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CALCULATING THE COLLISION RISK OF A MOBILE ROBOT WITH A FUZZY CONTROLLER IN AN ENVIRONMENT WITH DYNAMIC OBSTACLES
Kifayat Mammadova, Yegana Aliyeva, Aytan Aliyeva
Azerbaijan State Oil and Industy University [email protected]
[email protected] [email protected]
Abstract
Research on navigation of mobile robots in uncertain dynamic environments is of great importance. This article is focused on solving existing problems such as planning, optimization, failure in difficult situations, and error rate vector prediction under constantly changing uncertain conditions. The aim of the conducted research is to propose a multilayer fuzzy logic model based on decision making for robots to find safe path navigation by overcoming any kind of obstacles and to understand collision-free movement of mobile robots in an uncertain dynamic environment. In this study, fuzzy logic control prediction and multilayer solution priority rules are used to improve the quality of the next position based on the path length, safety and realization time. For this purpose, the article considers the topic of calculating the risk of collision with obstacles for planning the trajectory of a mobile robot with a fuzzy controller in an environment with dynamic obstacles.
Keywords: dynamic mobile robot, fuzzy approach, dynamic obstacle, membership function, risk assessment, fuzzy inference.
I. Introduction
In modern times, there is a great interest in robotic devices in all fields of industry, computer systems, scientific research centers, service works, household and agriculture. That is, they are considered necessary and an important factor in all fields. The replacement of humans in dangerous and military operations by robots has led to an increase in researchers' interest in mobile intelligent robots [1]. It is very important for mobile robots to plan the trajectory of an optimal free path by overcoming obstacles from their starting point to the final target point based on indicators such as time, distance and energy [2]. The goal of mobile robot is to understand the environment, interpret received data, improve knowledge about its position, overcome obstacles by controlling the turning angle and linear speed of the mobile robot, and to plan the route from the starting point to the final position [3]. Path planning in static environments is a well-studied and efficiently solved problem. However, the problem of robot trajectory planning in an environment presence several dynamic obstacles is still has not been solved. This is due to the need to add time dimension in trajectory planning [4].
Classical approaches to planning mobile robots are inadequate. Such approaches cannot overcome the challenges of dynamic environments or inadequate information about the environment [5,6]. Therefore, some reactive (i.e., taking action after the occurrence of events not taken into account in advance) approaches are presented. The application of these approaches allows to learn, justify the main task and use artificial intelligence methods in solving problems. In this work, artificial intelligence methods, including fuzzy logic and neural networks are form the
basis of navigation systems in mobile robots [7-9]. Fuzzy logic (FL) is a type of artificial intelligence that describes human thoughts and decision-making. Fuzzy logic allows interpretation of intermediate values between traditional values, such as yes/no, true/false, high/low and etc.. Fuzzy logic approaches have been used in many studies on mobile robot navigation [10].
A number of approaches have been proposed in various research studies related to the preplanning and analysis of the robot's motion trajectory. Comparing the approaches proposed in this study with other intellectual and heuristic methods is an effective way to solve the previously mentioned problems and make effective decisions in complex situations. Especially, the combination of our approach with fuzzy logic is one of the most important types of intelligent hybrid systems. In the fuzzy logic control system, the priority and prediction rules of the multilayer decision-making approach are used to improve the quality of the next position related to the working time, the safety conditions in risky situations, and the trajectory. Fuzzy systems are still used because planning is still difficult when there are some dynamic obstacles. This is because planning is included as an additional dimension to the search space explored by the planner. Moving obstacles and their velocity vectors are not known in advance. Other approaches to planning mobile robots are not powerful enough and cannot overcome the challenges of mixed complex situations where the environment is dynamic and unknown.
In this research work, a multilayer decision-based fuzzy logic approach is proposed for obstacle avoidance based on velocity vector of dynamic obstacle and priority behavior. Planning the navigation of a mobile robot in a complex dynamic environment and performing sensor-based planning tasks are the essence of this approach. To make the proposed approach more efficient and intelligent, it is considered appropriate to use fuzzy logic controllers. These controllers evaluate the robot's next-state prediction procedure in the action area and enable optimal decision-making. The best next step of a mobile robot is selected based on several proposed criteria for solving sensor-based planning problems in both risky and complex dynamic uncertain environments. These criteria are prepared by the rules of the proposed approach and transformed into fuzzy variables to create a fuzzy reasoning system.
