(Microsoft Word - \307\315\343\317 \325\310\307\315 \346\344\346\32164- 73) Al-Khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal,Vol. 13, No. 3, P.P. 64- 73 (2017) A Cognitive Hybrid Tuning Control Algorithm Design for Nonlinear Path-Tracking Controller for Wheeled Mobile Robot Ahmed S. Al-Araji* Noor Q. Yousif** *,**Department of Control and Systems Engineering/ University of Technology *Email:ahmedalaraji76@gmail.com (Received 13 December 2016; accepted 7 March 2017) https://doi.org/10.22153/kej.2017.03.004 Abstract This research presents a on-line cognitive tuning control algorithm for the nonlinear controller of path-tracking for dynamic wheeled mobile robot to stabilize and follow a continuous reference path with minimum tracking pose error. The goal of the proposed structure of a hybrid (Bees-PSO) algorithm is to find and tune the values of the control gains of the nonlinear (neural and back-stepping method) controllers as a simple on-line with fast tuning techniques in order to obtain the best torques actions of the wheels for the cart mobile robot from the proposed two controllers. Simulation results (Matlab Package 2012a) show that the nonlinear neural controller with hybrid Bees-PSO cognitive algorithm is more accurate in terms of fast on-line finding and tuning parameters of the controller lead to obtaining smoothness with small spikes control action as well as minimizing tracking pose error of the wheeled mobile robot than the performance of nonlinear back-stepping technique. Keywords: Bees Algorithm, Nonlinear Controller, Matlab Package, Particle Swarm Optimization, Wheeled Mobile Robots. 1. Introduction In general, a wheeled mobile robot system considers a multi-input multi-output nonlinear dynamic and time variant system and the one of fundamental problems in the control engineering is the motion control design to track the desired path [1] because the mobile robot has many applications in various directions life such as: science; education; industry; entertainment; security and military therefore the mobile robot is still active region of research [2]. In the recent years, different types of control algorithms and controllers that they are based on mathematical model of the wheeled mobile robot and they are proposed to solve the motion control of the robot’s wheels in order to follow the desired trajectory with high performance of the controllers in terms of generating optimal action that lead to minimizing tracking pose error during tracking reference path, such as nonlinear neural PID controllers [3 and 4], fuzzy logic and PID controllers [5 and 6], neural networks controllers [7 and 8], back-stepping controllers [9 and 10], adaptive sliding mode controllers [11 and 12] and neural predictive controllers [13 and 14]. The motivation for this research is taken from [2, 3, 4 and 10] which are focusing on the problems of the mobile robot in terms of tracking and stabilizing the mobile robot on the desired path as well as how can generate best torque control action without saturation state and no spike action? The main core of the contribution of this research is described as follows: • Using a different types of controllers with high computational accuracy that have been derived of the control laws in order to generate best torque action and lead to minimizing tracking pose error of the wheeled mobile robot. • Cognitive hybrid Bees-PSO optimization algorithm proposed to show the ability in the fast search in local (PSO) and global (Bees) Ahmed S. Al-Araji Al-Khwarizmi Engineering Journal, Vol. 13, No. 3, P.P. 64- 73 (2017) 65 regions in order to find and tune on-line the best parameters of the two types of controllers. • Adding a dynamic disturbance to investigate the robustness performance of the proposed controllers. • Changing the initial pose state to verify the adaptation performance of the proposed controllers. • Tracking a variable radius continuous trajectory to validate the capability of the proposed controllers. The organization of this paper can be described as follows: in Section two, is a description of the dynamic wheeled mobile robot model. Section three is deriving the different types of the proposed nonlinear controllers. In section four, the proposed of a cognitive hybrid Bees-PSO optimization algorithms is explained. Section five is presented the performance of the proposed controllers through simulation results. In section five, the conclusions are drawn. 