Navigation control of mobile robot using interval type-2 neural fuzzy controller optimized by dynamic group differential evolution

Abstract

This study developed and effectively implemented an efficient navigation control of a mobile robot in unknown environments. The proposed navigation control method consists of mode manager, wall-following mode, and towards-goal mode. The interval type-2 neural fuzzy controller optimized by the dynamic group differential evolution is exploited for reinforcement learning to develop an adaptive wall-following controller. The wall-following performance of the robot is evaluated by a proposed fitness function. The mode manager switches to the proper mode according to the relation between the mobile robot and the environment, and an escape mechanism is added to prevent the robot falling into the dead cycle. The experimental results of wall-following show that dynamic group differential evolution is superior to other methods. In addition, the navigation control results further show that the moving track of proposed model is better than other methods and it successfully completes the navigation control in unknown environments.

Introduction

Mobile robots have been used to solve many problems in recent years, and it helps human solving many problems such as environmental exploration, object handling, and navigation.^1–3 To achieve these missions in a complex environment, the navigation technology of mobile robot is a very important topic and the design of controller becomes a major subject.

Mobile robots detect the obstacles through the sensors to avoid collision in the navigation control. Therefore, obstacle avoidance is an essential element for the navigation commission. Recently, many novel designs^4–6 for intelligent robot control have been developed to improve obstacle avoidance in robot navigation. For example, researchers have used fuzzy logic control (FLC) and neural network (NN) to apply in robot navigation control. Al-Sahib and Ahmed ⁴ applied the information detected by the sensors directly into the designed fuzzy controller, and it makes the robot successfully perform the obstacle avoidance. Dutta ⁵ combined FLC, NN, and self-adaptive learning to adjust the parameters in fuzzy neural network (FNN). Though the adopted type-1 fuzzy models in Al-Sahib and Ahmed ⁴ and Dutta ⁵ could complete the navigation control, its performance was not acceptable, and the uncertainty of input data owing to the environmental noise in real state affects the control. In recent years, the type-2 fuzzy system has been proposed to perform fuzzy inference and obtain a better performance than the type-1 fuzzy system. Liang and Mendel ⁷ then used Karnik–Mendel algorithm to implement the order reducing of type-2 fuzzy system. Kim and Chwa ⁸ then used the type-2 neural fuzzy network (Type-2 FNN) to improve performance. But the computation in Liang and Mendel ⁷ and Kim and Chwa ⁸ is more complex. This study utilized interval type-2 neural fuzzy network combined with the center of sets (COS) ⁹ to reduce computational complexity. Therefore, in this study, an efficient interval type-2 neural fuzzy controller is proposed for navigation control of mobile robot.

Training of the parameters is the main problem in designing a type-1 or type-2 neural fuzzy network. Backpropagation (BP) training^5,8 is commonly adopted for solving this problem. Since the steepest descent approach is used in BP training to minimize the error function, the algorithms may reach the local minima very quickly and be not easy to find the global solution. The aforementioned disadvantages lead to suboptimal performance, even with a favorable neural fuzzy network topology. Therefore, training technologies that can be used to adjust the system parameters and find the global solution while optimizing the overall structure are required. Recently, many evolutionary algorithms^10–14 for optimization have been used to adjust the parameters in NN or FNN. Evolutionary algorithms, which are also called biologically inspired computation, originated from the observation of natural phenomenon. These simulate biological behavior of some creatures such as particle swarm optimization (PSO), ¹⁰ ant colony optimization (ACO), ¹¹ differential evolution (DE), ¹² and artificial bee colony (ABC). ¹³ DE has been commonly used to solve the optimization problems in recent years. It is superior in simple structure, less setup parameters, and optimization ability, but the disadvantage is unstable convergence and easy to fall into local optimum. In this study, a novel algorithm, dynamic group differential evolution (DGDE), is proposed to improve the drawbacks of traditional DE.

The aim of this study is to improve the navigation control of a mobile robot and successfully reach the goal in the unknown environment so as to increase the exploration benefits. The proposed navigation control method consists of mode manager, wall-following (WF) mode, and towards-goal (TG) mode. The mode manager was designed to enable the robot to switch between two behavior modes (1) TG and (2) WF according to the relation between robot and environment. An interval type-2 fuzzy neural network controller (IT2NFC) with DGDE algorithm is proposed to implement the WF control of mobile robot. In the proposed DGDE, the concepts of grouping as well as modified mutation are also used to increase solution stability and precision. In addition, a mechanism for calculating the distance between robot and goal was used to prevent the robot from falling into a dead cycle. Several evolutionary algorithms are implemented to compare and analyze the navigation performance by two evaluation indexes. One evaluating index is the total moving distance of the robot in whole navigation. The other evaluating index is the time required to complete the navigation.

Reinforcement learning of WF control

WF control of a mobile robot requires a good controller to maintain proper distance between robot and wall as well as being adaptive to every kind of environments. This study developed an IT2NFC with DGDE learning algorithm, and four evaluation methods are implemented for the comparisons of the robot’s WF.

