Improved artificial potential field method based on robot local path information

Abstract

The artificial potential field (APF) is an important method for robot path planning. However, some information in APF is not fully utilized in practical applications. In this paper, an improved artificial potential field (IAPF) method is presented, in which the local path information is defined and used. And the calculation formulas for various forces in IAPF are given, which include repulsive force (R-force) of obstacle on the robot, the attractive force (A-force) of target on the robot, and the resultant force of R-force and A-force. Then, based on the local path information, a method for solving the robot falling into local optimality problem is proposed and used into IAPF. Finally, IAPF is respectively simulated and discussed in general scenario, complex scenario, and scenarios with the same and different size of circular obstacles. The results show that IAPF has higher efficiency than traditional artificial potential field (TAPF) method and can overcome the local optimality problem. At the same time, IAPF is compared with dynamic window method in the scenarios with the same and different size of circular obstacles. The results show that IAPF is more efficient than the dynamic window approach (DWA) for robot path planning.

Introduction

The artificial potential field (APF) method proposed by Khatib in 1986 has become an important method for robot path planning. ¹ The basic principle of APF is to establish a virtual force field, which is composed of the attraction potential field (A-PF) generated by the target and the repulsion potential field (R-PF) generated by all obstacles in the environment. ² When the robot is located in this environment, the target exerts a attractive force (A-force) on the robot, all the obstacles exert repulsive forces (R-forces) on the robot. And the resultant force of the two kinds of forces produces the movement direction for the robot, so that the robot can move from the starting point to the target point. ³ Compared with path planning algorithms, such as A* algorithm, ⁴ ant colony algorithm, ⁵ particle swarm algorithm, ⁶ and Dijkstra algorithm, ⁷ APF has the advantages of simple mathematical model, small algorithm computation, fast response time, and good dynamic obstacle avoidance effect.^8–10 Nowadays, APF has been applied in many fields, such as path planning of unmanned aerial vehicle swarms, ¹¹ path planning of unmanned vehicles, ¹² path planning of mobile robots in warehousing and logistics, ¹³ and the path planning of smart factory robots, and so on. ¹⁴

Although APF has many advantages and achieved many applications in robot path planning, APF also has the following shortcomings: ¹⁵

When the A-force of the target and the R-forces of all obstacles are equal in magnitude and opposite in direction, the robot cannot determine the next moving direction, which results in stopping or constantly oscillating, namely falling into a local optimality.

If the position of an obstacle in the environment is too close to the target, when robot moves to the vicinity of the target the R-force of the obstacle will be greater than the A-force of the target, which causes the target to be unreachable.

In view of the shortcomings of APF, many scholars have made extensive researches on its solution. Some scholars improve the traditional artificial potential field (TAPF) algorithm itself to solve the existing unreachable target and local optimality problem. Zhao et al. aimed at the problems of unreachable target and easily falling into local optimality, proposed an improved strategy. The improved strategy can make the R-forces of obstacles around the target on the robot tend to zero, and make the robot walk along the edges of the obstacles. The results show that the strategy can overcome the local optimality problem and target unreachable problem existing in TAPF, and plan a feasible path for the robot. ¹⁶ Wang et al. proposed a path planning algorithm based on the improved artificial potential field (IAPF) method. Firstly, the algorithm introduces the concept of minimum safe distance and relative distance between the robot and the target, and uses the Gaussian transformation method to optimize the traditional potential function, which solves target unreachability problem when target point is near obstacles. Second, the evaluation criterion of the local minimum point is established and the local optimality problem is solved using A-force field rotation and virtual obstacle filling strategies. ¹⁷ Wang et al. proposed method to solve the unreachable target and local optimality problems of APF by introducing a virtual target point and changing the repulsion field function, which can satisfy the real-time path planning of the robot. ¹⁸ Pan et al. solved the unreachable target problem by adding the relative distance between the robot and the target into the function of the R-force field, and proposes a method of setting intermediate target points in path planning to solve the local optimality problem. ¹⁹ Wang et al. proposed left-turn potential field method and virtual target point method based on TAPF. The left-turn potential field method makes the robot turn at the local minimum point and jump out. The virtual target point method sets the virtual target point in a suitable position when the robot is in the local minimum point while ignoring the target point and obstacles to the robot, so that it can turn around and jump out of the local minimum. Simulation and testing proved that the mobile robot can bypass the local minimum point and reach the target point successfully by the method proposed. ²⁰ Fan et al. proposed an IAPF method, in which a distance correction factor is added to the repulsive potential field function to solve the unreachable target problem and a regular hexagon guidance method is proposed to improve the local minimum problem. ²¹ Jiang et al. introduced the distance index between the robot and the obstacle into the A-force field function, which solves the unreachable target problem of TAPF. And escape force is defined to eliminate local minimum problem. The experimental results verify the effectiveness and feasibility of the method proposed. ²² Yang et al. aiming at the problems of local minimum, modified the direction and influence range of the A-force field and increased the virtual target and evaluation function. And the problem of unreachable targets is solved by increasing gravity. The results show that the IAPF method can solve the above shortcomings of TAPF. ²³ Liang et al. solved the local optimality problem of TAPF by adjusting the R-force potential field function and used virtual target points to solve the target unreachable problem. ²⁴ Zhao et al. optimized the local minimum problem of TAPF by improving the R-force field function, and applied it to robot collision avoidance strategy and verified to be effective. ²⁵ Pang et al. proposed an APF-based navigation algorithm that avoids getting trapped in local minimum by changing the R-force potential field function. The results show that the robot can effectively avoid obstacles and reach the target point. ²⁶ Yang et al. proposed an IAPF method for the problems of local minimum, target unreachability, and low path smoothness. The IAPF method takes the area with fewer obstacles as the preferred area, builds a multi-target model based on the size and influence range of obstacles, and improves the potential field function. The results show that the sum of IAPF path planning and optimization time is further improved. ²⁷ Xiao et al. improve the APF method from two aspects. The hyperbolic tangent function of the distance from the target to the robot is used to improve the field to tackle the problem of the unreachable target point near obstacles. A behavior-based strategy based on collision cone is proposed to cope with the local minimum problem. Simulation results prove that the improved APF method is effective. ²⁸

