Path planning with obstacle avoidance for soft robots based on improved particle swarm optimization algorithm

1. INTRODUCTION

With the development of robotics, modern applications of robots have been extended to various complex situations and environments, such as medical surgery, search and rescue in confined spaces, industrial equipment maintenance^[1-3], etc. In these complex environments, the flexibility, suppleness, joint limitations, and other properties of robots have high requirements, and traditional rigid robotic arms are difficult to use in these unstructured environments, while flexible robotic In contrast, flexible robotic arms are characterized by high self-load ratio, high flexibility, and lightweight structure, and thus can achieve bending motion in any shape within the elastic deformation range, thus enabling dexterous operation and strong adaptability in narrow spaces and spatial environments with complex obstacle distribution^[4].

The main difference between soft robots and rigid robots is that the materials used to manufacture soft robots are mainly soft materials. However, the elasticity of soft actuators manufactured based on material deformation also makes their control strongly nonlinear, with an infinite number of degrees of freedom during large deformations, which makes it difficult to establish an accurate model^[5]. In recent years, scholars at home and abroad have conducted much research on the design, modeling, control, and obstacle avoidance of flexible robotic arms. IS Godage^[6]proposed a curve parametric (CP) kinematic model based on segmented constant curvature, which gives an accurate positive kinematic model by iterative methods and can simulate the bending motion of most of the flexible continuous arms. However, the Piecewise Constant curvature (PCC) method has numerical singularities when applied to the inverse solution of the soft body continuous arm, which cannot simulate the straight arm state of the continuous section to converge and takes a long time to compute. The inverse kinematics problem is also challenging, and conventional inverse methods are computationally overloaded or unsolvable for soft robots. Neppalli et al.^[7,8] finally obtained a multi-segment kinematic inverse solution by applying single-segment inverse kinematics to a multi-segment backbone for direction compensation. However, this method makes it difficult to avoid obstacles while maintaining access to known end joints and requires additional manual assistance to operate the robot arm movement.

At the same time, reliable obstacle avoidance algorithms are indispensable to make flexible robots better able to accomplish the corresponding tasks in tight spaces. Various practical obstacle avoidance planning methods have been proposed, such as artificial potential field methods^[9], neural network algorithms^[10], genetic algorithms^[11], and machine learning^[12], but the above obstacle avoidance algorithms are mainly used for traditional rigid robotic arms, and in order to enhance the ability of flexible robots to traverse obstacles, Khatib et al.^[13] implemented a multi-stage extended Kalman filter in a simulation for a flexible continuum. Palmer et al.^[14]. characterized the macroscopic features of the robot arm by constructing a spatial curve and optimized the path curve to make the soft robot follow the desired trajectory to the target point. However, this method makes the motion of the robot unsmooth.

In this paper, based on the above-mentioned research, the kinematic coupling problem of the soft robot is addressed by deriving the forward and inverse kinematic models of the soft robot and proposing a self-growing control method based on the bionic vine tip, which avoids the kinematic coupling by discrete the soft robot body into several joint segments and only controls the motion of the soft robot tip joints. In order to avoid obstacles and smooth the motion of the soft robot, this paper uses the particle swarm algorithm to plan the path of the soft robot tip joints, and the planned path is used as the tracking path of the soft robot tip joints, and the rest of the joints use the information of the previous moment of the tip joints to complete the action so that the soft robot can pass the obstacles in the restricted space, and finally, the end-fitting strategy is designed to make the soft robot accurately reach the target point. Finally, the end fitting strategy is designed to make the soft robot reach the target point accurately and realize the trajectory planning of the soft robot in a complex environment. The main contributions are as follows: 1. The second section introduces the structure of the soft robot, establishes a kinematic model of the soft robot based on segmented constant curvature, and verifies the correctness of the model in the Matlab simulation environment; 2. In the third section, an improved particle swarm optimization (PSO) path planning algorithm was constructed to design the optimal planning path, and random peaks and bubbles were set to model obstacles; 3. The fourth section proposes a path tracking algorithm based on tip self-growth, which uses tip joint self-growth control software robots to track the planned path in the third section, thereby avoiding kinematic coupling and improving algorithm efficiency; 4. In the fifth section, the above algorithm was validated through Matlab simulation experiments. The experiments showed that the improved PSO algorithm proposed in this paper can effectively avoid obstacles. The path tracking algorithm based on tip self-growth further improves the algorithm efficiency in path following and can be used for obstacle avoidance planning of soft robots.