Description of the mobile robot

The mobile robot, Pioneer3-DX, is manufactured by Adept MobileRobots LLC, USA and shown in Figure 1. This robot has the characteristics of high load, high endurance, and high scalability. Pioneer3-DX supports cross-platform libraries and software development kits as well as robot motion control, client–server models, and many equipped libraries to be applied in different areas or to be combined with various peripherals.

Figure 1.

Pioneer3-DX mobile robot.

The Pioneer3-DX mobile robot has eight sensors, one on each side, and the other six sensors are separated by 20° on the front side to provide with object detection, distance measuring, obstacle avoiding, characteristic identification, position, and navigation. Only one sensor of the sonar sequence is stimulated at one time, and the frequency of distance measuring is set to 25 Hz, that is, every sensor activates in 40 ms, and distance measuring is from 10 cm to 5 m. Some basic specifications of Pioneer3-DX are listed in Table 1.

Table 1.

Specification of Pioneer3-DX.

Length/width/height (cm)	44/39/23
Weight (kg)	9
Maximum translation speed (mm/s)	1400
Wheels’ diameter/width (mm)	195/47

WF control using IT2NFC with DGDE learning algorithm

This study proposes IT2NFC with a DGDE learning algorithm to efficiently implement the WF control of a mobile robot.

The proposed interval Type-2 neural fuzzy controller

The whole framework and operation process of IT2NFC are described in this section, and the proposed controller is to extend our previous research.^15,16 Because the computational complexity of normal interval Type-2 fuzzy system is higher in defuzzification, the nonlinear combination output of functional link neural network (FLNN) ¹⁷ is added into the consequent of corresponding fuzzy rule. Each rule can be expressed as

$$\begin{array}{c}\mathrm{Rule}j:\mathrm{IF}{x}_1\mathrm{is}{\tilde{A}}_{1j}\mathrm{and}{x}_2\mathrm{is}{\tilde{A}}_{2j}\dots \mathrm{and}{x}_i\mathrm{is}{\tilde{A}}_{\mathrm{ij}}\hfill \\ \mathrm{THEN}{y}_j=\sum _{k=1}^M{\omega }_{\mathrm{kj}}{\phi }_k\hfill \\ ={\omega }_{1j}{\phi }_1+{\omega }_{2j}{\phi }_2+\cdots +{\omega }_{\mathrm{Mj}}{\phi }_M\hfill \end{array}$$

(1)

where $x_i$ are the input variables, $y_j$ are the local output variables, ${\tilde{A}}_{\mathrm{ij}}$ are the interval type-2 fuzzy set, i is the number of input variables, ${\omega }_{\mathrm{kj}}$ are the linking weights of local output, ${\phi }_k$ are the basis trigonometric functions of input variables, M is the number of basis function, and j is the jth fuzzy rule.

IT2NFC is a five-layer network structure and is shown in Figure 2. $u^{\left(l\right)}$ represents output of the lth layer node. The operation of each layer is as follows:

• Layer 1 (input layer): The input data are imported into the next layer, and it is only the transmission of information without any computation

  $$u_i^{\left(1\right)}=x_i$$

  (2)

• Layer 2 (membership function layer): It is mainly the computation of fuzzification and every node is defined as the interval type-2 fuzzy set. It is shown in Figure 3 that Gaussian membership function with uncertain mean value, $[m_{\mathrm{ij}1},m_{\mathrm{ij}2}]$ , and fixed standard deviation (STD), ${\sigma }_{\mathrm{ij}}$ , and it is defined as

$$u_{\mathrm{ij}}^{\left(2\right)}=\exp (-\frac{{[u_i^{\left(1\right)}-m_{\mathrm{ij}}]}^2}{{\sigma }_{\mathrm{ij}}^2})\equiv N(m_{\mathrm{ij}},{\sigma }_{\mathrm{ij}},u_i^{\left(1\right)}),m_{\mathrm{ij}}\in [m_{\mathrm{ij}1},m_{\mathrm{ij}2}]$$

(3)

where $m_{\mathrm{ij}}$ and ${\sigma }_{\mathrm{ij}}$ represent the mean value and STD of ith input with jth Gaussian membership function. Because of footprint of uncertainty (FOU), it can be shown as the upper bound membership function ${\overline{u}}_{\mathrm{ij}}^{\left(2\right)}$ and the lower bound membership function ${\underset{¯}{u}}_{\mathrm{ij}}^{\left(2\right)}$ . The content of the membership is

$${\overline{u}}_{\mathrm{ij}}^{\left(2\right)}\left(u_i^{\left(1\right)}\right)=\{\begin{array}{c}\hfill N(m_{\mathrm{ij}1},{\sigma }_{\mathrm{ij}};u_i^{\left(1\right)}),\hfill \\ \multicolumn{1}{c}{1,}\\ \multicolumn{1}{c}{N(m_{\mathrm{ij}2},{\sigma }_{\mathrm{ij}};u_i^{\left(1\right)}),}\end{array}\begin{array}{c}\hfill u_i^{\left(1\right)}<m_{\mathrm{ij}1}\hfill \\ \multicolumn{1}{c}{m_{\mathrm{ij}1}\le u_i^{\left(1\right)}\le m_{\mathrm{ij}2}}\\ \multicolumn{1}{c}{u_i^{\left(1\right)}>m_{\mathrm{ij}2}}\end{array}$$

