Tracking control of mobile robot using ANFIS

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The field of trajectory-tracking control of mobile robot has been the focus of active ... International Conference on Mechatronics and Automation. August 7 - 10 ...
Proceedings of the 2011 IEEE International Conference on Mechatronics and Automation August 7 - 10, Beijing, China

Tracking Control of Mobile Robot Using ANFIS Masoud Imen

Mohammad Mansouri

Mehdy Aliyari Shoorehdeli

Department of Mechatronics Engineering, Science and Research Branch Islamic Azad university, Tehran, Iran [email protected]

Department of Control Engineering, Khajeh Nasirodin Toosi University of Technology, Tehran, Iran [email protected]

Department of Mechatronics Engineering, Khajeh Nasirodin Toosi University of Technology, Tehran, Iran [email protected]

Abstract- This study addresses a nonlinear trajectory tracking control problem for a kinematics Model of nonholonomic mobile robot with considering next 2 time path curvature. The tracking control of mobile robot using two cascade controllers is presented. The first fuzzy controller produces a variable which shows curvature of the path and is considered as one of the inputs of the second fuzzy controller. Adaptive Neuro Fuzzy Inference System (ANFIS) is applied as second stage controller for the solution of the path tracking problem of mobile robots. The inputs value to fuzzy logic layer are VC, C, dR & d the robot current linear velocity, trajectory curvature, distance from the robot actual position to the next desired position, and difference between the angles of the d and the robot actual heading, respectively. A gradient descent learning algorithm is used to adjust the parameters. That present controller is compared with previous work to confirm its effectiveness. Keyword- Fuzzy Controller, ANFIS, Cascade Tracking Control, Look-ahead Curvature.

Controller,

I. INTRODUCTION The field of trajectory-tracking control of mobile robot has been the focus of active research in the past decades , for both theoretic research and practical applications. During recent years, the interest in mobile robots has grown significantly because of the Mobile robots are increasingly used in industry, in service robotics, for domestic needs (vacuum cleaners, lawn mowers, pets), in difficult to access or dangerous areas (space, army, nuclear-waste cleaning, mining, forestry, agriculture) and also for entertainment (robotic wars, robot soccer), etc [1]. Various approaches and strategies have been proposed for these challenges of mobile robots with nonholonomic constraints. According to [2], if a system has constraint equations that involve velocities, accelerations, or derivatives of system coordinates, the constraint equations are said to be nonholonomic, or kinematic, and the mechanical system is said to be a nonholonomic system. An extensive review of nonholonomic control problems can be found in [3]. Two main approaches to controlling mobile robots are posture stabilization and trajectory tracking. The aim of trajectory tracking is to controlling robots to follow a given time varying trajectory (reference trajectory). It is a fundamental motion control problem and has been intensively investigated in the robotic community. To solve these problem, many researchers investigate various tracking control designs [4,5,8]. [9] used a Lyapunov function to design a local asymptotic tracking controller. Global tracking was explored by dynamic feedback linearization techniques in [10], backstepping techniques in [11,12] and sliding mode

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techniques in [13], Furthermore, adaptive control [14], fuzzy control [15], and neural network control [16 ], etc. Tracking control design for wheeled and for tracked mobile robots can be respectively found in [6]. In [17], the application of MPC (Model-based predictive control) schemes is used to control a WMR in the problem of trajectory tracking. Among available options, the neural networks have great potential for its self-learning ability. The cerebellar model articulation controller (CMAC) as a kind of neural network is developed by [18]. In [19] a nonlinear model of the WMR is used for trajectory tracking. The problem is solved considering unknown obstacles in the configuration space. A neural network helps to solve the optimization problem. Also, in [33] the path following problem is solved by using a neural network to predict the future behaviour of a car-like WMR. Furthermore, most reported designs rely on intelligent control approaches such as Fuzzy Logic Control [20] and Neural Networks [21]. Intelligent control techniques, based on neural networks and fuzzy logic, have also been developed for path tracking control of mobile robots [22]. Even though these intelligent control strategies have shown their effectiveness, especially for nonlinear systems, they have certain drawbacks due to their own characteristics. While conventional neural networks have good ability for self-learning, they also have some limitations such as slow convergence, the difficulty in reaching the global minima in the parameter space, and sometimes even instability as well. In the case of fuzzy logic, it is a human-imitating logic, but lacks the ability for self-learning and self-tuning. Therefore, in the research area of intelligent control, fuzzy neural networks (FNNs) are devised to overcome these limitations and to combine the advantages of both neural networks and fuzzy logic [23]. This provides a strong motivation for using FNNs in the modeling and control of nonlinear systems. In this paper, the tracking control of mobile robot using two cascade controllers is presented. The first fuzzy controller produces a variable which shows curvature of the path and is considered as one of the inputs of the second fuzzy controller. Adaptive Neuro Fuzzy Inference System (ANFIS) is applied as second stage controller for the solution of the path tracking problem of mobile robots. A gradient descent learning algorithm is used to adjust the parameters. That presented ANFIS controller is compared with fuzzy controller in previous work to confirm its effectiveness. This paper is organized as follows: The trigonometric neural networks are described in section II. Section III explains

