can be seen from Figure 19, when the puncture needle tip is guaranteed to penetrate the prostate region of the human body through two insertion points, the position of the terminal particle implantation (blue point) is within the range of (−340 to −115 mm) in pitch posture and (−120 to 120mm) in horizontal posture, which is smaller than the boundary range of the operational space of the XY plane.

[IMAGE OMITTED. SEE PDF]

Through the simulation analysis of three structural design plans B, C, and D, it can be seen that plan C can achieve relatively better rotation angles of each joint and driving torque (energy consumption) of servo electric cylinder. Meanwhile, the range of the wrist end working space strictly conforms to the constraints of the operable space of the human lithotomy position, and the terminal particle implant can complete multiple pose changes in the limited surgical space, so as to achieve the task of one point and multiple needles, meeting the requirements of the prostate particle implantation robot on safe operation and reducing damage.

5. Experimental Analysis of Physical Prototype

To verify the effectiveness of a particle implantation robot for prostate cancer base on imitation human arm is designed in this paper, a physical prototype is designed according to the optimized connecting rod length of each arm and the structure plan C, and verification experiment of rotation position of each arm joint, joint rotation angle position error test and terminal attitude control error test are carried out, as shown in Figure 20. Among them, (a) terminal particle implantation device; (b) spatial freedom machine rack; (c) RAIHA robot; (d) control system.

[IMAGE OMITTED. SEE PDF]

In the design of the robotic arm base on imitation human arm, in order to facilitate mechanical processing, the size of plan C needs to be rounded. Therefore, the installation parameters of the servo electric cylinder in the robotic arm base on imitation human arm are selected: l₂ = 50 mm; l₃ = 100 mm; l₄ = 60 mm; s₁ = 225 mm; s₂ = 284 mm; s₃ = 185 mm. In addition, due to the initial position of the servo electric cylinder, in addition to the cylinder part, it also includes: cylinder head (30 mm), tension pressure sensor (15 mm), connection head (30 mm), and base (35 mm). Therefore, combined with the parameters of u₁, u₂, and u₃ and the existing models, the shoulder and arm servo cylinder is 150 mm; Upper arm servo electric cylinder 200 mm; lower arm servo electric cylinder 150 mm.

5.1. Design of Control System

To realize the aseptic isolation between the patient’s operating room and the doctor’s operating room, it is convenient for the doctor to operate and control it. Meanwhile, to facilitate the real-time acquisition, analysis of experimental data and the generation of graphic results, the control system adopts a two-layer structure of upper and lower computer. Among them, the YanWei EMBA-781 model industrial computer is used as the upper computer, its control operation and monitoring interface is written in C#2012. The upper computer is mainly responsible for sending control commands, executing logical algorithms, processing data, transmitting data, storing data, and monitoring data.

The lower computer system of a particle implantation robot for prostate cancer base on imitation human arm is composed of four sub-modules, which are analog input sub-module, analog output sub-module, terminal particle implantation overall driver sub-module and encoder acquisition sub-module. Among them, (1) the analog input sub-module is composed of the signal amplifier of the displacement (force) sensor, Advantech PCI-1747U analog input card and ADAM-3968, which is used to collect the data information of three displacement sensors and three tension pressure sensors on the robotic arm base on imitation human arm and transmit it to the upper computer, which can realize the mixed force/bit control; (2) the analog output sub-module is composed of Advantech PCI-1724 analog output card, ADAM-3962 and servo driver, which is used to drive three servo electric cylinders on the RAIHA robot, two servo electric cylinders on the spatial freedom machine rack and two servo torque motors; (3) the overall driver sub-module of the terminal particle implantation adopts the self-developed motion acquisition card (mainly composed of 1 STC8G chip, 4 ATmega328P-AU chip and its peripheral conditioning circuit). It is used to drive the internal/external needle motor to complete the puncture movement, drive the internal/external needle rotating motor, collect the data of the six-dimensional force sensor and displacement sensor on external needle (including signal amplifier, to realize the terminal force perception and displacement control), and collect the data of the MPU6050 gyroscope sensor (to realize the terminal space attitude control); (4) the encoder acquisition sub-module adopts the Henkai USB_AMC4E _V1.0 four-axis encoder counting card, which can collect the ABZ encoder signals of the three servo electric cylinders in the RAIHA robot, and complete the speed loop control and acceleration monitoring of the servo electric cylinder (realize the periodic adjustment of the trajectory movement of the terminal of the RAIHA robot).

