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  1. 1. D Narendra Kumar, S LalithaKumari , Veeravasantarao D / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.1376-1382Dynamic Adaptive Control of Mobile Robot UsingRBF Networks D Narendra Kumar#, S LalithaKumari #, Veeravasantarao D* # Department of Power and Industrial Drives, GMRIT, Rajam, Andra Pradesh, India * Indian Institute of Technology, Kanpur, IndiaAbstract In this paper, an adaptive neuro-control The robot studied in this research is a kind of asystemwith two levels is proposed for the motion simplenonholonomic mechanical system.control of anonholonomic mobile robot. In the Nonholonomic property is seen inmany mechanicalfirst level, a PD controller is designed to generate and robotic systems, particularly thoseusing velocitylinear and angular velocities, necessary to tracka inputs. Smaller control space comparedreference trajectory. In the second level, a neural withconfiguration space (lesser control signals thannetwork converts the desired velocities, provided independentcontrolling variables) causes conditionalby the first level, intoa torque control. The controllability ofthese systems. So the feasibleadvantage of the control approach is that, no trajectory is limited. Thismeans that a mobile robotknowledge about the dynamic model is required, with parallel wheels can’t movelaterally.and no synaptic weight changing is needed in Nonholonomic constraint is a differential equationonpresence of robot’s parameter’s variation (mass the base of state variables, it’s not integrable.or inertia).By introducing appropriate Rolling butnot sliding is a source of this constraint.Lyapunovfunctions asymptotic stability of state The control strategy proposed on this papervariables and stability of system is guaranteed. addressesthe dynamic compensation of mobileThe tracking performance of neural controller robotsand only requires information about the robotunder disturbances is compared with PD localization.The problem statement is presentedcontroller. Sinusoidal trajectory and lamniscate onsection 2 and the kinematic and dynamic modeltrajectories are considered for this comparison. ofthe considered robot, on section 3 and 4. Neural controller design as well as the main controlKeywords— Direct Adaptive Control, RBF systemdesign is presented on section 5 and 6. SomeNetworks, Trajectory tracking, Set point results and final considerations are also presented ontracking, Lyapunov stability section 6.I. INTRODUCTION II. PROBLEM STATEMENT Navigation control of mobile robots has The dynamics of a mobile robot is timebeen studied by many authors in the last decade, variant and changes with disturbances. The dynamicsince they are increasingly used in wide range of model is composed of two consecutive part;applications. At the beginning, the research effort kinematic model and equations of linear and angularwas focused only on the kinematic model, assuming torques. By transforming dynamic error equationsofthat there is perfect velocity tracking [1]. Later on, kinematic model to mobile coordinates, thethe research has been conducted to design navigation trackingproblem changes to stabilization. In thecontrollers, including also the dynamics of the robot trajectory tracking problem, the robot mustreach and[2], [3]. Taking into account the specific robot follow a trajectory in the Cartesian spacestartingdynamics is more realistic, because the assumption from a given initial configuration.The trajectory“perfect velocity