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modelling of the flexible joint manipulator. The proposed fuzzy logic controller is describing in section IV. Section V illustrates the implementation results. Finally, the conclusion is presented in section VI. II. ADAPTIVE NEURO FUZZY INFERENCE SYSTEM A. ANFIS Stracture In this section, type III ANFIS topology and the learning method used for this neuro- fuzzy network are presented. Both neural network and fuzzy logic [24] are model-free estimators and share the common ability to deal with the uncertainties and noise. Both of them encode the information in parallel and distribute architectures in a numerical framework. Hence, it is possible to convert fuzzy logic architecture to a neural network and vice versa. This makes it possible to combine the advantages of neural network and fuzzy logic. A network obtained in this way could use excellent training algorithms that neural networks have at their disposal, to obtain the parameters that would not have been possible in fuzzy logic architecture. Moreover, the network obtained in this way would not remain a black box, because this network would have fuzzy logic capabilities to interpret in terms of linguistic variables [25]. The ANFIS combines the two approaches, neural networks and fuzzy systems. If both these two intelligent approaches are combined, good reasoning will be achieved in quality and quantity. In other words, both fuzzy reasoning and network calculation will be available simultaneously. The ANFIS is composed of two parts. The first part is the antecedent and the second part is the conclusion, which are connected to each other by rules in network form. The ANFIS structure shown in Fig. 1 is a five layer network. It can be described as a multi-layered neural network as shown in Fig. 1. The first layer executes a fuzzification process, the second layer executes the fuzzy AND of the antecedent part of the fuzzy rules, the third layer normalizes the MFs, the fourth layer executes the conclusion part of the fuzzy rules, and the last layer computes the output of the fuzzy system by summing up the outputs of layer four. The feed forward equations of the ANFIS with two inputs and two labels for each input shown in Fig. 1 are as follows: wi = μ Ai ( x ) × μ Bi ( x ) , i = 1,2 (1)

wi =

wi w1 + w 2

, i = 1,2

f1 = p1x + q1y + r1 ½ w1f1 + w2 f2 =w1f1 + w2 f2 ¾Ÿ f = f2 = p2x + q2 y + r2 ¿ w1+w2

ANFIS has high ability of approximation that will depend on the resolution of the input space partitioning, which is determined by the number of MFs in the antecedent part for each input. Usually, the MFs are used as bell-shaped with maximum grade equal to 1 and minimum grade equal to zero such as:

μ Ai ( x) =

1 x − ci 1+ ai

(4)

2 bi

­° x − c i μ Ai ( x) = exp®− ai °¯

2 bi

½° ¾ °¿

(5)

where {ai,bi,ci} are the parameters of MFs which are affected in shape of MFs. B. Learning algorithms Subsequent to the development of ANFIS approach, a number of methods have been proposed for learning rules and for obtaining an optimal set of rules. For example, [26] have proposed to merge Min–Max and ANFIS model to obtain neuro-fuzzy network and determine optimal set of fuzzy rules. [27] have presented application of Levenberg-Marquardt method, which is essentially a nonlinear least-squares technique, for learning the ANFIS network structure. In another paper [28], has presented a scheme for input selection and [25] have used Kohonen’s map for training. In this paper Gradient Decent (GD) is used to updating all parameters Of ANFIS structure.