5.2. Verification Experiment of Rotation Position of Each Arm Joint

To verify the effectiveness of the motion of the prostate particle implantation robot is designed in this paper, experiments are carried out to verify the rotation position of each arm joint. During the experiment, the limit position measurement method is adopted and combined with the spatial image visual recognition method, the limit position of each joint angle is measured [23]. Figure 21 shows the results.

[IMAGE OMITTED. SEE PDF]

In Figure 21, a–c are the verification of the vertical expansion height, and the height difference is 174.4 mm. Where, a is the initial position (upper arm joint angle is 60°, lower arm joint angle is 90°), b is the contraction position (upper arm joint angle is 30°, lower arm joint angle is 60°), c is the elongation position (upper arm joint angle is 60°, lower arm joint angle is 135°), and the height difference is in the range of (70–210). f–h is the verification of the ultimate angle of the upper and lower arms, and as can be seen from Figure 21, the rotation range of the joint of the upper arm is 29.6°–122.3°. i–k are the limit positions of the side swing rotation angle, and the inner and outer rotation angles of the side swing rotation are 16.3° and 43.7°. d–e are the tests of pitch and swing of the wrist join. As can be seen from the Figure 21, the prostate particle implantation robot is designed in this paper can realize the free and flexible adjustment of the puncture posture of the puncture needle in the limited and narrow space of the human lithotomy site, its workspace range can meet the requirements of the surgical space range shown in Figure 2.

Through the verification experiment on the rotation position of each arm joint of the prostate particle implantation robot, the results were compared according to Equation (17). It is evident that the prostate particle implantation robot designed in this paper can meet the surgical requirements for wide-range adjustment of puncture postures in the effective yet confined operational space of the human lithotomy position.

In addition, according to the operable space of the human lithotomy position shown in Figure 2, the prostate particle implantation robot with rectangular coordinate, parallel and mixed-link structure needs to occupy more limited and narrow space (width distance) between the patient’s legs than the prostate particle implantation robot with serial structure. Therefore, in order to further explain the structural size advantages of the serial prostate particle implantation robot designed in this paper compared with other serial robots, and it is easier to complete the multiposture puncture task. According to the comparison table of serial structure size shown in Table 4, the prostate particle implantation robot designed in this paper is smaller than other serial structure types in terms of structure size, and the structure size of the body is consistent with the arm of an Asian adult, which can realize the same multiposture flexible movement as the human hand. It can complete complex prostate particle implantation operation according to specific motion trajectory and space attitude planning. Among them, the representative robot is Denmark’s UR5 series robot (which can be used for prostate puncture surgery, but is currently in the research stage) and the cantilever tandem prostate particle implantation robot proposed and designed in literature [10].

Table 4 Comparative table of structural dimensions of series robots.

UR5 series robots	Cantilever series robot	Robotic arm base on imitation human arm

Through experiments and Table 4, it can be seen that the robot designed in this paper is smaller than other serial robots on the premise of flexible movement of various positions/attitude in space, and is more suitable for completing the task requirements of prostate particle implantation surgery in the limited and narrow working space of human lithotomy.

5.3. Experiment on the Influence of Rotary Joint Gravity Torque

Since the prostate particle implantation robot design in this paper belongs to a serial configuration, the gravity torque has a great influence on the rotation displacement angle of the rotating joint [32]. Therefore, to verify the tracking performance of the joint rotation position of the prostate particle implantation robot under the action of terminal gravity, experiments on the self-compensation performance of gravity torque and the position error of the joint rotation angle are carried out, and this experiment is verified by comparative analysis method.

The comparative experimental object selected in this study is the cantilever tandem prostate particle implantation robot is designed by the reference [10], and the experiment place is the laboratory where it is located. In addition, UR series serial robots are also representative, generally used for superficial biopsy and neurosurgical puncture. However, in terms of prostate particle implantation, it is still in the research stage due to its own insufficiency and limited bearing capacity (about 5 kg) [33].

