Improved Artiﬁcial Potential Field and Dynamic Window Method for Amphibious Robot Fish Path Planning

: Aiming at the problems of “local minimum” and “unreachable target” existing in the traditional artiﬁcial potential ﬁeld method in path planning, an improved artiﬁcial potential ﬁeld method was proposed after analyzing the fundamental causes of the above problems. The method solved the problem of local minimum by modifying the direction and inﬂuence range of the gravitational ﬁeld, increasing the virtual target and evaluation function, and the problem of unreachable targets is solved by increasing gravity. In view of the change of motion state of robot ﬁsh in amphibious environments, the improved artiﬁcial potential ﬁeld method was fused with a dynamic window algorithm, and a dynamic window evaluation function of the optimal path was designed on the basis of establishing the dynamic equations of land and underwater. Then, the simulation experiment was designed under the environment of Matlab2019a. Firstly, the improved and traditional artiﬁcial potential ﬁeld methods were compared. The results showed that the improved artiﬁcial potential ﬁeld method could solve the above two problems well, shorten the operation time and path length, and have high efﬁciency. Secondly, the inﬂuence of different motion modes on path planning is veriﬁed, and the result also reﬂects that the amphibious robot can avoid obstacles ﬂexibly and reach the target point accurately according to its own motion ability. This paper provides a new way of path planning for the amphibious robot.


Introduction
Over the last few decades, mobile robots have been widely used in many circumstances, which improves the working efficiency and reduces the cost of work to a great extent, and these advantages show their great developmental potential [1][2][3]. Amphibious robotic fish in mobile robots are the object of this study. This is not only an underwater robot with the efficient propulsion function of fish, but also an overland robot adapted to complex terrestrial environments.
The tasks of a mobile robot can be divided into transport tasks and reconnaissance tasks in general. No matter which kinds of works are completed, mobile robots all require the capability of fulfilling their mission autonomously under the fast modern development pace [4], and path planning as one of the crucial techniques ensures that the mobile robot can achieve independence in its work without manual intervention [5].
Path planning requires considering the maneuverability of robots and environmental constraints to find a trajectory from the initial point to the target point, and avoid all obstacles [6]. In essence, it is a method to optimize a system with complex constraints. In order to achieve this goal, many scholars have done a lot of research and put forward

Kinetic Analysis
It is worth noticing that the subject of this paper is the amphibious robot. The fin surface is driven by the fin rays, as they move up and down in turn in both environments. Its motion mode has its own characteristics in different environments. The propulsion force is generated by waving the fin surface underwater, and on the land, it is generated by the friction between the fin surface and the ground. The amplitude and frequency of the oscillation need to be further adjusted according to the different environments. The structure of the amphibious robot is generally divided into two parts: fish body and fin surface. The fish body includes the control system, the transmission system and various sensors on it, and the fin surface is regarded as the motion system.

Kinetic Analysis on the Land
The Newton-Euler equation is chosen to describe the dynamic equation, which can express two motion modes of rotation and translation, respectively, and it obtains the initial parameters that are taken as the input for algorithm. As mentioned above, the motion of the amphibious robot is powered by the fin surface waving, and the undulation is controlled by Appl. Sci. 2021, 11,2114 3 of 15 the swing of the fin with certain phase differences. Therefore, the specific Newton equation of kinetic analysis for the fin can be expressed as Equation (1).
where i is the fin number, . v i is the acceleration of the center of mass of the ith fin, T i is the constraining force, F a i is the active force, and F i is the external main force of the ith fin. The expression of the Euler equation is denoted as follows: where I C represents the inertia, ω i represents the angular velocity of the ith fin, . ω i represents the angular acceleration of the ith fin, L a i and L t i represent the main moment and the counter main moment, respectively, and M i is the external main moment of the ith fin.
When the amphibious robot is analyzed as a whole, the diagram of stress analysis can be formulated, as shown in Figure 1. The friction is produced by the contact between the fin surface and the ground, which creates the driving force to propel the amphibious robot forward. Thus, the analysis mainly focuses on the contact point, where G is gravity, F is the supporting force, and f . is the friction force. The summary formula can be expressed as Equations (3) and (4).
where λ is the wavelength of the fin surface, ..
x g and ..
z g are the inertia force in the x axis and the z axis, respectively, while x g and z g are the coordinates of the center of mass in the x axis and z axis, respectively, x is the abscissa of the contact point, and µ is the coulomb friction coefficient. surface. The fish body includes the control system, the transmission system and various sensors on it, and the fin surface is regarded as the motion system.