II. Mathematical modeling and kinematics of a mobile robot
Mathematical modeling of a mobile robot in a environment with dynamic obstacle includes a risk calculation based on finding the distance between the robot's platform and the nearest obstacle to the left, front, or right at each program iteration. The distance to the nearest obstacle is determined by a measuring sensor placed on the platform of the mobile robot.
Figure 1 provides a graphical representation of the distances between the platform of the mobile robot and the obstacles.
Figure 1: Distances between the platform of the mobile robot and the obstacles
In the computer modeling of the mobile robot, dynamic moving obstacles whose position in its working zone is unknown were considered. The result of the measurements performed by
distance measuring sensors in their Cartesian coordinate system is known at each program iteration. The modeling involves the problem of measuring the distance between all transmitters of the mobile robot and dynamic obstacles with unknown position during each program iteration. The distance between the mobile robot and obstacles is determined by the formula (1) [1,2]:
dsi = V (xm - XSi )2 + (ym - ySi )
(1)
there are (xm,ym) - coordinates of the obstacle; (xsi,ysi) - are the coordinates of the ith measuring sensor. If the obstacle is the left dynamic obstacle (or right , or front dynamic obstacle), (figure 2), the distance between the measurement sensor and the obstacle is calculated as follows:
(2)
There is (xci, yci) - are presented as the coordinates of the nearest point indicating the distance
from the ith measurement sensor to the obstacle (Fig. 2).
y
Figure 2: Graphical representation of the distance between a mobile robot and a dynamic obstacle
In practical terms, employing geometric techniques for distance measurement via sensors is not deemed significant. Currently, dynamic obstacles with unspecified positions may have varied shapes and placements. The sensor covering mounted on the mobile robot's working body is positioned to establish contact with obstacles.
The placement density of the measurement sensors should be arranged in such a way that the invariable (i.e., static state) zones of the sensor cover are not taken into account [7].
During the calculation of the distance between the mobile robot's platform and nearby obstacles, any objects within the sensing range of the distance sensors on the platform are considered obstacles. In computer modeling, the information (input data) received from all sensors represents the input parameters of the function. Based on the input data, the distance between each sensor and the nearest dynamic obstacle is determined.
In the process of developing the intelligent planning system of real-time displacement of a mobile robot in an uncertain environment, it is necessary to take into account a number of its characteristics, for example, the size and maximum speed of the platform of the mobile robot, the discretization period, the type of mechanical connection between the motor and the wheels.
There are two typical configurations of a mobile robot: a differential and a classic three-wheeled mobile robot (Figure 3) [8].
Figure 3: a) differential va b) a classic three-wheeled mobile robot configurations
The differential configuration of the mobile robot uses independent velocities on both wheels (left and right wheels, VL and VR) to move it to a specific point (x, y) in a two-dimensional plane and in a specific direction W.
The displacement of the three-wheeled mobile robot towards a target point and its direction are determined by utilizing a single controlled speed and angle applied to one wheel. The research focuses on a differential mobile robot as the subject of investigation.
In the differential mobile robot model, wL and wR represent the angular velocities of the left and right wheels. The radius of the driven wheels is denoted by r, and the distance between the two wheels is denoted by b. Assume that solid objects placed on the platform of a mobile robot move on a non-deformable horizontal surface. At any moment, the current coordinate (x, y) and rotation angle of the mobile robot in the Cartesian system are known (Fig. 3).
The movement parameters of the mobile robot in the working area are calculated by the following formulas [1,5]:
q = [x y ip <PL <PR
(3)
X -sin^ 0
y = -COS^ 0
A. . 0 1.
f]
(4)
Here: x, y - the current coordinates of the mobile robot. - rotation angle of left wheel;
- rotation angle of right wheel; ^ - turn angle of the mobile robot.