tracking” does not hold in practice. tracking problem is simpler than thestabilisationFurthermore, during the robot motion, the robot problem because there is no need to controlthe robotparameters may change due to surface friction, orientation: it is automatically compensatedas theadditional load, among others. Therefore, it is robot follows the trajectory, provided that thedesirable to develop a robust navigation control, specified trajectory respects the non-which has the following capabilities: i) ability to holonomicconstraints of the robot. Controller issuccessfully handle estimation errors and noise in designed in twoconsecutive parts: in the first partsensor signals, ii) “perfect” velocity tracking, and iii) kinematic stabilization isdone using simple PDadaptation ability, in presence of time varying control laws, in the second one, direct adaptiveparameters in the dynamical model. control using RBF Networks has been used for Artificial neural networks are one of the exponentialstabilization of linear and angularmost popular intelligent techniques widely applied in velocities. Uncertainties inthe parameters ofengineering. Their ability to handle complex input- dynamic model (mass and inertia) havebeenoutput mapping, without detailed analytical model, compensated using model reference adaptiveand robustness for noise environment make them an control.ideal choice for real implementations. 1376 | P a g e
  2. 2. D Narendra Kumar, S LalithaKumari , Veeravasantarao D / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.1376-1382III. KINEMATIC CONTROL angular displacement,𝑠 and𝜃, so that 𝑠 = 𝑣and 𝜃 = In this paper the mobile robot with 𝜔. One could easily design a control system basedondifferential driveis used(Fig. 1). The robot has two the block diagram on Fig. 2, if s and 𝜃 aredriving wheels mounted on thesame axis and a free measurableand 𝑠 𝑟𝑒𝑓 and 𝜃 𝑟𝑒𝑓 are defined. Thisfront wheel. The two driving wheels controllercan be based on any of the classic designareindependently driven by two actuators to achieve techniquesfor linear systems where the controllerboth thetransition and orientation. The position of receives the error signal and generates the input tothe mobile robot inthe global frame {X,O,Y} can be the plant (a PD,for example).defined by the position of themass center of themobile robot system, denoted by C, oralternativelyby position A, which is the center of mobilerobotgear, and the angle between robot local frame 𝑥 𝑚 , 𝐶, 𝑦 𝑚 andglobal frame.A. Kinematic model Kinematic equations [9] of the two wheeledmobile robot are: Fig. 2 Kinematic Control system block diagram 𝑥 cos⁡ (𝜃) 0 As the design of such a controller is simple, 𝑦 = sin⁡ 𝑣 (𝜃) 0 , (1) thismodel has been used for the control system 𝜔 design, despite of two problems that still hold: the 𝜃 0 1 linear displacementsalong a trajectory is practically 𝑣 𝑟 𝑟 𝑣𝑅 unmeasurableand 𝑠 𝑟𝑒𝑓 is meaningless. However,And = 𝑟 − 𝑟 𝑣𝐿 , (2) 𝜔 𝐷 𝐷 these problemscan be contoured, as will be shown on the nextsection. B. Kinematic controller design The robot stabilisation problem can be divided into two different control problems: robot positioning control and robot orientating control. The robot positioningcontrol must assure the achievement of a desiredposition ( 𝑥 𝑟𝑒𝑓 ; 𝑦 𝑟𝑒𝑓 ), regardless of the robot orientation.The robot orientating control must assurethe achievement of the desired position and orientation(𝑥 𝑟𝑒𝑓 ; 𝑦 𝑟𝑒𝑓 ; 𝜃 𝑟𝑒𝑓 ). In this paper we only consider the positioning control. Fig. 3 illustrates the