2. SOFT BODY ROBOT MODELING

2.1. Soft robot structure

2.1.1. Soft robot structure

The design of the tip-growing soft robot designed in this paper is inspired by the principle of tip growth of vines and other plant rhizomes^[15]. In this approach, the tip extends outward and grows, based on the environment, allowing it to actively control the movement. In the process of movement, it does not rely on its previous posture but constantly growing new tissue structures to adapt to the changing environment; its growth principle is shown in Figure 1. The front end of the flexible body is driven by air pressure to continuously grow a new structure; the movement process does not depend on the environmental structure. The first generation of soft robots is shown in Figure 2(a). The overall structure of the flexible robot arm is shown in Figure 2(b), mainly composed of a control box base, flexible body, and steering module. The flexible body is made of flexible film material. The drive mechanism is in the drive box and drives the flexible body to grow in space by turning outward through air pressure, and three micro motors control each steering module. The theoretical bending angle of this structure ranges from 0 to 60 degrees.

Figure 1. Principle diagram of the growth of the soft robot.

Figure 2. (a) soft Snake Robot (b) Overall structure of the soft body robot.

2.1.2. Kinematic modeling of soft robots

Soft robots do not have the traditional linkages and joints of rigid robots; they cannot be modeled using the traditional D-H parametric method of rigid robots^[16]. In 2006, Jones et al. proposed a theoretical kinematic model based on segmented constant curvature^[17]. This paper uses the segmented constant curvature method for the kinematic analysis of soft robots. In order to realize the kinematic modeling of the self-growing soft robot, the model is appropriately simplified, and the following basic assumptions are made: (1) The self-growing soft body robot maintains a smooth curve with the same curvature at the steering joints during the steering process. (2) The self-grown soft body robot only changes along the length of the neutral axis, and the radial dimension remains unchanged. (3) Neglecting the effect of gravity of controllers such as servos in the steering motion module on the motion characteristics of the model; As shown in Figure 3, for the soft robot arm with multiple joints, there are multi-level mapping relationships for motion control, including soft drive space to configuration space mapping and configuration space to task space mapping.

Figure 3. Multi-level mapping relationship of soft body unit space.

2.1.3. Mapping of soft unit drive and configuration space

The geometric model of the Paragraph i segment of the soft body cell is given by drawing on the soft body robot kinematic model of Professor Walker, as shown in Figure 4. During the motion of the soft robot cell, it is driven by three drive lines at 120°, thus changing its soft arm bending angle $$ \theta $$, $$ \theta\in(0,\pi $$) and soft arm deflection angle $$ \phi $$, $$ \phi\in(0,2\pi $$) with the Z-axis. Therefore, the arc curve parameters [$$ \theta $$($$ L_i $$), $$ \kappa $$($$ L_i $$), $$ \phi $$($$ L_i $$)] are used to describe the soft body unit poses, where $$ L_i $$ = [$$ l_{i_1} $$, $$ l_{i_2} $$, $$ l_{i_3} $$] to describe the actual length of the drive line. The unit arc curve parameters can be derived by geometric operations as follows:

(1) $$ \begin{equation} \phi_i = \arctan\dfrac{\sqrt{3}}{3} \cdot \dfrac{(l_{i_2} + l_{i_3} - 2l_{i_3})}{(l_{i_2}-l_{i_3})} \end{equation} $$

(2) $$ \begin{equation} \theta_i = 2 \dfrac{\sqrt{{l_{i_1}^2+l_{i_2}^2+l_{i_3}^2-l_{i_1}l_{i_2}-l_{i_2}l_{i_3}-l_{i_1}l_{i_3}}}}{3r} \end{equation} $$

(3) $$ \begin{equation} \kappa_i =2\dfrac{\sqrt{{l_{i_1}^2+l_{i_2}^2+l_{i_3}^2-l_{i_1}l_{i_2}-l_{i_2}l_{i_3}-l_{i_1}l_{i_3}}}}{r(l_{i_1}+l_{i_2}+l_{i_3})} \end{equation} $$

Figure 4. Geometric model of segment i of the soft body unit.