(4)

and

$${\underset{¯}{u}}_{\mathrm{ij}}^{\left(2\right)}\left(u_i^{\left(1\right)}\right)=\{\begin{array}{c}\hfill N(m_{\mathrm{ij}2},{\sigma }_{\mathrm{ij}};u_i^{\left(1\right)}),u_i^{\left(1\right)}\le \frac{m_{\mathrm{ij}1}+m_{\mathrm{ij}2}}{2}\hfill \\ \multicolumn{1}{c}{N(m_{\mathrm{ij}1},{\sigma }_{\mathrm{ij}};u_i^{\left(1\right)}),u_i^{\left(1\right)}>\frac{m_{\mathrm{ij}1}+m_{\mathrm{ij}2}}{2}}\end{array}$$

(5)

Therefore, the output of the layer 2 can be represented as an interval $[{\overline{u}}_{\mathrm{ij}}^{\left(2\right)},{\underset{¯}{u}}_{\mathrm{ij}}^{\left(2\right)}]$ .

Figure 2.

Framework of IT2NFC.

Figure 3.

Uncertain mean value of interval type-2 fuzzy set.

• Layer 3 (firing layer): Each node in this layer is a rule node using product operation to obtain the firing strength, ${\overline{u}}_j^{\left(3\right)}$ and ${\underset{¯}{u}}_j^{\left(3\right)}$

$${\overline{u}}_j^{\left(3\right)}=\underset{i}{\Pi }{\overline{u}}_{\mathrm{ij}}^{\left(2\right)}\mathrm{and}{\underset{¯}{u}}_j^{\left(3\right)}=\underset{i}{\Pi }{\underset{¯}{u}}_{\mathrm{ij}}^{\left(2\right)}$$

(6)

where $\underset{i}{\Pi }{\overline{u}}_{\mathrm{ij}}^{\left(2\right)}$ and $\underset{i}{\Pi }{\underset{¯}{u}}_{\mathrm{ij}}^{\left(2\right)}$ are, respectively, upper and lower rule boundaries of firing strength.

• Layer 4 (consequent layer): Interval type-2 fuzzy system produces interval type-1 fuzzy set by degrading computation and outputs the value after defuzzification. Owing to the higher complexity of traditional type-2 degrading computation like iterative procedure of Karnik–Mendel algorithm, ⁷ we adopt COS for less complexity by

$$Y\left(x\right)=[y_l,y_r]=\frac{\underset{a^1}{\int }\cdots \underset{a^M}{\int }\underset{f^1\in [{\underset{¯}{f}}^1,{\overline{f}}^1]}{\int }\cdots \underset{f^M\in [{\underset{¯}{f}}^M,{\overline{f}}^M]}{\int }1}{\frac{\sum _{i=1}^M{f}^i{a}^i}{\sum _{i=1}^M{f}^i}}$$

(7)

Such a method is composed of firing strength of upper and lower bound, and it simplifies the process of degrading computation. The composition of firing strength and responding output are

$${\underset{¯}{u}}^{\left(4\right)}=y_l=\frac{\sum _{j=1}^R{\underset{¯}{u}}_j^{\left(3\right)}\left(\sum _{k=1}^M{\omega }_{\mathrm{kj}}{\phi }_k\right)}{\sum _{j=1}^R{\underset{¯}{u}}_j^{\left(3\right)}}$$

(8)

and

$${\overline{u}}^{\left(4\right)}=y_r=\frac{\sum _{j=1}^R{\overline{u}}_j^{\left(3\right)}\left(\sum _{k=1}^M{\omega }_{\mathrm{kj}}{\phi }_k\right)}{\sum _{j=1}^R{\overline{u}}_j^{\left(3\right)}}$$

(9)

where $\sum _{k=1}^M{\omega }_{\mathrm{kj}}{\phi }_k$ is the nonlinear composition of input variables $x=(x_1,\dots ,x_M)$ in FLNN and ${\omega }_{\mathrm{kj}}$ is the linking weight of respective node. ${\phi }_k$ is the functional expansion of input variables and the functional expansion is the basis function composed by the trigonometric function

$${\phi }_k=[{\phi }_1,{\phi }_2,{\phi }_3,\dots ,{\phi }_M]=[x_1,\sin \left(\pi x_1\right),\cos \left(\pi x_1\right),\dots ,x_N,\sin \left(\pi x_N\right),\cos \left(\pi x_N\right)]$$

(10)

where $M=3\times N$ , M means the number of basis function and N is the number of variables.

• Layer 5 (output layer): The degrading output of previous layer located between $[{\overline{u}}^{(4)},{\underset{¯}{u}}^{(4)}]$ and the average value of ${\overline{u}}^{(4)}$ and ${\underset{¯}{u}}^{(4)}$ is used to output y for the crisp value of neural fuzzy controller as defuzzification

$$y=\frac{{\overline{u}}^{\left(4\right)}+{\underset{¯}{u}}^{\left(4\right)}}{2}=u^{\left(5\right)}$$

(11)

Differential evolution

Differential evolution is an evolutionary computation developed by Storn and Price in 1997. The main processes of differential evolution (initialization, mutation, and recombination) are similar to those in genetic algorithm and it is shown in Figure 4.