positioning problem, where∆𝑙is the distance between the robot and the desired reference(𝑥 𝑟𝑒𝑓 ;𝑦 𝑟𝑒𝑓 ), in the Cartesian space.Fig.1 The representation of a nonholonomic mobile The robotpositioning control problem will be solvedrobot if we assure ∆𝑙 → 0. This is not trivial since the Where 𝑥 and 𝑦 are coordinates of the center 𝑙variable does not appear in the model of equation 1.of mobile robotgear, 𝜃is the angle that represents theorientation of thevehicle, 𝑣 and 𝜔 are linear andangular velocities of thevehicle, 𝑣 𝑅 and 𝑣 𝐿 arevelocities of right and left wheels, 𝑟 is awheeldiameter and 𝐷is the mobile robot base length.Inputsof kinematic model of mobile robot are velocities ofright and left wheels 𝑣 𝑅 and 𝑣 𝐿 . The mainfeature of this model for wheeledmobile robots isthe presence of nonholonomicconstraints, due to therolling without slippingcondition between the wheelsand the ground.Thenonholonomic constraints imposethat the systemgeneralized velocities cannot assume independentvalues. In order to reduce the model complexity [5], Fig. 3Robot positioning problemone couldrewrite it in terms of the robot linear and 1377 | P a g e
  3. 3. D Narendra Kumar, S LalithaKumari , Veeravasantarao D / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.1376-1382To overcome this problem, we can define twonewvariables, ∆λ and ϕ . ∆λ ¸ is the distance to R,thenearest point from the desired reference that liesonthe robot orientation line; ϕis the angle of thevectorthat binds the robot position to the desiredreference.We can also define ∆ϕ as the differencebetween the𝜙 angle and the robot orientation:∆𝜙 = 𝜙 − 𝜃.We can now easily conclude that: 𝛥𝜆 ∆𝑙 = (3) 𝑐𝑜𝑠⁡ (𝛥𝜙) Fig. 4Robot positioning controller So, if ∆𝜆 → 0 and ∆𝜙 → 0then∆𝑙 → 0. Thatis,if we design a control system that assures the ¸ C. Set point trackingand∆λ and ∆ϕ converges to zero, then the desired On Fig. 5 a simulation of the robotreference, xref and yref is achieved. Thus, the robot stabilization control problem is shown, where thepositioningcontrol problem can be solved by initial position of robot is different and the desiredapplying any controlstrategy that assures such position is fixed. A simple PD controller has beenconvergence. implemented as positioning controller. The block diagram in Fig. 2 suggests that Fig. 6 shows the linear and angular errorsthe systemcan be controlled using linear and angular convergenceto zero, thus, assuring the achievementreferences, sref and θref , respectively. We will ofthe control objective.generatethese references in order to ensure theconverge of∆𝜆and𝛥𝜙to zero, as required by equation3. In otherwords, we want 𝑒 𝑠 = ∆𝜆¸ and 𝑒 𝜃 = ∆𝜙. Different initial positionsThus, if thecontroller assures the errors convergenceto zero, therobot positioning control problem issolved.To make 𝑒 𝜃 = ∆𝜙 ,we just need todefine 𝜃 𝑟𝑒𝑓 = 𝜙 ,so 𝑒 𝜃 = 𝜃 𝑟𝑒𝑓 − 𝜃 = 𝜙 − 𝜃 = ∆𝜙 .For this, we make: 𝑦 𝑟𝑒𝑓 − 𝑦 Δ𝑦 𝑟𝑒𝑓 𝜃 𝑟𝑒𝑓 = 𝑡𝑎𝑛−1 = 𝑡𝑎𝑛−1 (4) 𝑥 𝑟𝑒𝑓 − 𝑥 Δ𝑥 𝑟𝑒𝑓 To calculate 𝑒 𝑠 is generally not very simple,because 𝑠 output signal cannot be measured andwecannot easily calculate a suitable value for 𝑠 𝑟𝑒𝑓 .Butif we define the 𝑅 point in Fig. 3 as thereferencepoint for the 𝑠 controller, only in this case Target positionit is truethat 𝑒 𝑠 = 𝑠 𝑟𝑒𝑓 − 𝑠 = ∆𝜆. So: Fig. 5 Robot stabilization for different initial 𝑒 𝑠 = Δ𝜆 = Δ𝑙 . cos Δ𝜙 = (5) conditions 2 2 Δ𝑦 𝑟𝑒𝑓 Δ𝑥 𝑟𝑒𝑓 + Δ𝑦 𝑟𝑒𝑓 . 𝑐𝑜𝑠 𝑡𝑎𝑛−1 − 𝜃 Δ𝑥 𝑟𝑒𝑓 The complete robot positioning controller,based on the diagram of Fig. 2 and the equations 4and 5, is presented on Fig. 4. It can be used as astand-alonerobot control system if the problem isjust to drive torobot to a given position (𝑥 𝑟𝑒𝑓 ;𝑦 𝑟𝑒𝑓 ),regardless of the final robot orientation. Controller 𝑣𝑑 𝑘 𝑠 𝑒 𝑠 + 𝑘 𝑠𝑑 𝑒 𝑠𝑢= 𝜔 = (6) 𝑑 𝑘 𝜃 𝑒 𝜃 + 𝑘 𝜃𝑑 𝑒 𝜃 Fig. 6 Linear and angular errors IV. DYNAMIC CONTROL In this section, a dynamic model of anonholonomic mobile robot with motor torques will be derived first. 1378 | P a g e