The bending angle $$ L_i $$ of the centerline of the soft body arm according to the arc length equation can be expressed as:

(4) $$ \begin{equation} \theta_i =L_i\kappa_i \end{equation} $$

where the relationship between the driver length and the bending and deflection angles is:

(5) $$ \begin{equation} \begin{aligned} \left\{ \begin{array} ll_{i_1}=\theta_i\times(\dfrac{1}{\kappa_i}-r\times cos\phi_i)\\ l_{i_2}=\theta_i\times(\dfrac{1}{\kappa_i}- \dfrac{1}{2}r\times cos\phi_i-\dfrac{\sqrt{{3}}}{2}r\times sin\phi)\\ l_{i_3}=\theta_i\times(\dfrac{1}{\kappa_i}-\dfrac{1}{2}r\times cos\phi_i)-\dfrac{\sqrt{{3}}}{2}r\times+sin\phi)\\ \end{array} \right. \end{aligned} \end{equation} $$

2.1.4. Mapping of soft unit conformation and task space

The mapping relationship from the soft robot configuration space to the task space can be improved by combining the arc curve parameters with the D-H parameters, thus establishing the homogeneous transformation matrix (HTM) to describe the spatial relationship between adjacent coordinate systems, as shown in Figure 4. Through geometric analysis, the coordinate system $$ O_i-X_iY_iZ_i $$ to the coordinate system $$ O_{i+1}-X_{i+1}Y_{i+1}Z_{i+1} $$ can be decomposed into three rotation processes and one translation process: (1) Rotate $$ \phi_i $$ about the $$ Z_i $$ axis (2) Rotate $$ \theta_i $$ about the new $$ Y_i $$ axis (3) Rotate along the new $$ Y_i $$ axis $$ ||{\bf{O_iO_{i+1}} }|| $$. (4) Rotate $$ -\phi_i $$ around the new $$ Z_i $$ axis, and combined with the arc curve parameters, the chi-square transformation matrix can be found as:

(6) $$ \begin{equation} \begin{aligned} {T}= \begin{bmatrix} c^2\phi_ic\theta_i+s^2\phi_i & c\phi_is\phi_ic\theta_i-c\phi_is\phi & c\phi_is\theta_i & \dfrac{L_{i} cos\phi_i(1-cos\theta_i)}{\theta_i}\\\\ c\phi_is\phi_ic\theta_i-c\phi_is\phi & c^2\phi_i+s^2\phi_ic\theta_i & s\phi_is\theta_i &\dfrac{L_{i} sin\phi_i(1-cos\theta_i)}{\theta_i}\\ -c\phi_is\theta_i & -s\phi_is\theta_i & c\phi_i&\dfrac{L_{i} sin\phi_i}{\theta_i} \end{bmatrix} \end{aligned} \end{equation} $$

According to the above HTM transformation, the mapping of the soft body robot arm configuration space to the task space can be found. From equation (6), the joint position of the robot arm, [$$ P_{x_i},P_{y_i},P_{z_i} $$], consists of three equations, and all of them can be expressed by the arc curve parameters $$ \theta $$, $$ \phi $$, and $$ L $$.