Figure 4.

Flowchart of DE.

• 1. Initialization. It is to setup the parameters of DE and to randomly initialize the target vector of solution space

$$X_{i,G}=[x_{i,1,G},x_{i,2,G},\dots ,x_{i,D,G}]$$

(12)

where $i=1,2,\dots ,\mathrm{NP}$ and NP means the number of population, G is the number of generation, and D is the dimension.

• 2. Mutation. Three randomly selected individuals as $x_{r1,G}$ , $x_{r2,G}$ , and $x_{r3,G}$ in the solution space are multiplied by the vector distance between $x_{r2,G}$ and $x_{r3,G}$ . Then, it combined with $x_{r1,G}$ and it is composited as a mutant vector $U_{i,G+1}$ which is

$$U_{i,G+1}=X_{r1,G}+F(X_{r2,G}-X_{r3,G})$$

(13)

where $U_{i,G+1}=[u_{i,1,G+1},u_{i,2,G+1},\dots ,u_{i,D,G+1}]$ . F is the mutation weighting factor and affects the weighting of each solution vector. Conventionally, its value is set between 0 and 2. The 2D descript of mutation is shown in Figure 5.

Figure 5.

2D depiction of mutation.

• 3. Recombination. In this operation, every mutation vector performs crossover operation with corresponding target vector and then produces a new trial vector, $V_{i,G+1}$ as $V_{i,G+1}=[v_{i,1,G+1},v_{i,2,G+1},\dots ,v_{i,D,G+1}]$ . The operation equation is defined as follows

$$v_{i,D,G+1}=\{\begin{array}{c}\hfill u_{i,D,G+1},\mathrm{if}\mathrm{ran}{\mathrm{d}}_D(0,1)\le \mathrm{CR}\hfill \\ \multicolumn{1}{c}{x_{i,D,G},\mathrm{otherwise}}\end{array}$$

(14)

where $\mathrm{ran}{\mathrm{d}}_D(0,1)$ is the random value of corresponding dimension between 0 and 1. CR is the crossover rate.

• 4. Selection. It is evaluated by the fitness value to select trial vector whether it could be the target vector of next generation. If the fitness value of trial vector $V_{i,G+1}$ is worse than current target vector, $X_{i,G}$ is reserved to the next generation

$$X_{i,G+1}=\{\begin{array}{c}\hfill V_{i,G+1},\mathrm{if}\mathrm{Fit}\left(V_{i,G+1}\right)>\mathrm{Fit}\left(X_{i,G}\right)\hfill \\ \multicolumn{1}{c}{X_{i,G},\mathrm{otherwise}}\end{array}$$

(15)

Proposed DGDE

Owing to the fast convergence in traditional DE, it often falls into the local optimum. This study proposed a novel DGDE to improve the local searching ability in traditional DE. The detailed explanations of DGDE are shown as follows:

• Step 1: Initialization

The study proposed DGDE to adjust parameters in IT2NFC for the WF control. All the parameters in IT2NFC are coded as individual chromosomes including Gaussian mean, $m_{\mathrm{ij}}$ , STD, ${\sigma }_{\mathrm{ij}}$ , type-2 average displacement, $d_{\mathrm{ij}}$ , and linking weight, ${\omega }_{\mathrm{kj}}$ , and shown in Figure 6.

Figure 6.

Individual chromosome.

• Step 2: Grouping

The fitness values of all the chromosomes are sorted in descending order and all the grouping number is coded as 0 initially (see Figure 7).

Figure 7.

Sorting the fitness values in descending order.

The best individual is set as the leader and is updated with grouping number, g. Then, taking the leader as the center, two thresholds, average distance and average fitness, are calculated. If an individual is satisfied with both the two thresholds, it will be set as the gth group

$$DI{S}^g={\displaystyle \sum _{i=1}^{NP}{\displaystyle {\sum }_{j=1}^D\sqrt{{\left(Leade{r}_j^g-X_j^i\right)}^2}}},\mathrm{if}{X}^i\mathrm{is\; coded}0$$

(16)

$$\mathrm{FI}{T}^g=\sum _{i=1}^{\mathrm{NP}}|\mathrm{Fit}\left(\mathrm{Leade}{r}^g\right)-\mathrm{Fit}\left(X^i\right)|,\mathrm{if}{X}^i\mathrm{is}\mathrm{coded}0$$

(17)

$$\mathrm{Average}\_\mathrm{Distance}\left(\mathrm{ADI}{S}^g\right)=\frac{\mathrm{DI}{S}^g}{\mathrm{NI}}$$

(18)

$$\mathrm{Average}\_\mathrm{Fitness}\left(\mathrm{AFI}{T}^g\right)=\frac{\mathrm{FI}{T}^g}{\mathrm{NI}}$$

(19)

where D means dimension, NP is the number of chromosomes, ${\mathrm{Leader}}_j^g$ is the gth leader’s location, $\mathrm{ADI}{S}^g$ and $\mathrm{AFI}{T}^g$ are the gth group distance and fitness threshold, and NI is the total number of grouping number 0 in the solution space (see Figure 8).