In addition, some scholars have studied the local optimality problem of the APF in the robot path planning, and proposed some improved methods. Lee and Park proposed a new concept of virtual obstacles to help robots escape local minimum. Virtual obstacles are placed around the local minimum to knock the mobile robot back from the local minimum. Simulation results show that the proposed method is effective. ²⁹ Li et al. provided an improved APF, in which additional forces and a virtual local minimum region are used to solve local minimum problem. And simulations in single-obstacle and multi-obstacle environments validated the effectiveness of the improved APF. ³⁰ Bing et al. proposed an IAPF method, which solves three typical local minimum problems in traditional APF by correcting the direction of R-force. The simulation results show that the improved APF is effective. ³¹ Zhang et al. improved the R-force function in the APF method and added a detour force to the R-force function to effectively solve the local optimality problem. The simulation results show that the method proposed is simple and effective. ³² Zhou and Li proposed an improved obstacle potential field function model considering the size of the robot and obstacles, and adaptively changed the weight of the obstacle potential field function to make the robot get rid of the local minimum. The simulation results show that the method can make the robot escape from the local minimum and complete the robot path planning. ³³ Rostami et al. proposed an improved obstacle avoidance planning algorithm based on APF method and Simulation results show that the proposed method has superiority. ³⁴ Zheng et al. designed an improved virtual obstacle method for local path planning by proposing new minimum criteria, new switching conditions, and new exploration forces to solve the local optimality problem that the robot may fall into. ³⁵ Liu and Dou proposed an IAPF, in which the local optimality problem of TAPF is overcome by optimizing the R-force potential field function and changing the direction of the R-force component on the coordinate axis. The simulation results show that the IAPF can effectively path planning and will not make the robot fall into local optimum. ³⁶ Szczepanski et al. proposed a novel APF algorithm powered by augmented reality, which is able to predict upcoming local minimum and enhances the perception of mobile robots to bypass local minimum. Experiments are carried out on a robot operating system, and the proposed method allows to generate shorter paths without getting stuck in local minimum. ³⁷ Azzabi and Nouri proposed a new R-force potential field function which can generate an additional escape force to help the robot get out of the local optimal position. The simulation results show that the method can effectively solve the local optimality problem of TAPF. ³⁸ Zafar et al. combined gray wolf optimization algorithm and APF for robot navigation control, in which virtual target points are used to avoid the robot from falling into local optimum. The results reveal the proposed method can find an optimal or near-optimal path for robot. ³⁹ Li introduced the distance between the robot and the target point into the potential field function in TAPF. Using the obstacle connection method, the robot can quickly get rid of the local minimum point and complete the path planning. The simulation results show that the method is effective. ⁴⁰ Ni et al. propose an IAPF method by setting a virtual target point and guiding the robot out of local minimum point through the double-circle strategy. The experimental results show that the IAPF can effectively solve the problem of local minimum in the TAPF. ⁴¹ Wang et al. proposed an enhanced approach called the stochastic artificial potential field (APF) method. By filtering the obstacle locations and introducing a randomly directed R-force within a specific angular range, the problem of unbalanced forces on the robot is addressed, thereby mitigating the issue of falling into local minimum. Experimental results demonstrate that the stochastic artificial potential field method effectively resolves the challenges of local minimum. ⁴²