2. Model of a Dynamic Wheeled Mobile Robot Fig.1 shows the schematic diagram of the cart wheeled mobile robot that it consists of a two DC motors which is driving the two wheels with one an omni-directional castor wheel that will stabilize the platform of the mobile robot [12 and 13]. The motion and orientation of the mobile robot depends on a two independent actuators (DC motors) for left and right wheels. r is the radius of the same of the two wheels and the distance between these wheels is L and c is the center of gravity of the mobile robot. Fig. 1. Mobile robot Platform model. In general, the global coordinate frame is defined as [ ]YXO ,, while the pose vector of the mobile robot in the surface is defined as: T yxq ),,( θ= … (1) The position coordinates (x,y) are at point cwhile θ is orientation angle that is measured with respect to global frame in the X-axis therefore the configuration of mobile robot can be described by these three generalized coordinates. To investigate the motion and orientation of the wheeled mobile robot, two conditions should be achieved; the first is pure-rolling and the second is without-slipping in order to make the mobile robot’s lateral velocity is equal to zero as equation (2) [3 and 4]. 0)(cos)()(sin)( =+− •• ttyttx θθ …(2) Then, the equations of the kinematics wheeled mobile robot in the world frame can be represented as follows [7 and 10]: )(cos) 2 )()(( )( t twltwrr tx θ + =& …(3) )(sin 2 ))()(( )( t twltwrr ty θ − =& …(4) L twltwrr t ))()(( )( − =θ& …(5) where wr(t) and wl(t) are the right and left angular velocities respectively. Based on Euler Lagrange formulation [9 and13], the dynamic model of the mobile robot can be described as follows: λθ θ τ τ θθ θθ τ θ − + − =+ •• •• •• 0 cos sin 22 sinsin coscos 1 00 00 00 R L LLr dy x I M M ..(6) where L τ is the torque of the left wheel. R τ is the torque of the right wheel. M is the mobile robot’s mass. I is the mobile robot’s inertia. λ is the constraint forces. d τ is bounded dynamic disturbances. The pose equations of the mobile robot in the simulation can be described as follows: ���� = ����� � ��� � …(7) � ��� = �.���������� ���� � …(8) ��������� = ���� � …(9) ��� �� ��� = ����� � …(10) ������� = ��������� × ∆! …(11) � �� ��� = ��� �� ��� × ∆! …(12) "#��� = �.�$%&�'(��� &�()����* � …(13) Ahmed S. Al-Araji Al-Khwarizmi Engineering Journal, Vol. 13, No. 3, P.P. 64- 73 (2017) 66 "+��� = �.�$%&�'(����&�()����* � …(14) )1()(cos)]()([5.0)( −+∆+= kxtkkwlkwrrkx θ ...(15) )1()(sin)]()([5.0)( −+∆+= kytkkwlkwrrky θ ...(16) )1()]()([)( −+∆−= ktkwlkwr L r k θθ …(17) where � is total linear force; � is total angular torque; ������ is linear acceleration; ��� �� is angular acceleration; ���� is linear velocity; � �� is angular velocity; "+ is angular velocity of the left wheel; "# is angular velocity of the right wheel; )(),(),( kkykx θ are the pose of the mobile robot at each step kth of the movement and t∆ is the sampling time. 3. Nonlinear Controller Methodology In general, a feedback control signal based on different types of the nonlinear controllers are very important in the structure of the proposed controller in terms of stabilizing and minimizing the tracking pose error of the wheeled mobile robot during the mobile robot’s pose is drifted from the reference path. Block diagram of the proposed structure of the nonlinear trajectory tracking controllers for wheeled mobile robot can be shown in Fig. 2. The first structure of the nonlinear controller equation is neural network that it performed as a nonlinear PID controller [3 and 4]. The nonlinear neural control structure has strong of adaptation performance, high dynamic characteristic and good robustness performance because the proposed structure is built on a traditional PID controller and employed the theory of the neural