Figure 8.

The best chromosome is set as leader.

The distance, $\mathrm{Di}{s}^i$ , and fitness, $\mathrm{Fi}{t}^i$ , between individual coded “0” and leader in each group are calculated to identify whether ungrouping individual belong to the group by the aforementioned thresholds

$$\mathrm{Di}{s}^i=\sum _{j=1}^D\sqrt{{({\mathrm{Leader}}_j^g-X_j^i)}^2}$$

(20)

$$\mathrm{Fi}{t}^i=|\mathrm{Fit}\left(\mathrm{Leade}{r}^g\right)-\mathrm{Fit}\left(X^i\right)|$$

(21)

If $\mathrm{Di}{s}^i<\mathrm{ADI}{S}^g$ and $\mathrm{Fi}{t}^i<\mathrm{AFI}{T}^g$ , it means the individual and the leader are similar. That is, they are classified into the same group and updated the group number as g. If any condition is not satisfied, no grouping occurs (see Figure 9).

Figure 9.

Similar individuals are grouping into the same group.

If any ungrouping individual exists, it will return back to step 1. The remained ungrouping individual with best fitness value will be defined as the new grouping leader, and the process repeats steps 1 to 3 until all individuals are grouped (see Figure 10).

Figure 10.

Continue grouping for the ungrouping individuals.

• Step 3. Mutation

Here, two novel mutation methods (called DGDE_ M-1 and DGDE_M-2) are proposed in the DGDE

$$\mathrm{DGDE}\mathrm{\_}\mathrm{M}\mathrm{-}1:X_{\mathrm{rL},G}+F(X_{r1,G}-X_{r2,G})$$

(22)

$$\mathrm{DGDE}\mathrm{\_}\mathrm{M}\mathrm{-}2:X_{\mathrm{best},G}+F(X_{\mathrm{rL},G}-X_{r1,G})+F(X_{r2,G}-X_{r3,G})$$

(23)

where $X_{\mathrm{best},G}$ is the individual with best fitness value and $X_{\mathrm{rL},G}$ is a randomly selected leader in all groups.

To prevent the conventional DE easily falls into the local optimum, a random selected leader is used as a base vector (i.e. DGDE_M-1), and the directional vector can efficiently increase the searching ability of the algorithm. In DGDE_M-2, the best individual is set as the base vector. A differential vector of two randomly selected individuals is added, and the improved vector after mutation is surrounded around the best individual. If the position of leader is better than others, the differential vector of random leader and random individual can effectively improve searching ability and this is beneficial to search in the solution space.

Steps 4 and 5 are the process of recombination and selection, and are the same as the traditional DE. The flowchart of DGDE is shown in Figure 11.

Figure 11.

Flowchart of DGDE.

Reinforcement learning is utilized to implement in the WF mode of mobile robot. Such a method needs neither designing control rules by experts nor collecting training data. The proper way is to define the fitness function for evaluating the performance of mobile robot in the environment. Flowchart of WF mode is depicted in Figure 12. There are four input signals ( $S_1$ , $S_2$ , $S_3$ , $S_4$ ) and two output signals (V_L, V_R) in IT2NFC. $S_i$ is the distance detected by sonar sensors, and the detectable limitation is 0.1–1 m. The rotation speed of mobile robot’s left wheel (V_L) and right wheel (V_R) are output signals. The rotation speed is limited around −5.24 to 5.24 rad/s for over-speed and the time steps per cycle are set as 500 ms.

Figure 12.

Flowchart of wall-following mode.

To be adapted for every situation, the mobile robot is trained in a $11\mathrm{m}\times 8\mathrm{m}$ training environment including line, curve, straight angle curve, and U-type curve as Figure 13.

Figure 13.

Training environment.

For maintaining WF and avoiding collision in the training process, three stop conditions are designed:

1. Mobile robot will collide wall if any distance of sonar sensor, $S_i$ , is less than 0.1 m as shown in Figure 14(a).

2. Mobile robot leaves the wall if the sensing distance, $S_i$ , is greater than 0.7 m as shown in Figure 14(b).

3. Total moving distance is more than one cycle of training environment to guarantee the mobile robot at least completes one cycle of the training environment.

Figure 14.

(a) Colliding with wall and (b) leaving wall.

The proposed DGDE is used to optimize the IT2NFC parameters. Each individual represents one IT2NFC solution and is used to control the mobile robot in the training environment. While the robot meets the stop condition, the fitness function is used to evaluate the WF performance of the robot in the training environment. Then, the next individual takes the initial position until the terminating condition of the algorithm is activated.

The defined fitness function includes four sub-fitness functions, the moving distance ( $S{F}_1$ ), the distance to wall ( $S{F}_2$ ), the angle to wall ( $S{F}_3$ ), and the moving speed ( $S{F}_4$ ).