The corresponding linear and angular velocities of the mobile robot are defined as: If we replace statements (5), (6), (7) with (4):
The current coordinate and rotation angle of the mobile robot - (x, y) and O are determined from the following equation [9]:
V =
m, =
VR+VL a)R+aiL
- = -r
2 2
d0L _ V-(fc/2)w
Mu = ■
dt
d®R dt
V+(b/2)w
X -rsin^/2 -rsin^/2
y = -rcosip/2 -rcosip/2
A -r/b r/b
n
(5)
(6)
(7)
(8)
Based on the obtained equation (9), the current coordinate and direction of the mobile robot over time is calculated as follows [9,10]:
x = ■
r cos ^(w^+w^)
r sin ^(w^+w^)
lp = ^R-^L r
(9)
here, WLi - relative rotation angle of the left wheel;
WRi - relative rotation angle of the right wheel.
By determining the displacement of the wheels of the mobile robot, the calculation of its current coordinate is performed taking into account the already known position of the robot. The relative rotation angle of each wheel is measured by means of position sensors placed on the axles of the electric motors.
xi = xi-i + Vi = Vi-i +
r cos 0(A0Ri+A0Li)
r sin 0(A0ri+A0l{)
(10)
= &Î-1 +
r(A0Ri+A0Li)
Equation (10) shows that the movement coordinates and direction of the mobile robot depend on the measurement accuracy and geometrical parameters (the radius of the wheel-r, distance between the two wheels -b) of the rotation angle of the wheels.
Table 1 lists the values of the parameters of the studied robot model.
Parameters Values
The mobile robot's platform size, m 0.50
Distance between drive wheels -b, m 0.40
The radius of the wheel -r, m 0.042
Engine type 12 V DC
Maximum engine speed, dovr/daq 120
Discretization period, s 0.028
Based on the data in Table 1, the values of the following parameters of the mobile robot can be calculated:
number of iterations
per
second:
i
0,028
= 36
the maximum displacement step of a wheel in each iteration:
100*2^*0,042
= 0,0122 m.
Finding a suitable collision-free path connecting the initial position p0 at the initial time t0 of the mobile robot with the final position pend at the time tend can be considered as a navigation problem (task). In the environment with dynamic obstacles, it is important to determine the velocity vector of each obstacle, especially the displacement of obstacles in different directions towards the moving robot, the number of iterations in one second and the maximum displacement step of one wheel in each iteration.
The main problem here occurs when the robot decides to move inside the dangerous area where 3 obstacles moving in an unknown direction move towards each other and collide with the next position. Since the robot has a different solution to avoid obstacles in each movement, it cannot choose how to avoid them. For example, the decision made by the robot to avoid an
obstacle moving to the right is to move to the left, but the decision made by the robot against obstacles moving to the left and forward direction is different from this decision [11]. Therefore, a collision will occur as long as the robot cannot predict the dangerous area, i.e. predict and change its state to another state with a lower risk of collision and obstruction. In order to solve the problem, the development and realization of the intelligent planning system of the predicted navigation of the mobile robot in an unknown dynamic environment was considered.
The main goal in this section is to guide a robot moving under uncertainty to choose the most optimal next step between positions without collision with dynamic obstacles. A fuzzy control system is developed to verify the forecasting process and decisions about the next position and to choose the best positions in terms of minimum risk. This control is activated after the next step finds a position away from the collision. The values of three fuzzy variables, including left dynamic (LeftD), front dynamic (FD), and right dynamic (RightD), are calculated by the control for each iteration.