Kinetic Analysis on the Land
The Newton-Euler equation is chosen to describe the dynamic equation, which can express two motion modes of rotation and translation, respectively, and it obtains the initial parameters that are taken as the input for algorithm. As mentioned above, the motion of the amphibious robot is powered by the fin surface waving, and the undulation is controlled by the swing of the fin with certain phase differences. Therefore, the specific Newton equation of kinetic analysis for the fin can be expressed as Equation (1).
where is the fin number, is the acceleration of the center of mass of the fin, is the constraining force, is the active force, and is the external main force of the fin.
The expression of the Euler equation is denoted as follows: where represents the inertia, represents the angular velocity of the fin, represents the angular acceleration of the fin, and represent the main moment and the counter main moment, respectively, and is the external main moment of the fin.
When the amphibious robot is analyzed as a whole, the diagram of stress analysis can be formulated, as shown in Figure 1. The friction is produced by the contact between the fin surface and the ground, which creates the driving force to propel the amphibious robot forward. Thus, the analysis mainly focuses on the contact point, where is gravity, is the supporting force, and is the friction force. The summary formula can be expressed as Equations (3) and (4).
where is the wavelength of the fin surface, and are the inertia force in the x axis and the z axis, respectively, while and are the coordinates of the center of mass in the x axis and z axis, respectively, is the abscissa of the contact point, and is the coulomb friction coefficient. From Equations (1)-(4), the velocity and acceleration of each fin and the amphibious robot can be acquired. This provides the initial parameters and dynamical equation of the amphibious robot on land for the algorithm.

Kinetic Analysis Underwater
Different from the motion mode on the land, the underwater motion mainly depends on the water propulsion on both sides of the fin surface. Besides the rotation motion, a roll motion should be taken into account as well. A 5-DOF motion model of the designed amphibious robot is shown in Figure 2, which is expressed in a Cartesian coordinate system without rotational motion around the z axis.

Kinetic Analysis Underwater
Different from the motion mode on the land, the underwater motion mainly depends on the water propulsion on both sides of the fin surface. Besides the rotation motion, a roll motion should be taken into account as well. A 5-DOF motion model of the designed amphibious robot is shown in Figure 2, which is expressed in a Cartesian coordinate system without rotational motion around the z axis. In Figure 2, α, β and γ are the rotation angles corresponding to the x, y and z axes, respectively. The expression of the kinetic equation is given as follows: where is the centroid vector of the amphibious robot at time t, is the rotation angle around the x axis at time t, is the velocity, is the angular velocity at time t, is the external moment, is the inertia, and is the distance from any point to the rotation centerline.
The basic motion parameters can be determined in combination with the designed amphibious robot through the above analysis, and the specific parameters are shown in Table 1. In Figure 2, α, β and γ are the rotation angles corresponding to the x, y and z axes, respectively. The expression of the kinetic equation is given as follows: where C t is the centroid vector of the amphibious robot at time t, θ x t is the rotation angle around the x axis at time t, V t is the velocity, ω t is the angular velocity at time t, M is the external moment, I C is the inertia, and r is the distance from any point to the rotation centerline.
The basic motion parameters can be determined in combination with the designed amphibious robot through the above analysis, and the specific parameters are shown in Table 1. Through the kinetic analysis of the motion modes in different environments, a more vivid, clear and accurate motion description is obtained, which can theoretically explain the basic principle of the motion propulsion mode of the amphibious robot. Meanwhile, these parameters also provide the motion information of the amphibious robot for the algorithm, so that the simulation process can be closer to the practical case and also deliver more convincing results.