In the experiment, both the cantilever tandem prostate particle implantation robot and the experimental prototype is designed in this paper used a pull wire rotary encoder, torque is measured using a rotary torque sensor. Meanwhile, the mass of the terminal particle implant at the terminal installation position of the two robots are about 6 ± 0.3 kg, and the output power and torque of the drive motor and servo electric cylinder selected for each arm joint are consistent. In the course of the experiment, the experimental object of the rotary joint is the rotary joint of the upper arm, and the current detector is installed at the driving motor to analyze the compensation effect of the device on the gravity torque (since the magnitude of force and torque correspond to the driving current one by one, this method can equally measure the magnitude of force and torque, and judge the compensation effect of the gravity torque). In the experiment, the input angle signal applied to the upper arm joint is S(t) = 30sin(0.5t + π/10), and the running time t = 50 s.

The experimental results of self-compensation of gravity torque are shown in Figure 22 (In Figure 22b, A as the comparison robot, and B as the robot is designed in this paper). As can be seen from Figure 22a,b, when the sinusoidal rotation changes at a 60° angle, the cantilever tandem structure robot has a large torque fluctuation noise in the rotating joints of its upper arm, and the mean torque fluctuation of the joint of the upper arm is 32.41 Nm. The torque fluctuation of the rotating joint of the upper arm of the prostate particle implanted robot is designed in this paper is close to 0.88 Nm, which indicates that the triangle-like driving structure of the robot designed in this paper is stable. In addition, it can also be seen from Figure 22b that when the terminal load mass is the same and the driving angle of the upper arm is the same, the output driving current of the upper arm joint of B is significantly smaller than that of A, and its weight torque self-compensation is about 55.2%.

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To further illustrate the driving effect of the self-compensation of the gravity torque of the prostate particle implantation robot is designed in this paper on its angular position, an experiment on the angular position error of joint rotation is conducted, and the experimental results are shown in Figure 23. As can be seen from Figure 23a, although the tracking curve of the upper arm joint of the cantilever tandem prostate particle implantation robot can track the input signal well, there is obvious error between the input and the tracking signal, and the tracking signal has obvious high-frequency vibration at the upper peak value. Because the upper arm joint is affected by gravity obviously, the overall tracking signal waveform is lower. Figure 23b shows the tracking curve of the rotation position of the upper arm joint of a particle implantation robot for prostate cancer base on imitation human arm. Compared with Figure 23a, the tracking signal of the upper arm joint in Figure 23b almost coincides with the input signal, and the vibration condition is significantly improved, and the dynamic performance is very well.

[IMAGE OMITTED. SEE PDF]

The experimental results show that: due to the gravity torque generated by the terminal load gravity of conventional tandem the rotating joint of prostate particle implantation robot, the precision of particle implantation is seriously affected, so the gravity torque compensation at the upper arm joint is required to ensure the tracking precision of its rotation signal [9]. However, after the compensation the prostate particle implantation robot has complex mechanical structure, high control difficulty, high power consumption, and easy failure.

The prostate particle implantation robot is designed in this paper can track the angular position signal of the rotating joint well without additional gravity torque compensation (mechanical structure compensation and control algorithm compensation). Under the condition that the output power and torque of the driving device are the same, the robot can effectively reduce the vibration noise of each rotating joint when it is disturbed by the gravity torque at the terminal. While ensuring good particle implantation accuracy, it can reduce the difficulty of control and reduce additional energy consumption.

Therefore, a particle implantation robot for prostate cancer base on imitation human arm is designed in this paper not only retains the structural advantages of the series robot’s space motion range and the wide range of terminal attitude change, but also overcomes the shortcomings of the series structure, realizes the motion stability of the series robot, and meets the requirements of precise puncture of prostate particle implantation surgery.

5.4. Experimental of Terminal Attitude Control Performance

To verify the attitude control performance of a particle implantation robot for prostate cancer base on imitation human arm is designed in this paper, the experiment of the terminal attitude control performance is carried out, and the method of comparison experiment is used to verify the performance.