(7) $$ \begin{equation} \begin{aligned} \left\{\begin{array}{l} \mathrm{p}_{x_{i}}=\frac{L_{i_{1}} \cos \phi_{i}\left(1-\cos \theta_{i}\right)}{\theta_{i}} \\ \mathrm{p}_{y_{i}}=\frac{L_{i_{1}} \sin \phi_{i}\left(1-\cos \theta_{i}\right)}{\theta_{i}} \\ \mathrm{p}_{z_{i}}=\frac{L_{i_{1}} \sin \phi_{i}\left(1-\cos \theta_{i}\right)}{\theta_{i}} \end{array}\right. \end{aligned} \end{equation} $$

2.1.5. Inverse kinematic model of the soft body unit

To establish the mapping from the task space to the configuration space of a soft robot and thus the inverse kinematics model of a soft arm. However, it has been a challenging problem to realize the inverse kinematics solution of the soft robot arm by solving the nonlinear equations through HTM transformation. In order to improve the reliability of the solution and reduce the difficulty of the inverse kinematics solution so as to avoid the possible singular solution or even no solution in the inverse kinematics solution process. Assuming that the length of each segment of the flexible arm is equal, only two variables, bending and deflection, are considered, and the solution formula is as follows:

(8) $$ \begin{equation} \begin{aligned} \left\{\begin{array}{c} \theta_{i}=\tan ^{-} 1\left(\frac{2 Z_{i} \times \sqrt{X_{i}^{2}+Y_{i}^{2}}}{Z_{i}^{2}-\left(X_{i}^{2}+Y_{i}^{2}\right)}\right), Z_{i}>\sqrt{x_{i}^{2}+y_{i}^{2}} \\ \theta_{i}=\pi+\tan ^{-} 1\left(\frac{2 Z_{i} \times \sqrt{X_{i}^{2}+Y_{i}^{2}}}{Z_{i}^{2}-\left(X_{i}^{2}+Y_{i}^{2}\right)}\right), Z_{i}<\sqrt{x_{i}^{2}+y_{i}^{2}} \end{array}\right. \end{aligned} \end{equation} $$

(9) $$ \begin{equation} \begin{aligned} \left\{ \begin{array}{l} \phi_i=\tan^{-1}\left(\dfrac{Y_i}{X_i}\right),x_i > 0 y_i > 0\\ \phi_i=\tan^{-1}\left(\dfrac{Y_i}{X_i}\right)+\pi,x_i < 0 \\ \phi_i=\tan^{-1}\left(\dfrac{Y_i}{X_i}\right)+2\pi,other \end{array} \right. \end{aligned} \end{equation} $$

In order to improve the reliability of the solution and reduce the difficulty of the inverse kinematics solution, so as to avoid the possible singular solution or even no solution in the process of inverse kinematics solution. Assuming that each segment of the flexible arm is of equal length and only two variables, bending and deflection, to verify the correctness of the kinematic model, its single-joint kinematic algorithm is simulated in the Matlab simulation environment with the following relevant parameters: the joint length is 140 mm, the diameter of the circle where the drive line is located is 10 mm, the bending angle is [0, $$ \pi $$], and the rotation angle is [0, 2$$ \pi $$] so that the number of sampling is step=60 times, and the results of each drive line length change and end position change are derived. The simulation results are shown in Figures 5 and 6.

Figure 5. Single joint end position change curve.

Figure 6. Surrounding model of obstacles.

PATH PLANNING BASED ON PARTICLE SWARM ALGORITHM

3.2. Obstacle environment modeling

Three-dimensional obstacles generally have irregular geometrical shapes, so this paper uses the regular body envelope of the obstacle to approximate the modeling. This modeling method enlarges the obstacle domain to a certain extent but makes the obstacle domain greatly simplified and effectively improves the efficiency of planning. At a certain moment, as shown in the red joint $$ P_{n-1}P_n $$, the connecting rod will still collide with the obstacle if only the obstacle envelope is used as the judgment basis for path planning.

Figure 7. Surrounding model of obstacles.