Furthermore, some other robot path planning algorithms are considered to overcome the shortcomings of APF, and good results achieved. Based on APF, Wu et al. used genetic algorithm as the framework, combined the artificial potential field algorithm with the crossover of genetic algorithm to get IAPF. The research results show that the length of the path obtained by the IAPF is shorter. ⁴³ Janabi-Sharifi and Vinke proposed an algorithm combining APF method and simulated annealing algorithm, which can escape from local minimum by using the escape ability provided by simulated annealing algorithm. The simulation results show that the performance of the proposed method for the robot path planning is good. ⁴⁴ Vadakkepat et al. proposed an evolutionary APF method, which combined the APF with the genetic algorithm to derive the most dominant field function. In order to avoid the local minimum of the evolutionary APF method, an escape force algorithm is proposed. Simulation results show that the method has good robustness and effectiveness for robot path planning with non-stationary targets and obstacles. ⁴⁵ Park et al. proposed a path planning method combining the simulated annealing method with APF, and in which the simulated annealing technique provides the ability to get out of any possible local minimum. The simulation results show that the simulated annealing method can overcome the local optimality problem. ⁴⁶ Zhu et al. presented and evaluated the APF method with simulated annealing. The results indicate simulated annealing is a powerful technique for escaping local minimum when applied to local and global path planning. ⁴⁷ Abdalla et al. proposed a new algorithm combining IAPF with fuzzy logic, which can effectively solve the local optimality problem of TAPF. The experimental simulation proves that the algorithm can carry out effective path planning for the robot, and the path is smoother and more efficient. ⁴⁸ Hu et al. proposed an IAPF method based on the dynamic window method, which uses the evaluation function to evaluate the local minimum points around the robot, and selects the optimal point as the next path point. In addition, it is also proposed to connect the continuous points of the planned path to optimize the path under the premise of leaving enough safe space between the robot and the obstacle. The results show that the improved APF can better solve the local optimality problem and the optimization algorithm can plan shorter paths in shorter computation time. ⁴⁹ Shen and Li proposed an IAPF method, which uses the pre-planned path nodes generated by the rapid exploration random tree algorithm as a new source of gravity to weaken the effect of the target point. The simulation results show that the method can effectively avoid local minimum and has good adaptability. ⁵⁰ Xi et al. proposed an IAPF method, in which the bias target rapid exploration random tree algorithm is combined to determines the local minimum point in the APF method. The simulation results show that the proposed method can avoid the phenomenon that the APF method falls into a local minimum point while taking into account the global and real-time performance. ⁵¹ Ma et al. combined IAPF with the rapidly exploring random tree for driverless vehicle path planning, in which the repulsion function is used to solve the local minimum of APF and the rapidly exploring random tree to solve the path oscillation. The results show that the method is feasible and an optimal path can be obtained at the same time. ⁵² Dai et al. proposed a combined path planning algorithm based on APF and simulated annealing, which can effectively overcome the local optimality problem of APF. Simulation results show that the algorithm can perform path planning for robots in various environments. ⁵³ Wu et al. proposed a combined path planning algorithm based on APF and beetle antenna search method. The combined algorithm can effectively overcome the local optimality problem of APF, and the simulation results prove its effectiveness and superiority. ⁵⁴ Lee et al. combined APF with motion primitives for unmanned aerial vehicles path, which can solve the local optimality problem of TAPF. Experiments show that the proposed algorithm can provide a smoother, more efficient and feasible path than by TAPF. ⁵⁵ Zhang introduce a new IAPF, which incorporates the A-star method in constructing the artificial potential field. The new IAPF can effectively addresses the issue of path planning for mobile robots and avoid local minimum solutions. The feasibility of the new IAPF is verified by simulation experiment. ⁵⁶ Wu et al. proposes a novel method based on a deterministic annealing strategy to improve the potential field function by introducing a temperature parameter. The annealing and tempering strategies prevent the robot from being trapped at the local minimum. Experiments show that the proposed method can solve path planning in different environments. ⁵⁷

The setting of virtual target points or obstacle points, adding repulsion force, and the combination APF with other algorithms proposed in the above literatures provide a good solution to solve the shortcomings of APF in robot path planning. However, these methods do not make full use of the information in the process of robot movement, such as the robot gradually approaches the obstacle from a distance, and then gradually moves away from the obstacle, which is called local path information in this paper. Based on the local path information, the IAPF is obtained and discussed. And the main contributions of this study are summarized as follows:

An improved APF is proposed, in which the local path information is defined and used. The calculation formulas of the R-force, the A-force, and the resultant force in IAPF are derived.

A method to solve the local optimality problem is presented, in which the local path information is used.

IAPF is simulated in various scenarios and compared to dynamic window method. The results indicate that the IAPF is superior to the traditional APF and dynamic window method in obtaining better path and efficiency.

The rest of this paper is organized as follows. Section “Improved artificial potential field method” presents the improved APF using the local path information, which includes the principle of IAPF, calculations of the A-force, the R-force, and the resultant force in IAPF. In Section “Method for solving the local optimality problem,” method based on local path information is proposed for solving the local optimality problem. Section “Simulation and analysis” verifies and analyzes the IAPF by simulation in different scenarios. Finally, the conclusions and suggestions for future work are presented in Section “Conclusion and future work.”