1. Moving distance sub-fitness function ( $S{F}_1$ ). This sub-fitness function evaluates the moving distance of mobile robot. If $T_{\mathrm{dis}}$ is closer to the predefined $T_{\mathrm{stop}}$ , this means that the mobile robot can successfully perform the total moving distance in the training environment

   $$S{F}_1=1-\frac{T_{\mathrm{dis}}}{T_{\mathrm{stop}}}$$

   (24)

   While mobile robot’s moving distance is greater than $T_{\mathrm{stop}}$ , it is set $T_{\mathrm{dis}}=T_{\mathrm{stop}}$ that means this condition is satisfied. However, the sub-fitness function $S{F}_1$ is 0 for the collision-free control.

2. Distance to wall sub-fitness function ( $S{F}_2$ ). The distance between the mobile robot and the wall is measured for keeping a fixed value. If the distance maintains a predefined value, RD will be zero

   $$\mathrm{RD}\left(t\right)=|S_4\left(t\right)-d_{\mathrm{wall}}|$$

   (25)

   Here, $d_{\mathrm{wall}}$ is a predefined fixed value (d_wall = 0.4 m) depicted in Figure 15(a), and the sub-fitness function $S{F}_2$ is defined as average of $\mathrm{RD}(t)$ during the total moving distance

   $$S{F}_2=\frac{\sum _{t=1}^{T_{\mathrm{total}}}\mathrm{RD}\left(t\right)}{T_{\mathrm{total}}}$$

   (26)

3. Angle to wall sub-fitness function ( $S{F}_3$ ). The angle between the robot and the wall is evaluated as depicted in Figure 15(b). When mobile robot is in parallel with the wall, $\theta$ is 90° and $\theta$ is defined as follows

   $$\mathrm{\theta }\left(t\right)=\mathrm{co}{\mathrm{s}}^{-1}\left(\frac{x{\left(t\right)}^2+S_4^2-S_3^2}{2\times S_4\times x\left(t\right)}\right)$$

   (27)

   where $\theta$ is the angle between sonar sensor $S_3$ and $S_4$ . Then, x(t) is obtained by the cosine theorem

$$x\left(t\right)=\sqrt{S_4^2+S_3^2-2S_4{S}_3\cos ({40}^{\circ })}$$

(28)

In order to keep the mobile robot parallel to the wall, the average of $|\theta (t)-90|$ is defined as follows

$$S{F}_3=\frac{\sum _{t=1}^{T_{\mathrm{total}}}|\theta \left(t\right)-90|}{T_{\mathrm{total}}}$$

(29)

• 4. Moving speed sub-fitness function ( $S{F}_4$ ). It is to evaluate the mobile robot’s moving speed for maintaining the predefined speed

$$S{F}_4=1-\frac{V_{\mathrm{average}}}{V_{\mathrm{hope}}}$$

(30)

where $V_{\mathrm{average}}$ is the average moving speed of mobile robot and $V_{\mathrm{hope}}$ is a predefined speed (default is 0.6 m/s). If the average moving speed is greater than the predefined speed $V_{\mathrm{hope}}$ , we set $V_{\mathrm{average}}=V_{\mathrm{hope}}$ .

By summation of the four sub-fitness functions, $S{F}_1,\dots ,S{F}_4$ , with weighted coefficients ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ , the fitness function of the WF control is $F(·)$

$$F(·)=\frac{1}{1+({\alpha }_{1}S{F}_1+{\alpha }_{2}S{F}_2+{\alpha }_{3}S{F}_3+{\alpha }_{4}S{F}_4)}$$

(31)

Here, the weighted coefficients are $[{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4]=[0.45,0.45,0.05,0.05]$ . Higher weighted coefficients represent higher importance of the fitness function. The first two weighted coefficients of the controlling parameters are more important and are set to 0.45 for successful learning.

Figure 15.

Demonstration of (a) $d_{\mathrm{wall}}$ and (b) $\theta$ .

Experimental results of WF control

This study implemented two mutation methods, DGDE_M-1 and DGDE_M-2, and compared the efficiency and stability with other evolutionary algorithms. The initial parameters of DGDE are set in Table 2. In order to observe the stability of each algorithm, the experiments were repeated for ten times.

Table 2.

Initial parameters of DGDE.

NP	CR	F	Generation	Rule
30	0.9	0.5	3000	5–7

Initially, setup parameters are total number of parameters (NP), crossover rate (CR), mutation weighting factor (F), and number of Fuzzy rules (Rule). Since the best number of fuzzy rules is difficult to determine, 5, 6, and 7 rules are used in the experiments. The performance evaluation includes the best fitness value (Best), the worst fitness value (Worst), the average fitness value (Average), the STD, and the number of successful runs. Experimental results are shown in Table 3. Here the number of success means that the number of successful runs one cycle for mobile robot in the training environment. Though the controller with fewer rules means that it needs fewer coding dimension consumption, less computation time, and less memory, the experimental result shows that the performance is worse. Thus, six fuzzy rules are adopted in this study.

Table 3.

Performance evaluation of different rules.

