A new approach of real-time path planning based on belief space is proposed, which solves the problems of modeling the real-time detecting environment and optimizing in local path planning with the fusing factors. Initially, a double-safe-edges free space is defined for describing the sensor detecting characters, so as to transform the complex environment into some free areas, which can help the robots to reach any positions effectively and safely. Then, based on the uncertainty functions and the transferable belief model (TBM), the basic belief assignment (BBA) spaces of each factor are presented and fused in the path optimizing process. So an innovative approach for getting the optimized path has been realized with the fusing the BBA and the decision making by the probability distributing. Simulation results indicate that the new method is beneficial in terms of real-time local path planning.

Recently, the development and application of autonomous robots are with growing interest in industrial and military fields. As we all know, navigation is one of the key technical problems for autonomous robots, and the most important factor of navigation is map building based on the sensor system, especially when the autonomous robots are working in an entire unknown environment. The environment is reconstructed by merging the information transferred from the sensor system during the motion. To build a practical map, one of the most difficult problems is due to the poor environment information of the sensor system which has inherent wide radiation cone and the phenomenon of multiple reflections. Thus, how to describe these uncertainties and filter out inaccurate and conflicting information and how to construct the environment view are the hot issues.

In these few years, there are about three types of approaches of constructing the environment view that appeared in exoteric literatures. The first type is the occupancy grid mapping method [

In order to describe the uncertainties, or filter out the conflicting information detected by sensors, the probabilistic algorithms [

There is no doubt that the optimization problem is quite important for autonomous robots path planning. So many evolutionary optimizing techniques like genetic algorithm [

In this paper, a novel real-time path planning approach based on the belief space is introduced. As the transferable belief model (TBM), which is popular in these years, can be used to describe a highly flexible model to manage the uncertainty information in the multisensor data fusion problems. In particular, many applications of TBM have been presented in mobile vehicles and other areas [

The rest of the paper is organized as follows. In Section

Sonar is far from being an ideal sensor, mainly due to the width of the radiation cone and to the multiple reflections phenomenon. The former does not allow determining the exact angular position of the obstacle on the fixed angle

The uncertainty model has been set up by fuzzy measure approach. A single reading

That describe, respectively, how the degree of certainty of the assertions “empty” and “occupied” varies with

Since the intensity of the waves decreases to zero at the borders of the radiation cone, the degree of certainty of each assertion is assumed to be higher for points close to the beam axis. This is realized by defining an angular modulation function [

In order to weaken the confidence of each assertion as the distance from the sensor increases, the parameter

TBM is a model for describing quantified beliefs based on belief function. Beliefs can be held at two levels: (1) a “credal” level where beliefs are entertained and quantified by belief functions; (2) a “pignistic” level where beliefs can be used to make decisions and are quantified by probability functions. The relation between the belief function and the probability function when decisions must be made is derived and justified [

In TBM, the actual value

The mass

Belief function is defined as

The value

Plausibility function is defined as

The value

Combination rules: in general Bayesian theorem, the sensor detection

Decision making function is defined as

In the TBM, when a decision has to be made, a probability functions

We will build a simulation environment about the detection process of the robot sensor for testing the new approach of the real-time path planning process. The robot sensor is an initiative sensor, the angle of the detecting is 180°, and the distance of detecting is

In Figure

The sketch map of the sensor detection simulation.

Figure

Simulation results of the sensor detection at time

The simulation of the sensor detecting at time

The obstacles environment

The state of detecting the obstacles at time

The storage information of the detected obstacle at time

In the process of the path planning, the position information of the obstacle, and the robot, the information of the whole target point and the local target points must be described at each time, so it needs a uniform reference frame. There are two reference frames in the process of path planning of this paper: the reference frame of the robot movement and the reference frame of the sensor detection, so it needs the transition of the reference frame. In this paper, the reference frame of the robot movement is a vertical coordinate; the start point

The transformation of the detecting space coordinates of the sensor.