Improved artificial potential field method

The principle of IAPF

As shown in Figure 1, there are three consecutive robot positions in the process of robot path planning by the APF algorithm, which are set as P₁, P₂, and P₃, and three obstacles that have repulsive effect on the robot and are set as Ob ₁ , Ob ₂ , and Ob ₃ . As can be seen from Figure 1, from P₁ to P₂, robot moves closer to obstacle Ob ₂ , P₂ is the closest to Ob ₂ , and then starts to move away from Ob ₂ at P₃. This is consistent with the trajectory trend of the obstacle avoidance of the agent in reality, that is, it gradually approaches the obstacle and then gradually moves away from the obstacle. The APF algorithm simulates the process of robot obstacle avoidance well. However, in reality, once the agent starts to move away from the obstacle, it will not approach the obstacle again. The APF algorithm does not simulate this situation. To this end, this paper defines the process of the robot gradually approaching the obstacle and then gradually moving away from the obstacle as the local path information. Based on the APF algorithm, the local path information is used in robot path planning and the IAPF is obtained. IAPF has the characteristics of APF and uses local path information, which should better simulate the obstacle avoidance process of robot.

Figure 1.

The position relation between robot and obstacles in the process of path planning by the APF.

In IAPF, the obstacle that the robot starts to move away from is ignored. That is although the robot is still within the influence range of the obstacle, the R-force of the obstacle on the robot is no longer considered, and the obstacle is called the ignored obstacle. According to the robot position and the influence range of obstacles in Figure 1, the R-forces on the robot under the IAPF algorithm at position P₁, P₂, and P₃ can be obtained, as shown in Figure 2. In order to more clearly express the difference between IAPF and APF, the R-forces on the robot at the three positions under APF are given, as shown in Figure 3. According to Figures 2 and 3, the R-forces on the robot at P₁ and P₂ are the same as that of the IAPF and APF. However, under IAPF, robot is only repulsed by obstacles Ob ₁ and Ob ₃ at P₃, which is the result of using the local path information that robot starts to move away from Ob ₂ , and the Ob ₂ is the ignored obstacle.

Figure 2.

The R-forces on the robot under the IAPF at position P₁, P₂, and P₃.

Figure 3.

The R-forces on the robot under the APF at position P₁, P₂, and P₃.

In addition, in IAPF, the local path information can be obtained by calculating the distances of the robot from the obstacles in each step and comparing it with the distances of the previous step. Further, the local path information must correspond to the ignored obstacle, so the ignored obstacles are also considered as part of the local path information. In the process of robot path planning by IAPF, the local path information is stored separately, and its format is ( ${\mathrm{Distance}}_j^i$ , $\mathrm{Ig}\mathrm{\_}\mathrm{Obstacl}{\mathrm{e}}_i$ ). ${\mathrm{Distance}}_j^i$ indicates the distance to the obstacle i at the jth step, and $\mathrm{Ig}\mathrm{\_}\mathrm{Obstacl}{\mathrm{e}}_i$ indicates the ignored obstacle.

In summary, the flowchart of robot path planning based on IAPF is shown in Figure 4.

Figure 4.

The flowchart of IAPF.

The calculation of the A-force in IAPF

In IAPF, the A-force is calculated by the A-PF function, which is the same as the TAPF method. The A-PF function between the robot and the target can be expressed as:

$$U_{\mathrm{att}}(X)={\displaystyle \frac{1}{2}{K}_{\mathrm{att}}(X-X_{\mathrm{G}}{)}^2}$$

(1)

Where $U_{\mathrm{att}}$ is the A-PF, $K_{\mathrm{att}}$ is the gain coefficient of the A-PF, X represents the current position of the robot, and $X_{\mathrm{G}}$ represents the position of the target.

According to the negative gradient of the A-PF function, the A-force of the target on the robot can be calculated as:

$$F_{\mathrm{att}}(X)=-K_{\mathrm{att}}(X-X_{\mathrm{G}})$$

(2)

Where $F_{\mathrm{att}}$ is the A-force generated by the target on the robot and its direction is from the robot to the target, $(X-X_G)$ is the distance from the robot to the target.

The calculation of the R-force in IAPF

In the traditional APF, the R-force of obstacle on the robot can be calculated as followings:

$$F_{\mathrm{trep}}(X)=\{\begin{array}{c}{K}_{\mathrm{rep}}\left({\displaystyle \frac{1}{(X-X_{\mathrm{O}})}-{\displaystyle \frac{1}{{\rho }_0}}}\right){\displaystyle \frac{1}{{(X-X_{\mathrm{O}})}^2}\frac{\mathrm{\partial }(X-X_{\mathrm{O}})}{\mathrm{\partial }X},(X-X_{\mathrm{O}})\le {\rho }_0}\\ 0,(X-X_{\mathrm{O}})>{\rho }_0\end{array}$$

(3)

Where $F_{\mathrm{trep}}$ represents the R-force of the obstacle on the robot and its direction is from obstacle to the robot, $K_{\mathrm{rep}}$ is the gain coefficient of the R-PF, X is the position of robot, Xo is the position of obstacle, $(X-X_{\mathrm{O}})$ is the distance from the robot to obstacle, $\mathrm{\partial }(X-X_{\mathrm{O}})/\mathrm{\partial }X$ represents the partial derivative of $(X-X_{\mathrm{O}})$ with respect to X, and ${\rho }_0$ represents the influence radius of the obstacle.