Fitness values	Fuzzy rules
	DGDE_M-1			DGDE_M-2
	5	6	7	5	6	7
Best	0.920140	0.922009	0.919523	0.921140	0.922401	0.920824
Worst	0.911234	0.916940	0.910223	0.910587	0.917553	0.910721
Average	0.916057	0.919452	0.915998	0.917312	0.919648	0.917735
STD	0.002970	0.001660	0.002547	0.002354	0.001056	0.002806
No. of success	9	10	8	10	10	9

Different algorithms are applied in WF mode to identify the performance in Table 4. The experiments were repeated for 10 times for each algorithm. The average fitness values of 10 times in proposed DGDE_M-1 and DGDE_M-2 are 0.919452 and 0.919648, respectively. The large average fitness value and the small STD in Table 4 represent that the mobile robot can implement better the WF mode and higher stability. In addition, the proposed DGDE_M-1 and DGDE_M-2 in the repeated 10 experiments successfully perform the whole WF control, whereas the other methods will sometimes hit obstacles. According to the data in Figure 16 and Table 4, the proposed DGDE performs better in WF mode, shorter time in cycling the environment, and lower STD than other algorithms. Therefore, the proposed method also possesses higher stability. And there shows several moving tracks of mobile robot with DGDE in training environment in Figure 17.

Table 4.

Wall-following performance of different algorithms.

Algorithm	Evaluation				No. of success
	Fitness value
	Best	Worst	Average	STD
DGDE_M-1	0.922009	0.916940	0.919452	0.001660	10
DGDE_M-2	0.922401	0.917553	0.919648	0.001056	10
JADE 18	0.921810	0.910873	0.916329	0.003222	9
Rank-DE 19	0.915885	0.852167	0.900978	0.016270	6
PSO 10	0.919911	0.910513	0.916044	0.002795	7
ABC 13	0.918024	0.907403	0.911947	0.004461	7

Figure 16.

Training fitness curves of wall-following control using different algorithms.

Figure 17.

Moving track in training environment applied by different algorithms. (a) DGDE_M-1, (b) DGDE_M-2, (c) JADE, (d) Rank-DE, (e) PSO, and (f) ABC.

Two testing environments are set to verify whether different algorithm can successfully process WF after training. The performance evaluation includes the defined fitness function in equation (31) (FIT), the total moving distance of the robot in whole WF (T_DIS), the time required to complete the whole WF (T_TIME), and the average distance between the robot and the wall in the whole WF (A_DIS). The defined fitness function in equation (31) is used to evaluate the WF performance of a mobile robot. If a learning algorithm is converged, the robot does not move suddenly near the wall or suddenly away from wall. That is, the total moving distance of the robot in whole WF will be reduced. If the time required to complete the whole WF is short, the moving speed of mobile robot is fast. Then, the proposed algorithms are compared with other algorithms in Tables 5–7 and Figures 17–19, and the result shows that the proposed algorithm is superior to others.

Table 5.

Performance analysis in the training environment.

Evaluation values	Methods
	Figure 17
	DGDE_M-1	DGDE_M-2	JADE 18	Rank-DE 19	PSO 10	ABC 13
FIT	0.922009	0.922401	0.921810	0.915885	0.919911	0.918024
T_DIS (m)	39.72	40.14	40.74	40.48	40.66	40.59
T_TIME (s)	141	109	115	176	158	182
A_DIS (m)	0.4051	0.4068	0.4041	0.3937	0.4018	0.3963

Table 6.

Performance analysis in testing environment I.

Evaluation values	Methods
	Figure 18
	DGDE_M-1	DGDE_M-2	JADE 18	Rank-DE 19	PSO 10	ABC 13
FIT	0.815827	0.825023	0.816456	0.814484	0.811793	0.813140
T_DIS (m)	46.32	46.52	47.22	47.21	47.61	48.68
T_TIME (s)	135	129	136	195	193	176
A_DIS (m)	0.4104	0.4047	0.4016	0.3936	0.4026	0.3837

Table 7.

Performance analysis in testing environment II.

Methods Evaluation values	Figure 19
	DGDE_M-1	DGDE_M-2	JADE 18	Rank-DE 19	PSO 10	ABC 13
FIT	0.872589	0.883197	0.860414	0.865081	0.859318	0.861352
T_DIS (m)	56.36	57.10	58.13	58.14	59.71	59.06
T_TIME (s)	161	157	179	168	318	216
A_DIS (m)	0.4131	0.4004	0.4011	0.3947	0.4309	0.3802

Figure 18.

Moving track in testing environment I applied by different algorithms: (a) DGDE_M-1, (b) DGDE_M-2, (c) JADE, (d) Rank-DE, (e) PSO, and (f) ABC.

Figure 19.

Moving track in testing environment II applied by different algorithms: (a) DGDE_M-1, (b) DGDE_M-2, (c) JADE, (d) Rank-DE, (e) PSO, and (f) ABC.