A bubble film-wrapped obstacle modeling method is proposed to address the problems of the motion of soft robots. Due to the characteristics of soft body robots in the motion process, soft-body joints will certainly produce motion deflection. This paper is in the obstacle surrounded by another layer of bubble film attached as a non-collision point in particle swarm planning. The positions of joints $$ P_nP_{n-1} $$ and $$ P_{n+1}P_n $$ are shown in Figure 8, and the next moment is seen when the soft body joint safely crosses the obstacle while passing through the bubble film. This method makes the robot safer for path-tracking in an obstacle environment. Based on the above obstacle modeling, in order to make the planning path more accurate and reliable, this paper considers the size of the flexible robot and sets the size of the diameter of the flexible robot as D. Let d be the vertical distance from the center of the obstacle circle, $$ C_k $$ to the joint, $$ P_nP_{n-1} $$, and $$ R_k $$ be the radius encircled by the obstacle, then the threat cost function, $$ F_2 $$, of the joint is:

(11) $$ \begin{equation} F_2(R_i) = \sum\limits_{j=1}^{n-1} \sum\limits_{k=1}^{K}W_k \left\| {\bf{P_{ij}}P_{ij+1}} \right\| \end{equation} $$

(12) $$ \begin{equation} \begin{aligned} W_{k}\left\|\mathrm{P}_{\mathrm{ij}} \mathrm{P}_{\mathrm{ij}+1}\right\|=\left\{\begin{array}{l} 0, d_{k}>R_{k}+S \\ R_{k}+S-d_{k}, R_{k}<d_{k} \leq R_{k}+S \\ \infty, d_{k} \leq R_{k} \end{array}\right. \end{aligned} \end{equation} $$

Figure 8. Schematic diagram of robot movement under obstacle environment.

By considering constraints such as optimization, security, and feasibility of the path $$ R_i $$ according to Eqs. (10-12), the overall cost-cost function can be defined as:

(13) $$ \begin{equation} F(R_i) = \ \sum\limits_{k=1}^{3}W_kF_k(R_i) \end{equation} $$

where $$ W_k $$ is the weighting factor. The selection of the fitness function is particularly important when performing path planning for soft robots, and in this paper, the optimal overall cost ambiguity of soft robots in complex environments, such as obstacles, is used as the fitness function.

3.3. Elementary particle swarm optimization for soft robot trajectory planning

In this paper, considering the structural characteristics of the robot itself and the working environment requirements, only three scatter points need to be designed to avoid the bending angle being too large during the tracking process, which leads to the soft robot posture not being reached. Here, the three scatter points are regarded as one particle, the three-dimensional path is obtained by fitting the three-dimensional spline curve function, and the three-dimensional path length from the starting point to the target point is further solved as the particle adaptation value. In this paper, the simulation is carried out by MATLAB, and the effect is shown in Figure 9. The algorithm first obtains the initial positions and velocities of the particles randomly. It calculates the fitness based on the three scatter points of many particles to obtain the individual and global optimal solutions. Finally, the particle swarm position is updated iteratively by updating the equation:

(14) $$ \begin{equation} x_i=x_{i-1}+v_{i-1} \end{equation} $$

(15) $$ \begin{equation} v_i=w_{i-1}+c_1r_1(p_{best}-x_{i-1})+c_2r_2(g_{best}-x_{i-1}) \end{equation} $$

Figure 9. Path planning based on particle swarm optimization.

In the n-dimensional search space, the position of particle $$ i $$ ($$ i $$=1, 2, ..., $$ N $$) is denoted by $$ x_i(k) $$= $$ (x_{i1}(k) $$, $$ x_{i2}(k) $$, ..., $$ x_{in}(k) $$), and the parameter of the best position of an individual searched by particle $$ i $$ is $$ p_{best} $$=$$ (p_{i1}(k) $$, $$ p_{i2}(k) $$, ..., $$ p_{in}(k)) $$, and the parameter of the global best of the population is $$ g_{best} $$=$$ (g_{i1}(k) $$, ..., $$ g_{i2}(k) $$)...., $$ g_{in}k) $$), w is the inertial coefficient, which is the degree of trust on the current velocity direction. In this article, w is set to 1.2; $$ c_1 $$ and $$ c_2 $$ are the learning factors that regulate the step size in the learning process, $$ c_1 $$ = $$ c_2 $$=2; $$ r_1 $$ and $$ r_2 $$ are random values to increase the search randomness to avoid particles falling into local optimum. The path-planning process based on the particle swarm algorithm is shown in Figure 10. Through the basic PSO algorithm and soft body robots for path planning, simulation experiments revealed that the basic PSO algorithm can obtain the global optimal path. However, the algorithm itself is susceptible to falling into the local optimum, slower convergence, and other problems ^[19,20]. In order to solve these problems, this paper introduces an improved PSO algorithm with adaptive weights, referred to as (Adaptive Weighted PSO, AWPSO), aimed at enhancing the convergence of the speed and execution efficiency of algorithms.