Traditional Artificial Potential Field Method
The traditional artificial potential field method assumes that there is a virtual repulsion field around the obstacles, and there is also a virtual attraction field around the target point. The robot is affected by the attraction field to approach the target point, then avoids the obstacles under the repulsion field, and finally reaches the target point without any collisions.
The functions of the attraction field U att (q), repulsive field U rep (q) and resultant field U(q) are denoted as follows: where k att and k rep are the attraction coefficient and repulsive coefficient, respectively, q is the current position of the robot, q o is the position of the obstacle, q g is the position of the target point, and ρ 0 is the influence distance of the obstacle. When the distance between the obstacle and the robot is greater than ρ 0 , the robot cannot be affected. By deriving the distance from the above Equations (10)-(12), we can get the attractive force F att (q), repulsion force F rep (q) and resultant force F(q). Their functions are represented as follows:

Facing Problems
The traditional artificial potential field method mainly faces two problems; one is the local minimum problem, the other is the goal unreachable problem.
The first problem is shown in Figure 3. In these two cases, the resultant force of the robot at this position is 0. As the robot's movement is in the direction of the resultant force, the robot may stop moving so that it cannot arrive at the target point.
The second problem is shown in Figure 4. The goal is near the obstacle. While the robot is approaching the goal, the repulsion force is greater than the attractive force because the attractive force is directly proportional to the distance between the robot and the goal, which makes the robot unable to reach the target point and move away from the goal.

Facing Problems
The traditional artificial potential field method mainly faces two problems; one is the local minimum problem, the other is the goal unreachable problem.
The first problem is shown in Figure 3. In these two cases, the resultant force of the robot at this position is 0. As the robot's movement is in the direction of the resultant force, the robot may stop moving so that it cannot arrive at the target point.  The second problem is shown in Figure 4. The goal is near the obstacle. While the robot is approaching the goal, the repulsion force is greater than the attractive force because the attractive force is directly proportional to the distance between the robot and the goal, which makes the robot unable to reach the target point and move away from the goal.

Improved Attraction Field
A large number of scholars have improved and optimized the traditional artificial potential field method from various aspects. Most of them paid attention to modifying the robot at this position is 0. As the robot's movement is in the direction of the resultant force, the robot may stop moving so that it cannot arrive at the target point. The second problem is shown in Figure 4. The goal is near the obstacle. While the robot is approaching the goal, the repulsion force is greater than the attractive force because the attractive force is directly proportional to the distance between the robot and the goal, which makes the robot unable to reach the target point and move away from the goal.

Improved Attraction Field
A large number of scholars have improved and optimized the traditional artificial potential field method from various aspects. Most of them paid attention to modifying the