Navigation control

Recent research in mobile robot navigation has focused on known and unknown environments. Since limited information is available in an unknown environment, mobile robot is likely to collide and lead to the fail commission when compared to that in the known environment. Therefore, an effective navigation control method, mode manager, is used to switch the behavior mode according to the relation between the robot and environment. Two behavior modes, TG and WF, are utilized in the navigation control of mobile robot. When the robot closes to obstacles, the mode manager will switch to WF mode for avoiding collision. Otherwise mode manager switches to TG mode to assist robot approaching the goal as fast as possible. In this study, we do not consider to evaluate a fitness function for TG mode. The reason is TG mode performs the shortest distance between the mobile robot and the goal in an unknown environment. The detailed flowchart of navigation control is shown in Figure 20 and the relative pseudo code is also shown in Appendix 1.

Figure 20.

Flowchart of navigation control of mobile robot.

TG behavior

When a mobile robot is navigated in an unknown environment, the goal position information is available. The mobile robot moves toward the goal by adjusting the moving direction according to the relative position between the robot and the goal. There is an angle, ${\theta }_{\mathrm{TG}}$ , between moving direction and goal as shown in Figure 21. It is defined by

$${\theta }_{\mathrm{TG}}={\theta }_{\mathrm{Robot}}-{\theta }_{\mathrm{Goal}}$$

(32)

• ${\theta }_{\mathrm{Robot}}$ is the angle between mobile robot and x axis and ${\theta }_{\mathrm{Goal}}$ is the angle between goal and x axis.

Figure 21.

${\theta }_{\mathrm{Robot}}$ and ${\theta }_{\mathrm{Goal}}$ .

Mode manager

The mobile robot is divided into four areas, $R_1$ , $R_2$ , $R_3$ , and $R_4$ (Figure 22) to identify the moving direction. The mode manager identifies the goal position in advance, and it continuously identifies the goal direction located in one of the four areas $(R_i)$ . If the mode manager detects an obstacle, the mode manager switches to WF mode. Otherwise, the robot remains in TG mode and moves toward the goal. However, if the goal is located in the area $R_4$ , the robot switches to WF mode until the goal’s area is changed or the robot reaches the goal.

Figure 22.

Four sections of mobile robot.

The main purpose of TG mode is to make the robot approach the goal. The smallest distance between the robot and the goal is recorded during the navigation control. When the distance between the robot and the goal is longer than the recorded smallest distance during WF mode, the robot will fall into the dead cycle (see Figure 23). Therefore, a modified mode manager is proposed to escape the dead cycle in Figure 24. The mode manager counts the distance between the robot and the goal in each time step and records the smallest distance. If the current distance between the robot and the goal is smaller than the recorded smallest distance, the mode manger switches to TG mode. Therefore, when the mode manager is added into the navigation control, the mobile robot successfully escapes the dead cycle as shown in Figure 25.

Figure 23.

Dead cycle.

Figure 24.

Flowchart of mode manager.

Figure 25.

Escaping dead cycle of mobile robot.

Experimental results of navigation control

Public-approved navigation experimental environments are used to demonstrate the strength of the proposed navigation control method shown in Figure 26. Some other evolutionary algorithms are implemented to compare and analyze the navigation performance by two evaluation indexes. One evaluating index is the total moving distance of the robot in whole navigation. As the stability of the WF mode increases, the total moving distance decreases. The other evaluating index is the time required to complete the navigation or the moving speed in total. The analysis of the total moving distance and the time required to complete the navigation is listed in Table 8. In Table 8, the ABC ¹³ has a smaller total moving distance than our method, whereas the ABC ¹³ has a larger time required to complete the navigation than our method. The proposed method is superior in navigation control than other methods.

Figure 26.

Navigation control results of proposed method with mutation DGDE-1 (left figure) and DGDE-2 (right figure): (a) bug trap, (b) back_and_forth, B&F, (c) T, and (d) rooms_easy.

Table 8.

Performance analysis of different algorithms.

Algorithm	Environments
	Figure 26(a)		Figure 26(b)		Figure 26(c)		Figure 26(d)
	Distance (m)	Time (s)	Distance (m)	Time (s)	Distance (m)	Time (s)	Distance (m)	Time (s)
DGDE_M-1	35.75	107	57.66	171	56.0	159	174.6	497
DGDE_M-2	35.49	103	57.54	166	55.77	153	173.4	482
JADE 18	36.01	141	57.46	216	56.4	224	175.1	692
Rank-DE 19	35.66	147	57.41	240	55.97	242	174.0	748
PSO 10	35.76	140	57.56	224	56.06	217	174.8	674
ABC 13	35.58	145	56.91	231	55.84	236	174.9	743

Conclusion

This study proposes an efficient control method to implement navigation control of a mobile robot. According to the relationship between the robot and the environment, a novel mode manager is used to switch WF mode and TG mode. An interval type-2 neural fuzzy controller with the evolutionary algorithm is used in the mobile robot as an adaptive developing controller without designing rules by experts or collecting training dataset. The proposed DGDE uses the group concept, which substantially increases the searching ability and convergence speed of DE. The experimental results show that the proposed method of mobile robot is superior to others in WF mode and navigation control. The proposed model is successfully implemented as a navigation controller on Pioneer 3-DX mobile robot in an unknown environment.