In the process of robot path planning by IAPF, let the distance from the robot's current position (assumed to be the jth step) to the obstacle Obs as $L_{\mathrm{Obs}}^j$ . Based on the principle of IAPF, if $L_{\mathrm{Obs}}^j\le L_{\mathrm{Obs}}^{j-1}$ , and $L_{\mathrm{Obs}}^j\le {\rho }_0$ , it means that the robot enters the influence range of the obstacle Obs and is approaching the Obs. The R-force of the obstacle Obs on the robot should be calculated. However, if $L_{\mathrm{Obs}}^j>L_{\mathrm{Obs}}^{j-1}$ , it means that the robot starts to move away from the obstacle Obs, and the R-force of the obstacle on the robot will not be calculated.

Thus, based on formula (3), the calculation formula of the R-force of the obstacle Obs on the robot is obtained as:

$$F_{\mathrm{rep}}(X)=\{\begin{array}{c}{K}_{\mathrm{rep}}\left({\displaystyle \frac{1}{(X-X_{\mathrm{Obs}})}-{\displaystyle \frac{1}{{\rho }_0}}}\right){\displaystyle \frac{1}{{(X-X_{\mathrm{Obs}})}^2}{\displaystyle \frac{\mathrm{\partial }(X-X_{\mathrm{Obs}})}{\mathrm{\partial }X},L_{\mathrm{Obs}}^j\le L_{\mathrm{Obs}}^{j-1}\mathrm{and}{L}_{\mathrm{Obs}}^j\le {\rho }_0}}\\ 0,L_{\mathrm{Obs}}^j>L_{\mathrm{Obs}}^{j-1}\end{array}$$

(4)

Where $F_{\mathrm{rep}}$ represents the R-force of the obstacle Obs on the robot and its direction is from Obs to the robot, $(X-X_{\mathrm{Obs}})$ is the distance from the robot to Obs, and $\mathrm{\partial }(X-X_{\mathrm{Obs}})/\mathrm{\partial }X$ represents the partial derivative of $(X-X_{\mathrm{Obs}})$ with respect to X.

In formula (4), the local path information is used, that is $F_{\mathrm{rep}}(X)$ is zero when robot starts to move away from obstacle Obs at the nearest place of Obs. Even the robot is still within the influence range of the obstacle. Therefore, the amount of calculation can be reduced and the efficiency of path planning can be improved.

The calculation of resultant force in IAPF

Moving towards the target, the robot is affected by the A-force of the target and the R-force of the obstacles in the environment. According to formula (4), the resultant force, $F_{\mathrm{repall}}$ , of R-forces of all obstacles on the robot can be calculated as:

$$F_{\mathrm{repall}}(X)=\sum _{i=1}^n{F}_{\mathrm{rep}}^i(X)$$

(5)

Where n is the number of obstacles in the environment.

According to formula (2) and formula (5), the resultant force $F_{\mathrm{all}}$ of the robot movement can be obtained, and its expression is as follows:

$$F_{\mathrm{all}}(X)=F_{\mathrm{att}}(X)+F_{\mathrm{repall}}(X)$$

(6)

Assuming that the coordinate of the obstacle i in the environment is ( $x_i$ , $y_i$ ), the coordinate of the robot is ( $x_{\mathrm{rob}}$ , $y_{\mathrm{rob}}$ ), and the coordinate of the target is ( $x_{\mathrm{tar}}$ , $y_{\mathrm{tar}}$ ), then the component of the R-force generated by the obstacle on the robot in the x direction, $F_{\mathrm{repx}}^i$ , can be calculated as:

$$F_{\mathrm{repx}}^i=F_{\mathrm{rep}}\times \cos {\displaystyle \frac{x_i-x_{\mathrm{rob}}}{\sqrt{{(x_i-x_{\mathrm{rob}})}^2+{(y_i-y_{\mathrm{rob}})}^2}}}$$

(7)

From formula (7), the component in the x direction of the resultant force of R-force from all obstacles in the environment, $F_{\mathrm{repallx}}$ , can be obtained as:

$$F_{\mathrm{repallx}}=\sum _{i=1}^n{F}_{\mathrm{repx}}^i$$

(8)

The component of the A-force generated by the target on the robot in the x direction, $F_{\mathrm{attx}}$ , can be calculated as followings:

$$F_{\mathrm{attx}}=F_{\mathrm{att}}\times \cos {\displaystyle \frac{x_{\mathrm{tar}}-x_{\mathrm{rob}}}{\sqrt{{(x_{\mathrm{tar}}-x_{\mathrm{rob}})}^2+{(y_{\mathrm{tar}}-y_{\mathrm{rob}})}^2}}}$$