The comparative experimental object selected in this study is the parallel prostate particle implantation robot is designed in the literature [34], and the experiment place is the laboratory where it is located. This prostate particle implantation robot adopts a parallel structure, so the prototype can realize the position tracking of the driving joint well without using gravity torque compensation, and the bearing capacity is strong. Therefore, the control performance of its terminal attitude is better than that of the serial particle implantation robot is designed in literature [10].

In the experiment, the MPU6050 gyroscope is fixed in the needle tip position of the puncture needles of two robots. Then, according to the kinematic planning, the same attitude motion signal is input to control the position of the needle tip. Finally, the spatial attitude angles of the terminal puncture needles of the two robots were recorded (rotation angles of the three directional axes. Where, the X axis is the direction axis of height, the Y axis is the direction axis of side swing, and the Z axis is the direction axis of horizontal progress). The experimental results are shown in Figure 24.

[IMAGE OMITTED. SEE PDF]

As can be seen from Figure 24a, under the condition that all joints of the parallel prostate particle implantation robot are well driven, although the spatial attitude angle of the terminal puncture needle changes relatively smoothly, there are obvious errors in the attitude rotating around Y, and obvious vibration and noise occur at the upper/lower limit points. The reason for this error is that the parallel mechanism of the robot is particularly complex, which leads to the inevitable movement uncoordination in the control process. In addition, the displacement direction of X and Y axis adopts a single drive mode, so the displacement signal tracking in these two directions is stable. As can be seen from Figure 24b, the tracking spatial attitude signal of a particle implantation robot for prostate cancer base on imitation human arm can accurately respond to the given spatial attitude signal, and its attitude control stability is better than that of the parallel particle implantation mechanism.

The results of the terminal attitude control performance experiment show that the humanoid arm structure of a particle implantation robot for prostate cancer based on imitating the human arm is designed in this paper can complete the accurate terminal attitude control, effectively improve the puncture positioning accuracy, and meet the task requirements of prostate particle implantation surgery.

6. Conclusion

Aiming at the problems and requirements of the existing prostate particle implantation robot during the puncture operation in the limited and narrow working space of the human lithotomy site, this paper proposes a particle implantation robot for prostate cancer based on imitating the human arm, and conducts experimental research. The conclusions are summarized as follows:

• 1.

  During prostate particle implantation surgery, a particle implantation robot for prostate cancer based on imitating the human arm is designed in this paper. This robot retains the structural advantages of the serial structure with a wide range of spatial motion and terminal attitude change, and also has the same motion stability as the parallel structure. It is smaller in volume than the existing series structure, the range of joint angle and space posture is large, and simple and convenient operation.

• 2.

  This robot belongs to a series structure base on bionic improvement, which can control the position of the load-bearing joint well without mechanical and controlled gravity torque compensation, and can overcome the influence of its own gravity torque, and the gravity torque self-compensation is about 55.2%. Meanwhile, While ensuring good particle implantation accuracy, it can reduce the difficulty of control and reduce additional energy consumption.

• 3.

  This robot has good motion stability, can accurately control the attitude of the terminal puncture needle, and the effect is better than that of the parallel puncture robot, effectively improve the puncture positioning accuracy, and meet the task requirements of prostate particle implantation surgery in the narrow operable space of the patient’s lithotomy site.

Data Availability Statement

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

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

Bing Li makes substantial contributions to conception, design, and writing. Junwu Zhu and Bing Li contribution to the analysis of the simulation and experimental data processing. Lipeng Yuan contributions to provide software and hardware technical support for the control system of the experimental prototype in this paper. Bing Li and Lipeng Yuan contributes to the system debugging of the test prototype.

Funding

This study is partly supported by the National Natural Science Foundation of China (Grant 51675142).

Acknowledgments

First, The author would like to thank the Helen People’s Hospital, Suihua City, Heilongjiang, China for their support. Then, The author would like to thank the relevant physicians in imaging and urology for their guidance on this paper. Lastly, this research is partly supported by the National Natural Science Foundation of China (NSFC), Project no. 51675142. Part of the fund is allocated by the key project ZD2018013 at the Natural Science Foundation of Heilongjiang, China.

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