Improved Attraction Field
A large number of scholars have improved and optimized the traditional artificial potential field method from various aspects. Most of them paid attention to modifying the potential field function, and there are some reasons for this, as follows: (1) the expression of the potential field function is easy to modify; (2) the potential field is more in line with actual physics, so it is easy to understand; (3) the modification of the potential field can improve the efficiency of the algorithm in essence. Considering the following reasons, this paper focuses on the improvement of the attraction potential field.
First of all, the core of the local minimum problem is that the values of the attraction force and repulsion force are the same, and the directions are opposite, which can be improved by modifying one of them. As shown in Figure 3a, it can be found that modifying the value of the force field cannot solve this problem. No matter how the values of attraction force and repulsion force are modified, it can only aggravate the instability of the robot's motion, inducing vibration. This paper chooses to modify the direction of the attraction potential field to solve this problem. The general idea is as below: taking the goal as the center, a quadrilateral is generated within a certain distance. When the value of the resultant force is 0, the four corners of the created quadrilateral will be regarded as interim goals for navigation, and choosing which one is the goal relies on the evaluation function. After the robot reaches a position at a certain distance d 0 from the goal, the four virtual goals will be invalidated, so that the robot can reach the real goal. In addition, if the target point cannot be reached, it means the goal is too close to the obstacles, so we could increase the attraction force to make it reach the goal. According to the traditional attraction potential function, the attraction force is inversely proportion to the distance between the robot and the goal. Therefore, we can assume the attraction force as a constant within a certain range d 0 from the goal. In order to avoid the situation where the value of the resultant force is still 0 when the robot is near the goal, we can increase d 0 to solve this problem. The improved attraction potential field (Equation (17)) and schematic diagram ( Figure 5) are as follows: goals for navigation, and choosing which one is the goal relies on the evaluation function. After the robot reaches a position at a certain distance from the goal, the four virtual goals will be invalidated, so that the robot can reach the real goal. In addition, if the target point cannot be reached, it means the goal is too close to the obstacles, so we could increase the attraction force to make it reach the goal. According to the traditional attraction potential function, the attraction force is inversely proportion to the distance between the robot and the goal. Therefore, we can assume the attraction force as a constant within a certain range from the goal. In order to avoid the situation where the value of the resultant force is still 0 when the robot is near the goal, we can increase to solve this problem. The improved attraction potential field (Equation (17)) and schematic diagram ( Figure 5) are as follows: Equation (18) mainly compares the angle value θ between attraction force and repulsion force. When the value of the resultant force is 0, the angle value is equal to 180°. As such, the greater the difference is between the real angle value and 180°, the higher the evaluation function value's virtual goal can be; the robot is much more likely to move in the direction of the virtual goal a with higher evaluation function value. Equation (18) mainly compares the angle value θ between attraction force and repulsion force. When the value of the resultant force is 0, the angle value is equal to 180 • . As such, the greater the difference is between the real angle value and 180 • , the higher the evaluation function value's virtual goal can be; the robot is much more likely to move in the direction of the virtual goal a with higher evaluation function value.

Combing with Dynamic Window Method
The essence of the dynamic window method [22] is to sample the velocity space (v, ω) of the robot according to its dynamic characteristics, and generate new velocity v and angular velocity ω under the influence of various factors over time. The series of velocity spaces (v, ω) are simulated and analyzed to obtain a variety of paths; then, the optimal velocity space (v, ω) is obtained through the evaluation function. Finally, the whole optimal path will be obtained by repeating the process above.