Figure 10. Flow chart of particle swarm optimization algorithm.

4. PATH TRACKING ALGORITHM BASED ON THE IDEA OF TIP SELF-GROWTH

Path tracking means the soft robot follows a path into the obstacle space and reaches the target point. The required path is collision-free, so this paper studies a path-tracking algorithm based on the idea of "tip self-growth" to make the soft robot move collision-free in complex environments after obtaining the motion path by combining the particle swarm algorithm. The principle is to imitate the motion mechanism of self-growing vine tops in nature; during the path-tracking process, the tip of the soft robot keeps tracking the path while its body matches the path. The computational effort of the kinematic algorithm is significantly reduced.

Efficiently and accurately matching the soft body joints to the desired path at each step is the key problem in path tracking, and the essence of this problem is to efficiently find the points that precisely match the joints under the joint length constraints. This problem can be expressed as the equation:

In the equation, L is the joint length, and the length remains approximately unchanged in states such as bending during motion. The tip self-growth algorithm was developed to improve the efficiency of the planning process. The soft arm tip tracks the path movement, and the algorithm only needs to calculate the motion process of the tip joint $$ P_{i+1} $$. The rest of the joints use the tip joint information of the previous moments to move forward. The algorithm is described in detail as follows: 1) Initialize the robot position so that the initial position $$ P_{i+1} $$ of the tip joint of the soft robot coincides with the starting point R(0) of the planned path. 2) The tip joint enters the workspace. Here, the path length $$ L $$ is divided into three steps, as shown in Figure 11. Make a circle of radius $$ L $$ with the joint point $$ P_i $$ on the moving space as the center and the intersection point obtained with the path as the position of $$ P_{i+1} $$ joint at this moment, and the corresponding coordinates of the three moments are shown as the red dots on the path. In the figure, $$ P^{"}_{i+1} $$$$ P^{"}_{i} $$ is the position of the joint point after three moves. When $$ P_{i} $$ reaches $$ R(0) $$ after three moves, the top joint $$ P^{"}_{i+1} $$$$ P^{"}_{i} $$ has wholly entered the working area.

Figure 11. Tip joint movement.

3) Tip self-growth process. After process 2), tip joint $$ P_{i+1} $$$$ P_i $$ has wholly entered the working area, and joint $$ P_i $$$$ P_{i-1} $$ can completely repeat the motion of the head joint in the previous stage. At this time, the tip joint continues to track the path; the $$ P_i $$ position point of the sub-tip joint will pass the path point of $$ P_{i+1} $$ in the previous stage, in turn, the movement process to $$ P_i $$ position as the center of the circle, in turn, to make a circle of radius $$ L $$, the intersection point obtained to make the path point of $$ P_{i+1} $$, as shown in the blue curve in Figure 12. Similarly, all $$ P_{i+1} $$ position points are solved according to the path iteration.

Figure 12. Tip self-growth motion.