In Figure

In an uncertainty and dynamic environment, the environment information for path planning is obtained from the sensor on the robot only, so the algorithm should have well real-time ability and it is also the first step of generating the robot’s motion. According to the sensor detecting model, we propose a method for searching the important information from the detecting information in this paper, which is called the double-safe-edges (DSE) information.

Obviously, the sensor edges can be searched directly from the sensor detecting information, and its distance and direction can be ensured according to the position of the obstacles.

In Figure

The sketch map of searching the sensor edges at time

In Figure

The result of the simulation of searching the sensor edges at time

The edges are based on the sensor as mentioned above. But the robot has its own safe area because of its special shape and kinematics, if it considers the sensor edges only, and the path planning must be failing. So it is necessary to consider the environment information and the robot’s safe area together.

In this paper the definition of the double-safe-edges has consider the environment information and robot's safe area.

When the sensor edges been found, the algorithm will search some points which considering the environment information and robot's safe area, searching start from the sensor-edges according to its directions, the tangent lines which from these points to the robot’s safe circle are tangent to the edges of the obstacles at the same time. The robot and obstacle are at the different sides of the tangent line. These points are the set of the double-safe-edges points and these lines are the set of the double-safe-edges.

In Figure

The sketch map of searching the double-safe-edges points at time

The result of the simulation of searching the double-safe-edges points at time

The success of finding the double-safe-edges means that the environment detected by the sensor in real-time has been analyzed and interpreted efficiently, the environment information has been simplified, the real-time has been increased, and the robot’s safe area and the kinematics have been considered, so it will be efficient in generating the motion commands at the next step.

There are three types of the double-safe-edges: S-DSE, M-DSE, and Z-DSE.

(1)

The sketch map of S-DSE at time

(2)

The sketch map of S-DSE at time

(3)

The sketch map of S-DSE at time

After searching the double-safe-edges, the algorithm has transformed the focus from the environment information to some double-safe-edges points, and these points can be used to generate the robot’s motion. We will describe the double-safe-edges free space in this part.

In Figure

The sketch map of the safe distance and direction of the double safe edge points at time

Although the environment information can be detected by the sensor in real-time, the robot did not know the whole environment information; thus, optimizing the whole path in real-time path planning cannot come true. But there are still some important factors to affect the selection of the path in local environment, and we consider the six local planning factors in this paper, the avoidance collision factor

In Figure

The sketch map of the analysis of the optimization in local path planning at time

In Figure

The sketch map of the base idea of the belief space at time

We note that the selection state space is

The selection of the local target point must satisfy every belief function distribution at the same time, so some local target points can be deleted and the set has been changed to

As the uncertainty model has been described, we further discuss the belief function distribution in Figure

The sketch map of the belief function distribution of the sensor detection at time

The detection position uncertainty function is

According to the analysis mentioned above, the detection area uncertainty distribution is from the coordinate axes to the opposition side, and the detection position uncertainty distribution is from

The point

So the position

Thus, the BBA of the “occupy” and “empty” in TBM can be defined as

This BBA space is defined in the belief space. They are the belief functions distribution according to the base idea of the belief space, denoted by

Figure

The sketch map of the belief function distribution of the safe distance at time

Then the “safe” and “dangerous” plausibility function can be defined as

Finally, the BBA of the “safe” and “dangerous” can be defined as

All these BBA spaces are defined in the belief space. They are the belief functions distribution according to the base idea of the belief space, denoted by

So other factors can be defined and described in the belief space, it is the base step of fusing these factors to find the optimization local target point.