network technique. The structure of the proposed nonlinear neural controller can be shown in Fig. 3. The proposed control law of the nonlinear neural controller for the right and left torques as follows: yxRR oBoBkBk 12111 )1()( ++−= ττ (18) θττ oBoBkBk yLL 141315 )1()( ++−= …(19) The outputs yx oo , and θo of the neural networks. Sigmoid function is used as nonlinear relationship as equation (20) [3 and 4]: 1 1 2 − + = − γγ nete o …(20) Where .,, θγ yx= γnet is calculated from these equations: )]1()([)]([)( 32 −++= kekeBkeBknet xxxx )]1()([ 4 −−+ kekeB xx …(21) )]1()([)]([)( 65 −++= kekeBkeBknet yyyy )]1()([ 7 −−+ kekeB yy …(22) )]1()([)]([)( 98 −++= kekeBkeBknet θθθθ )]1()([ 10 −−+ kekeB θθ …(23) where )(keγ is the input error signal. The control parameters B1, B2, … and B15 are on-line updated by using cognitive hybrid Bees- PSO optimization algorithm. Fig. 2. The proposed structure of nonlinear trajectory tracking controllers for mobile robot. Nonlinear Controller Equations Cognitive Hybrid Bees- PSO Algorithm )2( )1( )( − − ke ke ke γ γ γ )( )( k k L R τ τ + - Dynamic Disturbances [ ])1()1( −− kk LR ττ ∆ )1( )1( )1( + + + k ky kx θ Rotation Matrix )1( )1( )1( + + + k ky kx r r r θ Reference Velocities Eqs. Parameters Ahmed S. Al-Araji Al-Khwarizmi Engineering Journal, Vol. 13, No. 3, P.P. 64- 73 (2017) 67 Fig. 3. The structure of nonlinear neural controller [3 and 4]. The second structure of the nonlinear controller equation is back-stepping technique based on Lyapounov method [10]. The nonlinear back-steeping controller method structure can be shown in Fig. 4. Based on back-stepping technique in [9 and 10], the nonlinear angular velocity control law can be described by Eq. (24) and applying the proposed hybrid Bees-PSO tuning algorithm for finding and tuning the best value of the parameters (B1, B2 and B3) then solving equations (7 to 14) to find the right and left torques control action for wheeled mobile robot. ++−+ ++++ = r evBevBw L eBev r evBevBw L eBev w w ryrrxr ryrrxr L R ))sin( 2 cos( ))sin( 2 cos( 321 321 θθ θθ (24) 4. Cognitive Hybrid Bees-PSO Algorithm The purpose of the on-line cognitive hybrid Bees-PSO algorithm is to find and tune the optimal gains control for the proposed controller to generate the best and smoothness torque signal that will lead to minimizing the tracking pose error of the wheeled mobile robot during dynamic disturbance has been added. In general, Bees algorithm mimics the food foraging behavior of swarms of honey bees and this algorithm carries out by using local (neighborhood) search that has two types of Bees (Selected and Recruit Bees) and global (random) search that has also two types of Bess (Scout and Fittest Bees) [15 and 16] while the particle swarm optimization (PSO) is considered one of the algorithms types that has a capability to search for the optimal solution by simulating the movement and flocking of birds. In PSO algorithm, the particles that have a position and velocity are a population of individual in order to move around the search space to find best or optimal solution through evaluation each particle by using a fitness function [3 and 4]. 4.1. Hybrid Bee-PSO (HBPSO) Algorithm In this work, the proposed on-line optimization algorithm is a hybrid tuning control algorithm which consists of Bees algorithm [15 and 16] and PSO algorithm [3 and 4]. The main advantages for this proposed optimization algorithm are to solve the problem of local search of the Bees algorithm using PSO algorithm which has the high ability for local search through generating population (Recruit Bees) as particles and this will speed up the process of optimization while the problem of the global search in the PSO algorithm is solved by using Bees algorithm which has the capability of combining the (Fittest Bees) and a new population of (Scout Bees) generating in the global search. The proposed on-line HBPSO algorithm is applied as a powerful optimization algorithm to H )(ke x + )(kRτ )1( −k R τ Ox + + +1 +1 + +1 )1( −ke x B1 H )(ke y Oy + + +1 +1 +1 -1 + +1 )1( −ke y H )(keθ + )(kLτ )1( −k L τ O + + -1 +1 +1 +1 + +1 -1 )1( −ke θ -1 +1 B15 B11 B12 B13 