(2)

III. KINEMATICS MODELING The kinematics model derived from the popular differentially steered wheeled mobile robots, that is assumed that the vehicle moves without slipping on a plane, i.e., there is a pure rolling contact between the wheels and the ground. The two coordinates are exist for derivation model: Cartesian or polar coordinates. Hence, because of popularity of Cartesian coordinates, the modelling done in Cartesian coordinates (Fig.2). The kinematics model (or equation of motion) of the WMR is then given by [29].

­ x = v cos θ ° ® y = v sin θ Ÿ X = f ( X , u ) ° ¯θ = ω

(3)

To model complex nonlinear systems, the ANFIS carries out input space partitioning that splits the input space into many local regions from which simple local models (linear functions or even adjustable coefficients) are employed. The ANFIS uses fuzzy MFs for splitting each input dimension; the input space is covered by MFs with overlap; that is several local regions can be activated simultaneously by a single input. As simple local models are adopted in the ANFIS model, the

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Fig. 1 The equivalent structure of ANFIS ( type III ANFIS )

(6)

Where x , y represents the position of the mobile robot center, with respect to a global inertial coordinate system {O,X,Y}. Where the angle –  its orientation relatively to the x-axis. The V &  are linear and angular velocities respectively, that are considered as the inputs to the mobile robot. The discrete time representation of this model will be done by considering a sampling period dt, a sampling instant i and applying the Euler’s approximation to (6). Also, from “(6),” the mobile robot behavior is subject to an additional nonholonomic constraint: x (i )× sin θ (i ) − y (i ) × cosθ (i ) = 0 (7) This constraint means that the robot can not move in the direction of the wheel axis (i.e. y). A. Trajectory tracking As shown in Fig. 2, The reference trajectories can be described by a reference state vector Xr=[xr,yr,r]T and a reference control signal vector ur=[vr,r]T and have the same constraints ( kinematics model ) as “(6),”:

­ xr = vr cosθ r ªcosθ r ° «  ® y r = vr sin θ r Ÿ X r = « sin θ r ° «¬ 0 ¯θ r = ωr

0º 0»» ur 1 »¼

(8)

Where Xr is a known, pre-specified reference trajectory. It is usual in this case to associate to this reference trajectory a virtual reference robot, which has the same model than the robot to be controlled. Thus, in discrete-time we have: X r (i + 1) = f d ( X r (i ), ur (i )) (9) The problem of trajectory tracking can be stated as to find control law such that: X (i ) − X r (i ) = 0 (10) Thus, an error state Xe can be defined as follows:

ª cosθ X e = ««− sin θ «¬ 0

sin θ cosθ 0

0º 0»» 1»¼

ª xr − x º « y − y» « r » «¬θ r − θ »¼

(11)

B. Path tracker parameters The path tracking controller that was implemented here is based on the controller in [29, 30], but the inputs and outputs of the fuzzy logic controller (FLC), the rules and the path representation is major changed to [30], also the controller approach is changed to [29]. The form of the control law equation is as follows: ­V = f1 (Vc , C , dR , dθ ) (12) ® ¯ω = f 2 (Vc , C , dR , dθ ) Where V &  are the translational and rotational velocities of the robot, C the look ahead curvature (LAC) and it is a feed forward input, dR the distance from the actual position of the robot to the next desired position, d the difference between the angles of the line joining the current position to the next desired position and the actual heading of the robot, VC the current linear velocity as Fig. 3 shows.