In Figure

As the global target

The direction of the distribution

The direction is

So the path proportion function is

As the global target

so the path proportion function in two ways is

The sketch map of the belief function distribution of the optimization the path at time

The direction of the distribution is

The direction is

So the path proportion function is

So the path “optimization” and “nonoptimization” plausibility function can be defined as

As the same way, the BBA of the “optimization” and “nonoptimization” in TBM can be defined as

All these BBA spaces are defined in the belief space. They are the belief functions distribution according to the base idea of the belief space, denoted by

Figure

Consider the reached proportion functions in the same track radius:

Consider the reached proportion functions in the same detection area:

The sketch map of the belief function distribution of the dynamics of the robot at time

So the reached proportion function at certain detection time at local target point

So the path “reach” and “unreach” plausibility function can be defined as

Then the BBA of the “reach” and “unreach” in TBM can be defined as

All these BBA spaces are defined in the belief space. They are the belief functions distribution according to the base idea of the belief space, denoted by

Figure

The sketch map of the belief function distribution of the escaping the movement obstacle at time

The one side edge point of the obstacle ob is

The collisions function of the robot which moves from the position

So the path “safe” and “collisions” plausibility function can be defined as

Then, the BBA of the “safe” and “collisions” in TBM can be defined as

All these BBA spaces are defined in the belief space. They are the belief functions distribution according to the base idea of the belief space, denoted by

Suppose that, at any given time, the local target points set is

Figure

The structure of selecting the local target point belief space at time

In “credal” level each local target point has its own factors, so it has to filter the fusing local target point belief space to make sure of the whole factors at the same time. Each factor has its own belief space, the whole factors BBA depend on the selection state space of the factors, and this chain structure of the local target point belief space can transform the influence of the factors to the BBA function in the belief space. In the “pignistic” level it denotes the influence degrees of the factors using the probability functions; it is the final form of selecting the local target point.

It needs to fuse the belief space when the factors have been described to the BBA functions in the belief space; the details of the process at certain times are as follows.

The local target points set is selected according to the double-safe-edges free space, denoted as

The BBA of the correlation influence factors set can be calculated, denoted as

In belief space

The

In a real-time path planning process, the environment information requires to be detected at each time, so the simulation of the local target point in belief space should satisfy this character. In this paper, the maps of simulation have been made by the .bmp pictures beforehand, and the algorithm of the local target point in belief space has been written in the software by the program, so in the process of the simulation it shows the start point and the target point, the obstacles on the map, the particle and the self-safe area, the lines of detecting, the position of the robot, and the path.

There are about two different types of simulation that will be shown in this paper to prove the feasibility of the double-safe-edges space and the idea of selecting the local target point in belief space. Simulation I is the simulation of double-safe-edges space in two conditions; it will show the special map which can show the characters of the double-safe-edges space and the death area (U shape) which can show its flexible ability. Simulation II is the simulation of efficiencies of the selecting the local target point in belief space, and it will show the changes of belief in the process of the detecting path planning.

The simulation of the path planning in special environment which has lots of edges and corners.

The simulation of the path planning in U-shape environment which is always called dead area.

So this simulation has proved that the method of double-safe-edges space is a feasible method to describe the real-time detecting environment.

The simulation of the path in situation A for distance detecting.

The BBA state in the process of situation A.

Figure

The simulation of the path in situation B for distance detecting.

The BBA state in the process of situation B.

As can be seen from literature works that there are a lot of methods for robot path planning, but most of them do not work well in a complex real-time environment. In this paper, we are making some efforts for solving two problems in real-time detecting path planning: one is the expression the environment, and the second is how to optimize the path in local path planning. The double-safe-edges space has been presented to express the environment, and the simulation has proved the feasibility of this approach. Then, the belief space has fused the factors and the uncertainty of detection in real-time detecting path planning successfully, the simulation of the belief space is well running. So these achievements will help the researching of the real-time path planning effectively. Certainly, there are a lot of tough jobs such as the details of the system structure, or how to control the robot accurately. All these considerations should be further extended in our future work.

This work was supported by the National Natural Science Foundation of China (nos. 51379049 and 51109045) and the Fundamental Research Funds for the Central Universities of China (HEUCFX41302).