The motion equation of the amphibious robot can be summarized as follows: where x t represents the displacement of the amphibious robot in the x axis at time t, y t and z t similarly represent the displacement in the y axis and z axis, θ t represents the motion angle of the robot at time t, ∆t represents the time step, v and ω represent the velocity and acceleration, respectively, and the specific values can be obtained from the above kinetic function. α, β and γ represent the angle between velocity v and the x axis, y axis, z axis, respectively.
This algorithm needs to be constrained by the robot's motion ability and environment, which consists of three points: (1) the velocity constraint V m , which is the velocity and angular velocity within the range of the robot's own motion ability; (2) the acceleration constraint V a , which measures the influence of the acceleration range of the robot on the speed; (3) the obstacle constraint V d , which is to ensure that the robot will not collide with the obstacle under the range of braking ability. The expressions are as follows: where . v a and . ω a represent the maximum acceleration of the robot at time t, . v b and . ω b represent the maximum deceleration, and dist(v, ω) represents the nearest distance between the robot and the obstacle.
The evaluation function is given as Equation (23).
where heading(v, ω) is the angle between the velocity direction and the goal, velocity(v, ω) is the value of the velocity, and a, b, c are the influence factors of angle, distance and velocity, respectively. This paper concentrates on the path planning in an amphibious environment. The trajectory and motion state of the robot can be restored as much as possible by combining with the artificial potential field method and the dynamic window method. The impact on the whole trajectory can be observed more precisely when the motion state changes, and this provides the data support for the further adjustment and optimization later on. Figure 6 shows the flow-process diagram of the hybrid algorithm. Firstly, information related to the environment, obstacles and the robot itself must be obtained. The former two items of information are identified by sensors mounted on the robot, while the latter information is obtained by its own parameters, kinematics and dynamics equations. Since this paper mainly studies amphibious bionic robotic fish, it is necessary to determine whether the environment of the robot is land or underwater. This can be identified by mounting a pressure sensor on the top of the robot. Before the environment is confirmed, the robot is in a static state. Then, the corresponding motion parameters are selected through the controller based on the environment. After the above preparation work is completed, the potential field function is established and the path planning for the unknown environment is carried out; meanwhile, the constraint function also constrains the robot itself. In the process of movement, whether the resultant force of the potential field is counteracted is constantly judged. If it is counteracted, the virtual target is added, and the potential field function is re-established. This process is repeated continuously until the resultant force is not counteracted. On this basis, it is also necessary to constantly determine whether the environment of the robot has changed, and if so, the motion parameters will be transformed. Finally, when approaching the target point, it is necessary to determine whether the distance between the robotic fish and the target point is less than the influence distance of the target point. If it is within the influence range of the target point, it is necessary to delete the virtual target point so as not to affect the accuracy when the virtual target point is confirmed as the real target point.
ously until the resultant force is not counteracted. On this basis, it is also necessary to constantly determine whether the environment of the robot has changed, and if so, the motion parameters will be transformed. Finally, when approaching the target point, it is necessary to determine whether the distance between the robotic fish and the target point is less than the influence distance of the target point. If it is within the influence range of the target point, it is necessary to delete the virtual target point so as not to affect the accuracy when the virtual target point is confirmed as the real target point.