Fig. 2 Coordinate system of the WMR

Fig. 3 ANFIS controller parameters

The functions f1 and f2 are the control laws of a Sugeno type fuzzy controller. Sugeno controllers take in fuzzy inputs and discrete outputs. The parameters used in the controller as follows: C is obtained using an other fuzzy logic module, that which inputs are 1 & 2 in Fig. 3. The desired trajectory of Mobil robot (WMR) is described by a set of waypoints, that linked each other from N1 to Ni. Whatever, the distance of waypoint become bigger, the precision is decrease and speed of robot will be increase. At any time, the nearest node to the robot’s position is defined as N1 and the next one in the list of nodes on the trajectory is named N2 to Ni respectively. On the other hand, robot commonly must be moved to goal position. thus, the robot’s head will be always in the direction of the node N3. The angles between the lines N1N2 and N2N3, and between the lines N2N3 and N3N4 are Named angles 1 & 2, respectively. The path curvature is defined by the parameter C, that is function of angles 1 and 2. If the current velocity of robot, VC, is high and the C is increased or due to big error, robot needs to make a sharp turn d, then it must first slow down while turning smoothly. At any time, by knowing the current node, the robot’s current position, heading and velocity, All the parameters to the input of the controller can be calculated . The general structure of the control loop block diagram is shown in Fig. 4. The calculation module is calculated the controller parameters and determined the current position of the robot with respect to the trajectory, by takes in the position, heading, and current velocity of the robot. The fuzzy logic controller then determines V and  such that the robot follows the trajectory in a smooth and efficient manner. IV. NEURO FYZZY PATH TRACKING CONTROLLER The task of the path tracking “Adaptive Neuro Fuzzy Inference System” (ANFIS) controller is to direct the robot to follow the trajectory in a smooth and continuous manner at the best possible precision. It’s not mater, If the robot didn’t pass exactly through the waypoints on the trajectory, but at least must passes in their proximity and arrives to the final destination. For increase the precision of the robot path tracking can be make closer the waypoints are to each other, but at lower speed. The ANFIS PTCU composed of two part. The 1st part is a Fuzzy based module, that is used to determines the numerical value of the look-ahead curvature (LAC) [29] and the 2nd part

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Fig. 4 Control loop block diagram

is a path tracker module based on ANFIS, that determines the linear and angular velocity, that are controller outputs. Both modules are Sugeno type fuzzy inference systems of order zero. The document [7,31] contains a brief and practical introduction to ANFIS control. For a more detailed analysis on fuzzy control [32] provides a more in-depth theoretical study. The LAC to determine the value of C uses the angles 1 and 2. Fig. 5 shown The membership functions of this parameters [29]. The membership functions of 1 are Straight1 , High1 and VeryHigh1 and those of 2 are Straight2, High2 and VeryHigh2. The membership functions of C are singletons that take values between 0 and 5. If 1 and 2 are small, the value of C is indicated 0, that meaning is there is no path curvature and at the current state the robot will follow a straight line. The example rule: IF 1 is Straight1 AND 2 is High2 THEN C is cModerate The truth value for the rule is obtained by using the product operator: Ai = μ (straight1(α1 )) × μ (High2(α 2 )) (13) Where  is a value between 0 and 1, that indicates the truth value that an input value belongs to some membership function. The 2nd module of ANFIS PTCU is takes the value VC, C, dR, d and determines the V and  in the output. Fig. 6 shown the membership functions of each of the input and output

parameters. The inputs ranges are: 0dR7 m, -180d180, 0VC1 m/s, and the outputs ranges are: 0V800 mm/s, _3535  sec. The inference rules maps the input membership functions to the output membership functions. The behaviour of the controller is such that it changes linear velocity and angular speed in a smooth and almost continuous manner. When the curvature is sharp, the controller decreases the speed and produces the needed rotational speed in the right direction to make the turn smoothly. The weighted average method is used to defuzzi cation and calculate the outputs. V. NUMERICAL RESULTS This section presents the numerical results using the proposed ANFIS tracking control methodology. it was done under MATLAB 7.6 (R2008a). To demonstrate the effectiveness of the proposed Adaptive Neuro Fuzzy Inference System (ANFIS) controller, that introduced by myself, the kinematic Model of a WMR (differentially steered wheeled mobile robots) is considered. This controller based on ANFIS is introduced by aim to controlled the Path tracking problem. To investigate the performance and demonstrate the flexibility of designed path tracking controller, the act of tracking has been performed in two typical paths, first simple trajectory with the least complexity and then trajectory with

Fig. 6 Membership Functions of path tracker parameter. From top to down are VC, C, dR & d respectively.

Fig. 5 LAC parameters and theircorresponding membership Function.