B14 B2 B3 B4 B5 B6 B7 B8 B9 B10 Fig. 4. The nonlinear back stepping controller structure [10]. ),,,,,( 3,2,1 BLrwvqfv rrecc = θe ey ex [ ] 321 BBB [ ]rr wv L R w w t velocity ∂ ∂ [ ]Lr ang lin Acc Acc ( )IM L R τ τ Ahmed S. Al-Araji Al-Khwarizmi Engineering Journal, Vol. 13, No. 3, P.P. 64- 73 (2017) 68 find and tune the optimal stable parameters of the nonlinear neural controller in order to enhance the performance characteristics of the system by reducing the processing time as well as improve the response accuracy through minimizing the tracking error for mobile robot. The proposed HBPSO algorithm’s steps can be described as follows: 1. Scout Bees (n) are generated in the global (random) search as the initial population with randomly fifteen values of the controller’s parameters in the nonlinear neural controller while only three control parameters in the nonlinear back-stepping controller. 2. Calculated the fitness of the (n) Scout Bees by using the fitness equation (25) [3] based cost function. -.!/011 = 2 3 4567 �8�97�5� ...(25) Where: : > 0 to avoid division by zero. The proposed cost function is a mean square error as equation (26). =>? = 2 @ ∑ ��B�CD − B� %+�G�CD − G� %�H�CD − H� %@ �I2 + ��+./�CD − �+./� % + �J�CD − J� %� …(26) K: is the number of iteration. 3. In the local search, chosen the number of particles depends on the number of the highest fitness for the Scout Bees (population) as (m) Selected Bees. 4. The size of (patch size) of the local search is determined by applying the proposed equations (27). ∆L̅ = 0.025 × L̅5PQ × #R/STU�0,1� …(27) L̅�CX = L̅5PQ + ∆L̅ …(28) Where L̅ : is the vector of the fifteen parameters of the proposed controller. 5. Particles are generated by using equations (28) to search and find the best controller’s parameters. 6. Evaluated the proposed cost function of each particle using the mean square error equation (26). 7. Set to pbest for each particle in the current searching point. When the search is finding the best value of pbest then sets the pbest to gbest and store the number of the particle with the best value. 8. The pbest value is replaced by the current value if the value is better than the current pbest of the particle and if the best value of pbest is better than the current gbest, gbest is replaced by the best value and the particle number with the best value is stored. 9. Each particle is update by using Eqs. (29 and 30). )()( )( )( 22 )( )( 11 )()1( k m k k m k m k m k m BgbestrcBpbestrcBB −+−+Ω∆=∆ + … (29) )1()()1( ++ ∆+= k m k m k m BBB …(30) where popm ,.....3,2,1= ; )(k mB is the particle’s weight m at kth iteration; Ω : is the inertia weight factor; c1 and c2 are the positive values ; r1 and r2 are random numbers between 0 and 1; m pbest is best previous weight of m th ;Particle and gbest is best particle among all the particle in the population. 10. Return to step six if the current iteration number did not reach to the predefined maximum iteration number otherwise pick out the highest fitness for the particles as Fittest Bees (m). 11. Return to global search by assign the (n-m) remaining Bees to random search and generating a new population of Scout Bees. In general, the steps of the on-line cognitive hybrid Bees-PSO optimization algorithm for finding and tuning control parameters are repeated at 0.1 second (sampling time) for each k th sample based on Shannon theorem. 5. Simulation Results MATLAB package used to verify the proposed nonlinear controllers of the trajectory tracking for the dynamic model of the wheeled mobile robot. The Eddie mobile robot platform specifications are picked from [17]: M=12kg is the mobile robot’s mass; I=1.536kg.m 2 is the mobile robot’s inertia; r=0.075m is the radius of wheel and L=0.39m is the distance between wheels. The MATLAB simulation is carried out on- line cognitive tuning control algorithm with proposed controllers as shown in Figure (2) to track a reference pose with continuous variable radius path and 0.1 sec is sampling time. In this paper, two types of the nonlinear controllers are used (neural network and back-stepping technique) with hybrid Bees-PSO tuning control