Simulation Analysis
It is necessary to carry out a simulation to verify the effectiveness of the proposed hybrid algorithm. In this paper, the CPU of the experimental computer is i7-9750h, the memory of the computer is 16GB, and the operating system is windows 10. The algorithm is simulated and verified by the MATLAB 2019a.

Simulation of Improved Artificial Potential Field
In order to verify that the improved artificial potential field method can overcome the problems of local minimum and goals being unreachable, the traditional artificial potential field method and the improved artificial potential field method are compared and tested. In the process of several simulations, they performed the same, and all could reach the target point. However, in order to highlight the advantages of the improved algorithm, we specifically find this special case and record the coordinate points to present it.
The specific parameters are shown in Table 2. The simulation experiment is conducted for the problem of goal unreachable. The purpose is to verify whether the improved artificial potential field method can solve this problem and whether the execution time and path length can perform better than the traditional artificial potential field method. The results are shown in Figures 7-9, as follows: The simulation experiment is conducted for the problem of goal unreachable. The purpose is to verify whether the improved artificial potential field method can solve this problem and whether the execution time and path length can perform better than the traditional artificial potential field method. The results are shown in Figures 7-9, as follows:  It is very important to prove the conclusion with data. Table 3 is obtained by integrating the data in Figures 7-9. The main parameters are execution time and path length, due to these two data can well prove the performance of a path algorithm.  It is very important to prove the conclusion with data. Table 3 is obtained by integrating the data in Figures 7-9. The main parameters are execution time and path length, due to these two data can well prove the performance of a path algorithm.  It is very important to prove the conclusion with data. Table 3 is obtained by integrating the data in Figures 7-9. The main parameters are execution time and path length, due to these two data can well prove the performance of a path algorithm. Although the goal can be reached and the path lengths of both methods are the same, as shown in Figure 7, the improved algorithm is less time-consuming than the traditional algorithm, which means that the improved algorithm is more efficient. The advantages of the improved artificial potential field method can be obviously highlighted as given in Figures 8 and 9, as the robot cannot reach the goal by choosing the traditional artificial potential field method, and its execution time and path length are significantly longer than the improved method. As such, the results confirm that the improved artificial potential field method can solve the problem of goal unreachable, which reflects its effectiveness, efficiency and reliability.
The purpose is to verify whether the improved artificial potential field method can also solve this goal unreacheable problem, and whether the execution time and path length can perform better than the traditional artificial potential field method. The results are shown in Figures 10 and 11, as follows: The purpose is to verify whether the improved artificial potential field method can also solve this goal unreacheable problem, and whether the execution time and path length can perform better than the traditional artificial potential field method. The results are shown in Figures 10 and 11, as follows: The left side pictures represent the simulation results of traditional artificial potential field method, and the right side is the improved artificial potential field method. In the same environment, it can well show the excellent performance of the improved traditional artificial potential field method, which not only solves the local minimum problem, but also has shorter running path, and also proves the effectiveness of the improved algorithm. The purpose is to verify whether the improved artificial potential field method can also solve this goal unreacheable problem, and whether the execution time and path length can perform better than the traditional artificial potential field method. The results are shown in Figures 10 and 11, as follows: The left side pictures represent the simulation results of traditional artificial potential field method, and the right side is the improved artificial potential field method. In the same environment, it can well show the excellent performance of the improved traditional artificial potential field method, which not only solves the local minimum problem, but also has shorter running path, and also proves the effectiveness of the improved algorithm. The left side pictures represent the simulation results of traditional artificial potential field method, and the right side is the improved artificial potential field method. In the same environment, it can well show the excellent performance of the improved traditional artificial potential field method, which not only solves the local minimum problem, but also has shorter running path, and also proves the effectiveness of the improved algorithm.
It can be seen from Table 4 that the robot using the traditional artificial potential field method halts and vibrates when halfway, since the value of the resultant force is equal to 0 in the case of the local minimum problem. The improved artificial potential field method solves this problem well, which again verifies the superiority of the improved artificial potential field method compared with the traditional artificial potential field method.

Simulation of the Hybrid Algorithm
In order to enhance the simulation performance of the path planning for an amphibious robot, it is very important to take into consideration the impact of different motion modes on path planning. The dynamic window method is combined with the improved potential field method for simulation, because its algorithm requires the dynamics and kinematics equations of the robot as its own constraints, which is more in line with the actual situation. Figures 12-15 are the simulation results as follows, the right side figures show the curve which means the distance between the robot and the obstacle, and the left side is the whole process of the robot running.
Here, the numbers from 1 to 10 represent the obstacles' serial numbers. The horizontal axis represents the number of iterations, and the vertical axis represents the distance between the robotic fish and the obstacle. Every curve has this pattern: with an increase in time, the robot approaches the obstacle and then moves away from it, and finally, the fluctuation is caused by the constant correction of distance (essentially local minima problem). In the end, the vibration can approach and flatten quickly, indicating that at this time, it keeps a certain distance from the obstacle, and does not take a repeated road section. In other words, it can continue to move towards the target point. Meanwhile, it can also prove the correctness of the establishment of the artificial potential field equation. The bottom horizontal blue line indicates the minimum boundary distance, and our simulation is always within the safe range. In addition, the intersection of 10 lines with different distances in the figure is precisely due to the change of the environment, which changes the motion mode.