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great complexities like slopes and turns. The complex trajectory is a linear trajectory, that generated with composed of some various linear functions, and other ones are pendulous trajectories, that generated with Sinc Function. The initial conditions are taken as Pr (1) =[0 0 0]T, and the robot initial conditions for the linear and pendulous trajectories are given by P(1)=[0 5 30]T & P(1)=[0 1 0]T, respectively. Also the initial linear velocity is VC =0.1 m/s, and The learning rate ( ) of the adaptive rule is chosen as 0.009 & 0.03 for 1st and 2nd path respectively. The result of tracking act in the various trajectory and its error are shown in continue. The dark blue solid line stands for the reference trajectory and also the red line stands for robot Path tracking efforts. As it shows in Fig. 7 the tracking controller can follow the both complex and simple paths with great precision. According to the tracking errors graphs, the root mean square error of paths are 0.3930 and 0.1225 for 1st and 2nd path respectively. Therefore, the tracking controller shows an appropriate precision for both plain and complicated paths which demonstrates the acceptable performance of ANFIS base controller. In order to compare the ANFIS base controllers with fuzzy tracking ones, the experiment was redone with a fuzzy tracking

controller [7] on the same paths. The results are shown in Fig. 9. According to mentioned figures, unlike the ANFIS base controller fuzzy tracking one has some loss precision with simple paths. These kinds of controllers are not able to follow the extremes and semicircular curves along the simple path. The precision of following path is not good in 2nd path based on Sinc function, and the controller can tracking the path fairly acceptable after passing half of cycle of the reference trajectory, which shows some inability to pass the curves. On the other hand, by survey the performance of path tracking controller in complex trajectories, it is perceive, while the tracking precision is relatively good in the linear trajectory, in some parts of path, such as entrance to curves or suddenly veer, the tracking act is pendulous. In conclusion, having talked about different trajectories tracking, both ANFIS and FUZZY base, and compared RMS error of each ones (Table 1), ANFIS controllers show a better performance in following any kinds of path by any degrees of complexity and difficulty, also have flexibility to different conditions like sudden turns or change in directions. Figs 9 and 10 are shows the robot path tracking act in Fuzzy Method, and the robot linear and angular velocities, respectively.

Fig. 7 ANFIS app. : robot tracked path and references trajectories. In top is the linear trajectory and Sinc Functions trajectory is in down.

Fig. 9 Fuzzy app. : robot tracked path and references trajectories. The linear and Sinc Functions trajectories are in top and down respectively.

Fig. 8 ANFIS app. : Robot linear and Angular velocities. The linear trajectory is in top and in down is Sinc Functions trajectory. All figures in left hand are linear velocities and right hand are Angular velocities.

Fig 10 Fuzzy app. : Robot linear and Angular velocities. The linear trajectory is in top and in down is Sinc Functions trajectory. All figures in left hand are linear velocities and right hand are Angular velocities.

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[10]

Table 1 RMS Error RMS Error Linear Trajectory Sinc Fun. Trajectory

Fuzzy Tracking Error 0.7403 07748

ANFIS Tracking Error 0.3930 0.1225

[11]

[12]

VI. CONCLUSIONS Due to variety and diversity of Mobil Trajectory in real environments, the problem of the mobile robot tracking control is difficult. In this paper, two type of path tracking controllers has been introduced and discussed by intelligent approaches. Proposed ANFIS controller is mainly focused on future Path curvature to produce linear and angular velocities signal for smooth motion of robot. discovers that the designed ANFIS base controller has more abilities and better performance in tacking paths with any kinds of complexities, such as sudden turns, successive curves, and slopes in compares with the FUZZY one. According to the results of tracking on several trajectories with different difficulties and Complexities, that is shown in figures this paper investigates and compares the designed controller basted on ANFIS network and FUZZY logic and Of course that is required to explain, however the performance of FUZZY based controller was acceptable, the usage of learning ability of neural networks in ANFIS algorithm can improve the efficiency and performance of tracking the difficult trajectories. In conclude that designed ANFIS base control system shows great precision and tracking ability. In addition to online network, it has noticeable resistance to noise because of its intelligent algorithm.

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[18]

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