(4) End position adjustment. When the tip joint reaches the target point of the previous interpolation moment $$ t_{end-1} $$, the soft body joint, according to the previous growth of the complete joint $$ L $$, will lead to the soft body joint endpoint cannot stay in the target position. In this paper, an end position adjustment algorithm is proposed, as shown in Figure 13, for a soft robot tip joint $$ P_{i+1} $$$$ P_i $$ with a target position Goal. By the path tracking algorithm of tip self-growth, it is ensured that the amount of motion between the end position and the target point at the moment of $$ t_{end-1} $$ does not exceed S. The target position Goal falls within a sphere with $$ P_{i+1} $$ as the center and radius S, as shown in the shaded area in the figure. The position is updated by first making the end position coincide with the target position, then making a sphere of radius L with $$ P^{"}_{i+1} $$ as the center of the sphere, and using the intersection of the sphere and the soft body $$ P^{"}{_i} $$ as the new position of the tip joint. After the position update is completed for the tip joint, the rest of the joints of the soft body robot from the sub-tip end to the control box are updated sequentially. Figure 14 shows a schematic diagram of the final movement of the soft robot by growing only the displacement S ($$ S<L $$) through the end joints as an example of three soft joints. Through the bit shape of the soft robot at the moment of $$ t_{end-1} $$, the tip joint $$ P_3P_2 $$ is updated to the position $$ P^{'}_{3}P^{'}_{2} $$ with Goal as the target position according to the end adjustment algorithm, the joint $$ P_2 $$$$ P_1 $$ is updated to $$ P'_{2}P'_{1} $$ with $$ P^{'}{_2} $$ as the target position, and similarly, the joint $$ P_1 $$ is updated to $$ P^{'}_{1} $$. Through the end, joint horizontal growth, therefore, the starting end position $$ O $$ is updated to $$ O^{'} $$ ($$ OO^{'} < S $$). The soft robot can reach the specified position by completing the iteration from the tip to the control box through all joints.

13. End position adjustment.

Figure 14. 3-section soft joint end position adjustment.

Figure 15. Flowchart of tip self-growth algorithm.

This method only needs to calculate the location point of the tip joint $$ P_{i+1} $$ in each step, and the soft robot tip self-grows to complete the action following, significantly simplifying the calculation and improving path tracking efficiency.

5. SIMULATION EXPERIMENT

In order to verify the correctness and effectiveness of the soft continuous robot head traction algorithm proposed above, the algorithm is simulated and analyzed in the Matlab environment. First, the obstacle map is randomly generated using Matlab in a 3D space with a length and width of 100 cm and a height of 200 cm. The constrained space containing bubble film wrapping is generated by the method described in Section 3.2, as shown in Figure 18. The coordinates of the starting point are located at the point (0, 0, 20). Through the elementary particle swarm algorithm, the number of particles of the algorithm is set to 50, the number of iterations is 300, w=1.2, $$ c_1 $$=$$ c_2=2 $$, and the global path planning is carried out for the full length of the shortest path, 154.09cm. Using the "tip self-growth" process described in Section 4 to track the path, the experimental single joint length L=14cm was divided into 15 steps to move, and the average time was calculated to be only 0.0321s, and the simulation results obtained at the moment of $$ t_{end-1/2} $$ are shown in Figure 17. The algorithm gives full play to the advantages of the soft robot's high degree of freedom and easy obstacle avoidance.

Figure 16. Restricted area path planning map.

Figure 17. $$ t_{end-1/2} $$ moment path tracking.

Figure 18. End position adjustment.

At the moment of $$ t_{end-1} $$, the "tip" position reaches the target point. At this time, the end position adjustment algorithm is used, and the adjustment effect is shown in Figure 18, which shows that the robot crosses the obstacle without collision. The endpoint reaches the target position at the moment of tend. So far, the "tip self-growth" algorithm has been completed. To illustrate the reduced computational effort of the method in this paper compared to existing heuristics in the inverse solution of the soft robot kinematics and ensure the effectiveness of the soft robot motion in a constrained environment, the execution times of the five different methods are compared below. Here, 1) "Tip self-growth" method; 2) improved end-following; 3) fabrik method; 4) trajectory method; 5) weighted Jacobi matrix generalized inverse method Table 1 shows the total execution time of the solutions of the five methods, from which it can be seen that the execution time of the "tip self-growth"-based path-following algorithm is shorter than that of the heuristic algorithm, the numerical method based on the Jacobi matrix. The reason is that after getting the planned path, the "tip self-growth" algorithm only needs to solve the position of the tip joint step by step without solving the generalized inverse and multiplication of the matrix and performing geometric iterations, which makes the operation the fastest. Also, the growing soft robot structure studied in this paper can fully utilize the soft robot's flexibility and obstacle avoidance characteristics compared with other algorithms. The introduction of the end position adjustment algorithm can effectively ensure that the end of the robot reaches the target position with a fast running time.