(9)

Then the direction of the resultant force of the robot movement, that is, the angle θ between it and the x-axis can be obtained as:

$$\theta =\arccos \left({\displaystyle \frac{F_{\mathrm{repx}}+F_{\mathrm{attx}}}{F_{\mathrm{all}}}}\right)$$

(10)

Method for solving the local optimality problem

In APF, there is the problem of robot stopping at some position or oscillating near the position, that is, the local optimality problem. To solve the problem, a method is proposed and used in IAPF, in which the local path information is used. Firstly, search for the ignored obstacles in local path information and find the closest to the current position of the robot. Then, the R-force of the ignored obstacle is calculated and added to the resultant force on the robot. This will change the result of the resultant force on the robot, so the robot gets out of the local optimal position. The process includes two steps shown as follows.

(1) Find the ignored obstacle closest to the robot

According to the local path information stored during the robot path planning process, the ignored obstacle closest to the robot can be found and labeled as Ob_IC. Then set the position of Ob_IC as $X_{\mathrm{s}}$ , the distance from the robot to the obstacle can be calculated and set as $L_{\mathrm{s}}$ .

(2) Calculate the R-force of obstacle Ob_IC on the robot

The robot is inside the influence range of obstacle Ob_IC, that is, $L_{\mathrm{s}}\le {\rho }_0$ , as shown in Figure 5(a). In this case, the R-force of obstacle Ob_IC on the robot is calculated by formula (3), and then added to the calculation of the resultant force of R-forces. So, the magnitude and direction of the resultant force on the robot will change, the robot can get out of the local optimality.

The robot is outside the influence range of obstacle Ob_IC, that is, $L_{\mathrm{s}}>{\rho }_0$ , as shown in Figure 5(b). In this case, a circle is made by the current position of the robot as the center and ${\rho }_0$ as the radius. Then, a connecting line is obtained by connecting obstacle Ob_IC to the current position of the robot. An intersection point, red circle shown in Figure 5(b), is obtained between the connecting line and the circle. The position of red circle is selected as the position of new obstacle corresponding to obstacle Ob_IC, and set as $X_{\mathrm{new}}$ . Then the R-force of the new obstacle can be calculated by formula (3) and added to the calculation of the resultant force of R-force. So, the magnitude and direction of the resultant force on the robot will change, the robot can get out of the local optimality.

Figure 5.

Calculate the R-force of obstacle Ob _IC on the robot.

Simulation and analysis

Simulation in general scenario

In order to verify the performance of IAPF for robot path planning, a general scenario with 14 randomly distributed obstacles is selected and shown in Figure 6. In the scenario, the obstacles are regarded as mass points. The size of the scenario is 12 m × 12 m, the starting point and target point of the robot are set as (0,0) and (12,12), respectively. In addition, and the related parameters setting in simulation of IAPF are shown in Table 1, where L and V denote the step length and step frequency, respectively.

Figure 6.

General scenario.

Table 1.

Parameters setting in simulation of IAPF.

Parameter	Value
L	0.3 m
V	1 step/s
${\rho }_0$	5 m
$K_{\mathrm{att}}$	30
$K_{\mathrm{rep}}$	35

In the scenario, IAPF for robot path planning is simulated. At the same time, the TAPF is also simulated under the same scenario and parameters. The results of IAPF and TAPF are shown in Figure 7, in which Figure 7(a) and Figure 7(b) represent the paths obtained by IAPF and TAPF, respectively. In addition, the path planning time and length of the path by two methods are also obtained and shown in Table 2.

Figure 7.

Robot paths obtained by IAPF and TAPF in general scenario.

Table 2.

Path planning time and length of the path by IAPF and TAPF.

	Path planning time (s)	Length of the path (m)
IAPF	60	18
TAPF	64	19.2

It can be seen from Figure 7 that both IAPF and TAPF can plan a feasible path for the robot from the starting point to the target point. But it can be found that the two methods obtain different paths, and the path by IAPF is smoother than by TAPF. Furthermore, it can be seen that the two paths change greatly when they pass through the obstacle filled with green. In Figure 7(a), near the obstacle with green color, the robot moves more smoothly and follows almost straight path to the target point, while in Figure 7(b), the robot follows the curve to the target point. This is because in IAPF, the R-force of the obstacle will no longer be calculated in the consecutive steps after the robot leaves the obstacle, namely the use of local path information, which makes the robot less affected by the R-force of the obstacles and more affected by the A-force of the target point, so that it can reach the target point with a smoother and shorter path length.

In addition, it can be found from Table 2 that the time needed to obtain the path by IAPF is less than by TAPF. Also, the length of path by IAPF is less than that by TAPF. This indicates IAPF can get a better path and has higher efficiency than TAPF.