The first fuzzy variable (FV) is the distance to the left (right or front) dynamic obstacle, defined as intersection points (IP) between the sensor range and the obstacle. The robot learns to determine the safe distance to the left (right or front) dynamic obstacle when detected through this part of the control. The control measures the intersection points between the sensor layers and the obstacle. If the obstacle intersects the top layer of the sensor, it means that the obstacle is far from the robot. At the same time, if the obstacle crosses the middle layer of the sensor, it indicates that the distance between the obstacle and the robot is safe. However, if the intersection occurs with a lower layer, the robot is close to the obstacle. By extracting these values, we obtain the values (FV) to be used in future fuzzy control. This variable can be formulated as follows [11]:
Sensor layers (Rt = 0.2,0.4,0.6,0.8), Range quarters (qj = 1,2,3,4th) (11) IP = (n, Rx(jlj,■), ,■ ), xpolygon, ypolygon) (12)
Function [distance] = to obtain the obstacle distance from the interval (cross-interval, first layer radius, second layer radius, third layer radius, fourth layer radius) (13)
Function (distance) = Determining the direction of the obstacle from qj
(1stqj, 2nd Rj) (14)
LeftD (Rt, qj) = (distance, direction) (15)
LeftD £ [-1,1] (16)
When the minimum distance of LeftD is placed at the new position Ri(min), its maximum distance occurs at Ri(max). All values of LeftD are between these maximum and minimum levels. A zero value of IP indicates that the obstacle is outside the range of the sensor, and a positive value indicates that the obstacle is within the range of the sensor. Three ranging sensors are used to determine the exact position and direction of obstacles. For example, if the robot detects that the obstacle is in the first and second quarters, it means that the obstacle is in front of the robot. The second and third quadrants show the left side of the robot, and the third and fourth quadrants show the obstacles behind the robot. Finally, the fourth and first quadrants belong to the right side of the robot. Sometimes the obstacle is in the first quarter of the navigation. In this case, the sensor informs for all other quarters that the obstacle is in the right corner of the robot, and so on. Four linguistic variables were identified for LeftD, including Close, Medium, Far, and Very Far. Analogously, these linguistic variables were used for the right and front dynamic barriers.
Input parameters Left, FD, and Right, entered as control parameters, represent the distance to the left, right, and front dynamic obstacles, respectively. These elements are determined using the same equations applied for dynamic barriers.
The constituent functions of the left dynamic (LeftD), front dynamic (FD) and right dynamic (RightD) input variables are the same, and their structure is shown in Figure 4. Fuzzy subsets for the input and output variables of the mobile robot are defined as trapezoidal membership functions.
Upon detecting the obstacle's distance to the next position and its location, the controller organizes these obstacles into a matrix. dynamic obstacles. If the controller detects that the next position faces two or more dynamic obstacles, it determines the position of the three nearest dynamic obstacles.
The output fuzzy variable of this process is the risk for the next trajectory. Fuzzy rules are formulated to evaluate the efficiency of the next trajectory. An intelligent system has been developed with five linguistic grades for this speech, including Very High (VH), High (H), Normal (N), Low (L) and Very Low (VL). The output variable of the proposed fuzzy control can be expressed as:
7 V
ra :i .hi i «
\ / ; X- /
\ / V \
Y
/ \ / \ A
Figure 4: Obstacle function curves using the trapezoidal method Risk = Function (LeftD, FrontD, RightD), Risk £ [0,1]
(17)
XX M
X X
- W V
w AA
Figure 5: Trapezoidal membership function curves of risk
III. Structure of the issue
Trapezoidal membership functions (MF) are chosen in fuzzy logic depending on the structure of the issue:
1. Have simpler analytical structures.
2. Preserves the adaptability.
3. It gives the user more freedom in MF construction.
4. Preserves the novelty.
Also, 5 intervals are selected to evaluate the efficiency of the next trajectory by accurately identifying dangerous situations and to formulate fuzzy rules.
Determining the input and output values of the mobile robot. Designed fuzzy logic elements.
input variables:
Left dynamic = relationship between the robot and the left dynamic obstacle in the interval -
1- 1.
Front dynamic = -1- 1 relationship between the robot and the front dynamic obstacle in the interval -1- 1.
Right dynamic = relationship between the robot and the right dynamic obstacle in the interval -11.
Output variable:
Risk = Probability of collision risk of robot with obstacles in interval 0-1.
The goal is to guide the robot to choose the most optimal next step between positions, avoiding collisions. Fuzzy management is developed to verify the forecasting process and next position decisions and select better positions with minimum risk. This control is activated after the next step finds a position away from the collision. The values of three fuzzy variables, including left speaker, front speaker, and right speaker, are calculated by the control for each iteration. Tables 2 and 3 describe the normalized universe values of the input and output parameters.