Simulation of the Hybrid Algorithm
In order to enhance the simulation performance of the path planning for an amphibious robot, it is very important to take into consideration the impact of different motion modes on path planning. The dynamic window method is combined with the improved potential field method for simulation, because its algorithm requires the dynamics and kinematics equations of the robot as its own constraints, which is more in line with the actual situation. Figures 12-15 are the simulation results as follows, the right side figures show the curve which means the distance between the robot and the obstacle, and the left side is the whole process of the robot running.     Here, the numbers from 1 to 10 represent the obstacles' serial numbers. The horizontal axis represents the number of iterations, and the vertical axis represents the distance between the robotic fish and the obstacle. Every curve has this pattern: with an increase in time, the robot approaches the obstacle and then moves away from it, and finally, the fluctuation is caused by the constant correction of distance (essentially local minima problem). In the end, the vibration can approach and flatten quickly, indicating that at this time, it keeps a certain distance from the obstacle, and does not take a repeated road section. In other words, it can continue to move towards the target point. Meanwhile, it can also prove the correctness of the establishment of the artificial potential field equation. The bottom horizontal blue line indicates the minimum boundary distance, and our sim-  Here, the numbers from 1 to 10 represent the obstacles' serial numbers. The horizontal axis represents the number of iterations, and the vertical axis represents the distance between the robotic fish and the obstacle. Every curve has this pattern: with an increase in time, the robot approaches the obstacle and then moves away from it, and finally, the fluctuation is caused by the constant correction of distance (essentially local minima problem). In the end, the vibration can approach and flatten quickly, indicating that at this time, it keeps a certain distance from the obstacle, and does not take a repeated road section. In other words, it can continue to move towards the target point. Meanwhile, it can also prove the correctness of the establishment of the artificial potential field equation. The bottom horizontal blue line indicates the minimum boundary distance, and our simulation is always within the safe range. In addition, the intersection of 10 lines with dif- We divide these into two groups. Figures 12 and 13 are group 1, and Figures 14 and  15 are group 2. Among all the environmental parameters and the parameters of robot fish itself, the only difference lies in the transformation time of the dynamic equation. This is because, according to the improved equation, only this parameter is easy to control. Figures 12 and 14 show that there is no time step for the transformation progress; however, Figures 13 and 15 show the results of the transformation motion mode with 20 times steps. It can be seen that the robot can reach the goal in all cases, which proves that the hybrid algorithm is effective. As shown in group 1, the transformation time has little impact on path planning; however, in the second group, which has a local minimum problem, the transformation time has a significant impact on the efficiency and stability of the whole path planning. We set the duration of the conversion so that the wavelength is shorter, faster and more stable. Therefore, the hybrid algorithm increases the transformation time as much as possible to ensure that it can be more efficient and effective.

Discussion
The artificial potential field method is widely used in local path planning because of its simple and real-time performance. Many scholars have studied the improved algorithm to further improve its performance. It is usually used as follows: through the improvement of the gravitational field model to avoid the problem of encountering obstacles, when the distance is too far and the gravitational force is too large; on the basis of the original repulsion field and considering the influence of the relative position and velocity between the target and the robot, a new repulsion function is introduced to reduce the strength of the repulsion potential field of the obstacle near the target, and solve the problem that the obstacle near the target leads to an unreachable target. By adding a random perturbation, the robot can jump out of the local optimal value, similar to the local optimal value solution of the gradient descent method, or react in advance by introducing the method of predicted distance.
Dynamic path planning in a complex environment is required for the environmental adaptability of amphibian robotic fish. This paper combines the advantages of the two methods: on the premise of exploring in the unknown environment and reaching the target, at the same time, the robot itself is constrained to ensure that the path planned according to the robot's own conditions can be achieved by the algorithm. This fusion algorithm can restore the motion track and motion state of the robot as much as possible, and the result is more convincing. Meanwhile, it can more accurately observe the influence degree of the change of motion state on the whole motion track, which is convenient for targeted improvement.

Conclusions
In this paper, an improved path planning algorithm for an amphibious robot is proposed, which overcomes the shortcomings of the existing algorithm. The simulation experiment in this paper can be divided into two parts: one is to verify that the improved artificial potential field method has a greater ability to solve the target unreachable and local minimum problems than the traditional artificial potential field method. Secondly, the feasibility of the fusion algorithm is verified, and the performance of the fusion algorithm is improved by adjusting the conversion time. The simulation is suitable for describing the path planning of amphibious robot. The results verify the efficiency, instantaneity, and reliability of the hybrid algorithm. This provides a new approach to path planning for this kind of amphibious robot with a variable motion pattern.