Simulation in complex scenario

In order to verify IAPF for solving local optimality problem in APF, a complex scenario with the number of obstacles 3 times that of the general scenario (Figure 6) is selected, and shown in Figure 8. The size of the scenario is 12 m × 12 m, and the obstacles in the scenario are also regarded as mass points. The starting point and target point of the robot are set as (0,0) and (12,12), respectively. The parameters in simulation of IAPF and TAPF are set and shown in Table 3.

Figure 8.

Complex scenario.

Table 3:

Parameters setting in simulation of IAPF and TAPF.

Parameter	Value
L	0.2 m
V	1 step/s
${\rho }_0$	3 m
$K_{\mathrm{att}}$	20
$K_{\mathrm{rep}}$	35

In the complex scenario, both IAPF and TAPF for robot path planning are simulated, and the results are obtained, as shown in Figure 9. Figure 9 (a) represents the path obtained by TAPF, Figure 9 (b) and Figure(c) represent the paths obtained by IAPF and named as Path I and Path II, respectively.

Figure 9.

Simulation results of IAPF and TAPF in complex scenario.

It can be seen from Figure 9(a) that TAPF makes the robot fall into local optimality and fails to plan a path for the robot. However, in Figure 9(b), IAPF solves the local optimality problem of TAPF at the same position and makes the robot continue to move on until it arrives at the target. This is because the local path information is used in IAPF when the robot falls into a local optimality, and then the ignored obstacle is added, so that the resultant fore on the robot is no longer zero and the robot gets out of the local optimal point. In addition, IAPF can plan another feasible path for robot in the complex scenario, for example, Path II in Figure 9(c). This shows that in the complex scenario, IAPF can solve the local optimality problem of TAPF.

Simulation in the scenario with circular obstacles

In order to further verify the feasibility of IAPF for robot path planning, obstacles are no longer regarded as mass points as in the scenarios in Figures 6 and 8, but circular obstacles with radius. Figure 10 is a scenario with the size of 35 m × 35 m, and in which there are 30 randomly distributed circular obstacles with radius of 1.2 m. The starting point and target point of the robot are (0,0) and (30,30), respectively. The parameters setting in simulation of IAPF and TAPF are shown in Table 4.

Figure 10.

The circular obstacle scenario.

Table 4.

Parameters setting in simulation of IAPF and TAPF.

Parameter	Value
L	0.7 m
V	1 step/s
${\rho }_0$	3 m
$K_{\mathrm{att}}$	15
$K_{\mathrm{rep}}$	35

According to the scenario in Figure 10 and the parameters in Table 4, IAPF and TAPF are simulated, and the results are obtained, as shown in Figure 11 and Table 5. Figure 11 shows the paths obtained by IAPF and TAPF, and Table 5 shows the path planning time and path length of the two methods.

Figure 11.

Simulation results of IAPF and TAPF in the circular obstacle scenario.

Table 5.

Path planning time and length of the path by IAPF and TAPF.

	Path planning time (s)	Length of the path (m)
IAPF	70	49
TAPF	89	62.3

As can be seen from Figure 11, in the circular obstacle scenario, both IAPF and TAPF can plan feasible paths for the robot from the starting point to the target point. However, the path obtained by TAPF appears oscillation, the path obtained by IAPF is smoother. This is due to the use of local path information in IAPF to avoid the robot path oscillation. In addition, it can be seen from Table 5 that IAPF takes less time to obtain a path than TAPF, and the path length by IAPF is shorter than that by TAPF. This shows that IAPF can also obtain better path in circular obstacle scenario and is more efficient than TAPF.

Simulation in the scenario with different sizes of circular obstacles

Furthermore, a scenario with different size of circular obstacles is selected, as shown in Figure 12, to verify the performance of IAPF for robot path planning in the circular obstacle scenario. The size of the scenario is 30 m × 30 m, 16 circular obstacles of different sizes are randomly distributed, with radiuses of 0.8 m, 1.2 m, 1.6 m, and 2 m, corresponding to red, yellow, light green, and green obstacles, respectively. The starting point and target point of the robot are (0,0) and (30,30), respectively. And the parameters setting in simulation of IAPF and TAPF are shown in Table 6.

Figure 12.

The scenario with different sizes of circular obstacles.

Table 6.

The radius of circular obstacles and parameter settings in simulation.

	Parameter	Value
	L	0.7 m
	V	1 step/s
Parameters in IAPF and TAPF	${\rho }_0$	3 m
	$K_{\mathrm{att}}$	15
	$K_{\mathrm{rep}}$	35

According to the scenario in Figure 12 and the parameters in Table 6, IAPF and TAPF are simulated, and the results are obtained, as shown in Figure 13 and Table 7. Figure 13 shows the paths obtained by IAPF and TAPF, and Table 7 shows the path planning time and path length of the two methods.

Figure 13.

Simulation results of IAPF and TAPF in scenario with different sizes of circular obstacles.

Table 7.

Path planning time and length of the path by IAPF and TAPF.