parameters to show which of them is batter in term of generating optimal and smoothness torque control action and minimizing the tracking error with minimum number of iteration. The parameters of the optimization hybrid Bees-PSO algorithm that will define as follows: The Scout Bees (n) is equal to 10 at the global search; the Selected Bees is equal to 5; the particle is equal to 40 in the local search; the Fittest Bees Ahmed S. Al-Araji Al-Khwarizmi Engineering Journal, Vol. 13, No. 3, P.P. 64- 73 (2017) 69 is equal to 5; the iteration number (N) in the global search is equal to 3and the iteration number (P) in the local search is equal to 3. Case Study The reference pose trajectory for wheeled mobile robot can be described in below equations (31, 32 and 33): ) 20 2 sin()( t tx r π = …(31) ) 40 2 sin()( t ty r π = …(32) ) )())(())(( )( (tan2)( 22 1 txtytx ty t rrr r r ∆+∆−∆ ∆ = −θ … (33) The mobile robot has initial pose as ]4/,0,1.0[)0( π−=q . After applying the structure of the cognitive hybrid Bees-PSO tuning control algorithm with two different types of the nonlinear controllers and adding small values of the dynamic disturbances as the term that taken from [14] [ ]Tttd )2sin(01.0)2sin(01.0=τ to show the controller ability of robustness through the on- line adaptation the parameters of these controllers. The trajectory tracking for mobile robot model can be shown in Fig. 5 which it is clearly, the excellent tracking performance depends on MSE of the mobile robot pose when was applying the nonlinear neural controller than nonlinear back- stepping controller because the cognitive hybrid Bees-PSO tuning control algorithm was found and tuned fifteen parameters of the nonlinear neural controller that covered the nonlinear operation regions while in the nonlinear back-stepping controller has three parameters only and could not cover these regions especially when adding a bounded dynamic disturbances. This proposed control algorithm has a capability of perfect search in two search space (local and global) with minimum number of iteration in both search space. Fig. 5. The desired and actual trajectory for mobile robot. In Fig. 6 demonstrates the orientation tracking performance of the mobile robot with two types of controllers. Fig. 6. The desired and actual orientation for mobile robot. In this on-line tuning control algorithm, the proposed Mean Square Error (MSE) clearly improved the performance of these controllers by showing the pose error convergence for the mobile robot motion at 200 steps, as shown in Figs. 7 and 8. -1.2 -1 -0.5 0 0.5 1 1.2 -1.2 -0.5 0 0.5 1.2 X-axis (meter) Y -a x is ( m e te r ) Actual Path based Back Stepping Actual Path based Neural Desired Path 20 40 60 80 100 120 140 160 180 200 -4 -3 -2 -1 0 1 2 3 4 Samples (Sampling Time 0.1 Sec) O r ie n ta ti o n ( r a d ) Orientation based Back -Stepping Orientation based Neural Desried Orientation Ahmed S. Al-Araji Al-Khwarizmi Engineering Journal, Vol. 13, No. 3, P.P. 64- 73 (2017) 70 Fig. 7. On-line performance index for nonlinear neural controller. Fig. 8. On-line performance index for nonlinear back-stepping controller. Fig. 9 shows the effectiveness of the nonlinear neural controller response through generating best torque control action without saturation control action state to track the desired path in minimum time while the torque control action for the nonlinear back-stepping controller can be shown in Fig. 10 which has non smoothens and not high spikes and reach to the maximum saturation torque action value in data sheet of motor [17] is 3 N.m only at starting and for very short time. Fig. 9. Torque control action for nonlinear neural controller. Fig. 10. Torque control action for nonlinear back- stepping controller. The response of angular velocity of the left and right wheels of the platform Eddie wheeled mobile robot can be shown in Fig. 11 have smoothness response which are generated from nonlinear neural controller while the angular velocities of the left and right as shown in Fig. 12 have small spikes response that are generated from the nonlinear back-stepping controller. It is clear, that the