Table 2: Normalization table of universes of input parameters
Input parameters Range Normalized piece Linguistic variable
Left, front, right dynamic obstacles [-<*> - -1.00] [0 - 0.25] Close
[-0.99-0.00] [0.25-0.5] Medium
[0.01-0.50] [0.5-0.75] Far
[0.51-1.00] [0.75- 1.0] Very far
Table 3: Normalized universium table of the output parameter
Output parameters Range Normalized piece Linguistic variable
Risk [0-0.20] [0-0.20] Very low
[0.21-0.40] [0.21-0.40] Low
[0.41-0.60] [0.41-0.60] Normal
[0.61-0.80] [0.61-0.80] High
[0.81-1.00] [0.81-1.00] Very high
Defining fuzzy rules. The knowledge base consists of a set of fuzzy control rules expressed as a fuzzy conditional proposition. The knowledge base constitutes the rule set of the fuzzy controller. When building a knowledge base, it is important to define input variables, output
variables, types of fuzzy regulation rules etc. Fuzzy rules must be based on one of four basic conditions to obtain. For example, 1) if the fuzzy control rules are given in the form of a fuzzy conditional sentence, then there is a connection between the state variables included in the condition part and the control variables included in the sentence part; 2) in the case of fuzzy rules, the management activity of a person is observed and created by revealing the input-output connection; c) in the linguistic approach, the fuzzy model of the controlled process can be viewed as a linguistic description of its dynamic characteristics. d) the knowledge base of the fuzzy controller is formed through self-learning.
The fuzzy IF-THEN implication proposed by Mamdani was used in the research work. Each fuzzy control rule is given in the form of a fuzzy conditional sentence. These rules describe the relationship between the state variables included in the condition part and the control variables given in the judgment part. 64 fuzzy rules indicating the positions of the left, front and right dynamic obstacles and the dependence of the risk on these positions have been defined. (Figure 6).
Figure 6: Fuzzy-logic rules
There is some examples: Rule 1.
If (LeftDynamic is Close) and (FrontDynamic is Close) and (RightDynamic is Medium) then (Risk is VeryHigh)
If (LeftDynamic = -0.83) and (FrontDynamic = -0.77) and (RightDynamic = -0.26) then (Risk =
0.89)
Rule 2.
If (LeftDynamic is Close) and (FrontDynamic is Medium) and (RightDynamic is Far) then (Risk is High
If (LeftDynamic = -0.81) and (FrontDynamic = -0.32) and (RightDynamic = 0.07) then (Risk =
0.67)
Rule 3.
If (LeftDynamic is Medium) and (FrontDynamic is Medium) and (RightDynamic is Far) then (Risk is Normal)
If (LeftDynamic = -0.30) and (FrontDynamic = -0.32) and (RightDynamic = 0.25) then (Risk =
0.44)
Rule 4.
If (LeftDynamic is Far) and (FrontDynamic is VeryFar) and (RightDynamic is Far) then (Risk is Low)
If (LeftDynamic =0.44) and (FrontDynamic = 0.94) and (RightDynamic =0.24) then (Risk =
0.26)
Rule 5.
If (LeftDynamic is Medium) and (FrontDynamic is VeryFar) and (RightDynamic is VeryFar) then (Risk is VeryLow)
If (LeftDynamic = -0.28) and (FrontDynamic = 0.82) and (RightDynamic = 0.95) then (Risk = 0.082)
Figure 7 illustrates the logic inference system.
Figure 7: Logic inference system.
Figure 8 depicts the fuzzy spatial clusters of the mobile robot's left, front and right dynamic obstacles.
Figure 8: Fuzzy spatial clusters of the mobile robot's risk due to dynamic obstacles: a) Front dynamic-Right dynamic b) Front dynamic -Left dynamic c) Right dynamic -Left dynamic.
IV. Conclusion
A sensor-based algorithm is proposed for robot trajectory planning in various dynamic environments. A fuzzy system was used to evaluate the positions generated by the planner and select better steps that reduce the total path length with the help of an algorithm. In addition, it
keeps the robot away from possible local minima. This system uses priority rules and prediction of a multilayer approach to improve the quality of the next position by taking into account risky situations to develop three fuzzy control variables. Fuzzy variables will be calculated every time the planner creates a new collision configuration. They will serve as input to a fuzzy reasoning system to decide the overall risk of the next step. When the prediction shows that the risk of collision with many obstacles is too high, the mobile robot finds a new way from alternative solutions to avoid the risky regions through priority behavior. This algorithm makes the multilayer decision making process more efficient and effective by using the advantages of fuzzy logic.
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