	Path planning time (s)	Length of the path (m)
IAPF	72	50.4
TAPF	66	46.2

It can be seen from Figure 13 that both IAPF and TAPF can obtain feasible path for the robot in the scenario with different sizes of circular obstacles. However, it can be found that the path obtained by IAPF is smoother than that by TAPF. This is because the use of local path information in IAPF allows robot to quickly jump out the oscillation and reach the target point on a smoother path. In addition, it can be seen from Table 7 that IAPF takes less time to obtain the path than TAPF, and the path length by IAPF is shorter than that by TAPF. This shows that IAPF can obtain better path and is more efficient than TAPF.

Comparison of IAPF and dynamic window approach

In this section, IAPF is compared with the dynamic window approach (DWA), ⁵⁸ which is a classical robot path planning algorithm. The scenarios with the same size and different size of circular obstacles, named as Scenario A and Scenario B, respectively, shown in Figure 14, are selected to simulate the IAPF and DWA algorithms. In Scenario A, there are 25 circular obstacles with the same radius of 1 m. In Scenario B, there are 25 circular obstacles of different sizes are randomly distributed, with radiuses of 0.8 m, 1.2 m, 1.6 m, and 2 m, corresponding to red, yellow, light green and green obstacles, respectively. The starting point and target point of the robot in Scenario A are (0,0) and (30,30), respectively, while in Scenario B are (0,0) and (35,35). The parameters setting in simulation of IAPF and DWA are shown in Table 8, where L_A and L_B denote the step length in Scenario A and Scenario B, respectively.

Figure 14.

Scenarios for IAPF and DWA: (a) scenario with the same size of circular obstacles and (b) scenario with the different size of circular obstacles.

Table 8.

Parameters in simulation of IAPF and DWA.

	Parameter	value
	$L_{\mathrm{A}}$	0.8m
	$L_{\mathrm{B}}$	0.7m
IAPF	V	1step/s
	${\rho }_0$	3m
	$K_{\mathrm{att}}$	15
	$K_{\mathrm{rep}}$	35
	Robot initial state	[0 0 pi/2 0 0]
DWA	Robot kinematics model	[1.0, toRadian(20.0), 0.2, toRadian(50.0), 0.01, toRadian(1)]
	Evaluation function parameters	[0.05, 0.2, 0.1, 3.0]

According to the scenarios in Figure 14 and the parameters in Table 8, IAPF and DWA are simulated, and the results are obtained, as shown in Figure 15 and Table 9. Figure 15 shows the paths obtained by IAPF and DWA, and Table 9 shows the path planning time and path length of the two algorithms.

Figure 15.

Paths of IAPF and DWA in different scenarios.

Table 9.

Path planning time and length of the path by IAPF and DWA.

		Path planning time (s)	Length of the path (m)
Scenario A	IAPF	58	46.4
	DWA	84	58.9
Scenario B	IAPF	71	49.7
	DWA	96	51.8

As can be seen from Figure 15, in the Scenario A and Scenario B, both IAPF and DWA can plan feasible paths for the robot. However, it can be found that the path obtained by IAPF goes straight to the target in a straight line every time it bypasses the obstacle, while the entire path planned by DWA is relatively curved. In addition, it can be seen from Table 9 that IAPF takes less time to acquire the path than DWA, and the path length by IAPF is shorter than that of DWA. This indicates that IAPF can obtain better path and has higher performance than DWA.

Conclusion and future work

In this paper, the local path information is defined, which includes the process of the robot gradually approaching the obstacle and then gradually moving away from the obstacle, and the ignored obstacle. Based on the APF algorithm, the local path information is used and the improved APF is obtained, namely IAPF. Then the calculation formula of relative forces in IAPF are given, which include the R-force formular of the obstacles on the robot, the A-force formular of the target on the robot, and the resultant force formular. Especially, in the R-force formula, the local path information is used, which reduces the amount of calculation. In addition, a method to solve local optimality problem in APF is introduced based on local path information, which includes two steps: obtain the ignored obstacle closest to the robot and calculate the R-force of the obstacle on the robot.

After that, the general scenario is selected to verify IAPF. The simulation results indicate that compared to TAPF, the robot path obtained by IAPF is smoother and less length, and the planning time is less. Furthermore, IAPF is simulated in a complex scenario and it is verified that IAPF with the local path information can overcome the local optimality problem and presents superior performance than TAPF.

In addition, the scenarios with the same size and different size of circular obstacles are selected to further verify the IAPF. The results show that IAPF is superior to TAPF in terms of path planning time and path length. At the same time, the IAPF is compared with DWA, and results show that IAPF can get shorter path and takes less time than DWA. This shows that IAPF is very efficient in robot path planning.

In this work, an IAPF method is proposed and verified in the general scenario, the complex scenario, and the scenarios with the same size and different size of circular obstacles. However, in the actual scenarios, the obstacles still have various complex properties, and it is difficult for APF, even IAPF to plan a feasible path for the robot. Therefore, combining a newer path planning algorithm with APF will be reserved for our future work.