response between (60 to 80) samples and between (120 to 140) samples of the torque and angular velocity as shown in Figs. 9 and 11 respectively are non-smooth and have spikes because in this region the orientation in Fig. 6 is changing between -180 o to 180 o that needs to add 2pi for correcting the orientation and this big change will lead to small spikes and non- smooth response in this region. Figs. 13-a,b,c show the robustness and adaptation performance of the proposed controller in terms of keeping on minimum tracking pose error for the wheeled mobile robot and stabilizing the pose of the mobile robot when the mobile robot tries to drift from the desired path because the effect of the bounded dynamic disturbances to the system. Fig. 11. Angular Velocities of the left and right wheels for nonlinear neural controller. 20 40 60 80 100 120 140 160 180 200 0 0.005 0.01 0.015 0.02 0.025 Samples (Sampling Time 0.1 Sec) P e r fo r m a n c e I n d e x ( M S E ) 20 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Samples (Sampling Time 0.1 Sec) P e r fo r m a n c e I n d e x ( M S E ) 20 40 60 80 100 120 140 160 180 200 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Samples (Sampling Time 0.1 Sec) L e ft a n d R ig h t T o r q u e C o n tr o l A c ti o n ( N .m ) The Torque of Left Wheel The Torque of Right Wheel 20 40 60 80 100 120 140 160 180 200 -4 -3 -2 -1 0 1 2 3 4 Samples (Sampling Time 0.1 Sec) L e ft a n d R ig h t T o r q u e C o n tr o l A c ti o n ( N .m ) The Torque of Right Wheel The Torque of Left Wheel 20 40 60 80 100 120 140 160 180 200 -2 0 2 4 6 8 10 12 L e ft a n d R ig h t W h e e l A n g u la r V e lo c it y ( r a d /s e c ) The Angular Velocity of Left Wheel The Angular Velocity of Right Wheel Ahmed S. Al-Araji Al-Khwarizmi Engineering Journal, Vol. 13, No. 3, P.P. 64- 73 (2017) 71 Fig. 12. Angular Velocities of the left and right wheels for nonlinear back-stepping controller. Fig. 13. Pose error of the mobile robot: a) error in X-axis; b) error in Y-axis; c) orientation error. 6. Conclusions The simulation results on the on-line cognitive hybrid Bees-PSO tuning algorithm of different types of nonlinear controllers are presented in this paper for the dynamic wheeled mobile robot model which shows preciously and that the proposed hybrid Bees-PSO tuning control algorithm was better than the results in [3 and 4] in terms of the following capabilities: • Fast and stable on-line finding and tuning the parameters of the controller with minimum number of iteration in the local and global search. • Obtaining best torque control action, with small spikes as well as no saturation torque action state for the nonlinear neural controller. • Minimizing the pose error and no oscillation output for the wheeled mobile robot during motion in the reference path. • Strong adaptability and robustness performance when dynamic disturbances have been added to the mobile robot. 7. References [1] Leenaa N. and Sajub K.K., Modelling and Trajectory Tracking of Wheeled Mobile Robots. 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Accessed Sept. 2016. )2017( 64-73، صفحة 3د، العد13دجلة الخوارزمي الهندسية المجلم احمد صباح االعرجي 73 خوارزمية هجينة مدركة لتنغيم مسيطر الخطي لتتابع مسار لعجلة اإلنسان آلي متنقل تصميم **نور قاسم يوسف *أحمد صباح االعرجي الجامعة التكنولوجية /قسم هندسة السيطرة والنظم**،* ahmedalaraji76@gmail.com:البريد االلكتروني* الخالصة بشكل حي ومتصل للمسيطر الالخطي لتتابع مسار عجلة اإلنسان اآللي الحركي لتباع المسار ان هذا البحث يقدم خوارزمية تنغيم المسيطر المدركة يطر (العصبي المستمر المرغوب.ان الهيكلية المقترحة لخوارزمية االمثلية هي الهجينة (النحل مع حشد الجسيمات االمثلية) اليجاد وتنغم قيم كسب المس وبساطة تقنية التنغيم وبشكل حي ومتصل.وطريقة الخطوة الراجعة) وتتميز بسرعة لقد تم إثبات من ان افضل عزم مسيطر لعجلة اليمين واليسار لعربة اإلنسان اآللي تم توليدها بشكل حي ومتصل من خالل المسيطرين المقترحين. ر دقة من حيث ايجاد وتنغيم عناصر المسيطر بشكل خالل نتائج المحاكاة أن المسيطر العصبي الالخطي المقترح مع الخوارزمية الهجينة المدركة هي اكث تقليل الخطأ ألتتابعي لعجلة اإلنسان اآللي مقارنة مع أداء المسيطرة ذات تقنية الخطوة فضالعنحي ومتصل ويؤدي الى الحصول على فعل سيطرة ناعم الراجعة.