The next step of the research can be carried out in the following aspects: 1) to build a physical prototype and apply the obstacle avoidance planning algorithm to the real object for the mutual verification of simulation and physical experiments; 2) to consider the error in the physical prototype in the process of movement, design a closed-loop control system with feedback, and optimize it so as to reduce the error; 3) to study and analyze the dynamics model of the soft body robot so as to make the control effect better.

Next, the particle swarm algorithm with basic weights (PSO) and the improved PSO algorithm (AWPSO) are used for path planning of soft body robots to compare the actual results of the two algorithms. The improved PSO algorithm adopts the adaptive weighting PSO algorithm with $$ w_{max} $$= 1.5, $$ w_{min} $$= 1.0, $$ c_1 $$= $$ c_2 $$= 2. Both algorithms have a population of 50 particles and a maximum number of iterations of 300. The coordinates of the start point are located at the point (0, 0, 20), and the end point coordinates are located at the point (100, 90, 70). Thirty times of the two algorithms were performed for the path planning, and the obtained global optimal path was recorded. The global optimal path length is recorded, and the average value of the optimal individual fitness value with the number of iterations is shown in Figure 19. The mean and variance of the global optimal paths are calculated, and the results are shown in Table 2. From Figure 19 and Table 2, it can be seen that both PSO and AMPSO can realize global path planning, and the globally optimal path lengths planned by both of them are not much different. However, the improved algorithm proposed in this paper has a shorter computation time and higher efficiency, and the variance is slightly smaller than that of the former. Combined with the starting point and end point set by this path planning, it is found that the search environment of the algorithm is better, without many complex obstacles. Therefore, it can be concluded that the basic particle swarm algorithm and the improved algorithm proposed in this paper can realize the global path planning in a relatively simple environment, but the improved algorithm converges faster, and the stability of the algorithm is higher.

Figure 19. Fitness of path planning varies with the number of iterations.

6. CONCLUSIONS

The main contribution of this paper is to address the problem of obstacle avoidance planning for the tip self-growing soft robot. Firstly, the configuration and modeling of the soft robot are introduced, and the derivation between the multi-level kinematic mapping relations is completed. Further, the particle swarm algorithm is used to perform a path search in the free motion space of the robot to achieve collision-free path planning. Considering the position deviation of the real robot due to the flexible structure during the motion, the obstacle is modeled with a different design of bubble film wrapping to avoid a collision. The proposed tip self-growing path-following algorithm improves the algorithm's efficiency in path-following. It ensures that the flexible robot can reach the target position while passing through the restricted environment. The effect of the "tip self-growth" algorithm on path tracking is analyzed by MATLAB simulations. The feasibility of particle swarm algorithm-based obstacle avoidance planning for soft robots is verified.

Through theoretical analysis and simulation results to prove its effectiveness, particle swarm-based soft robot path planning can achieve effective obstacle avoidance, and the "tip self-growth" algorithm in the solution is also relatively simple. Therefore, this method can be used for soft robot obstacle avoidance planning while also establishing a foundation for real-time control of the soft robot.

The next step of the research can be carried out in the following aspects: 1) to build a physical prototype and apply the obstacle avoidance planning algorithm to the real object for the mutual verification of simulation and physical experiments; 2) to consider the error in the physical prototype in the process of movement, design a closed-loop control system with feedback, and optimize it so as to reduce the error; 3) to study and analyze the dynamics model of the soft body robot so as to make the control effect better.

DECLARATIONS