  4. 4. D Narendra Kumar, S LalithaKumari , Veeravasantarao D / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.1376-1382A. Dynamic model nonlinearities are not known completely they can be The dynamic equations of motion can be approximated either by neural networks or by fuzzyexpressed as [10] systems. The controller then uses these estimates to 𝐴𝜃 𝑅 + 𝐵𝜃 𝐿 = 𝜏 𝑅 − 𝐾1 𝜃 𝑅 (7) linearize the system. The parameters of the 𝐵𝜃 𝑅 + 𝐴𝜃 𝐿 = 𝜏 𝐿 − 𝐾1 𝜃 𝐿 (8) controller are updated such that the output tracking Where error converges to zero with time while the closed 𝑀𝑟 2 𝐼 𝐴 + 𝑀𝑑2 𝑟 2 loop stability is maintained. The design technique is 𝐴= + + 𝐼0 popularly known as direct adaptive control technique. 4 4𝑅2 A. Function approximation 𝑀𝑟 2 𝐼 𝐴 + 𝑀𝑑2 𝑟 2 The control problem becomes difficult 𝐵= − 9 4 4𝑅 2 when 𝑔(𝑋) is unkown because the fact that the Here M is the mass of the entire vehicle, 𝐼 𝐴 approximation of 𝑔(𝑋)can be zero at times whichis the moment of inertia of the entire vehicle makes controller unbounded. For simplicity, weconsidering point A,𝐼0 is the moment of inertia of the have considered 𝑓(𝑋) as unknown function and dθR dθL 𝑔 𝑋 as known function. Radial basis functionrotor/wheel and 𝑑𝑡 and 𝑑𝑡 are angularvelocities of the right and left network (RBFN) is used to approximate 𝑓(𝑋). Fig. 7wheelrespectively. τR , τL are right and left wheel shows RBF network. The weight update law of the K RBF network is derived such a way that the closedmotor torques. A1 = 0.5. loop system is Lyapunov stable and the output tracking error converges to zero with time.B. State space model In the equation (12), 𝑓(𝑋) can be Substitute θR , θL as ωR , ωL respectively in approximated as 𝑓 𝑋 = 𝑊 𝑇 ∅(𝑋) using a radialequations (7), (8) and convert these velocities into basis function network. Then the control lawlinear and angular velocities using equation (2). 𝑈 = 𝑔 −1 (𝑋) −𝑓 (𝑋) + 𝑋 𝑑 + 𝐾𝑒 will stabilize theThen the state space model will become system (equation 10) in the sense of Lyapunov 𝑣 𝑣 𝜏𝑅 = 𝐴𝑋 + 𝐵𝑈 𝜏 (10) provided 𝑊 is updated using the update 𝜔 𝜔 𝐿 2 𝑋 −𝐶 Where 𝐴 𝑋 , 𝐵 𝑈 are functions of parameters A and B. 𝑇. − . law𝑊 = −𝐹∅𝑒 Where ∅(𝑋) = 𝑒 2𝜎C. Feedback linearization Φ(X) The above model (equation 10) is similar to Xa general state space model of nonlinear system as 𝑓(𝑋)follows V W V 𝑋 = 𝑓(𝑋) + 𝑔 𝑋 𝑈 (11) E E When the nonlinearities𝑓(𝑋) and 𝑔(𝑋) are C Ccompletely known, feedback linearization can be T Tused to design controller for a system, where the O Ocontroller may have a form [8]: R R 𝑈 = 𝑔 −1 (𝑋) −𝑓 𝑋 + 𝑋 𝑑 + 𝐾 𝑒 (12) Here, 𝑒 = 𝑋 𝑑 − 𝑋 where 𝑋 𝑑 represents Fig. 7 Multi input-output RBF networkdesired state vector. The above mentioned controllaw makes the closed loop error dynamics linear as B. Weight update lawwell as stable thus the error converges to zero with Let us assume that there exists an ideal weighttime. 𝑊 such that the original function 𝑓(𝑋) can be But these nonlinear parameters are represented as𝑓 𝑋 = 𝑊 𝑇 ∅(𝑋).unknown in reality. So neural network models are Control 𝑈in the system (equation 11) we get,used to estimate these functions and use it in control 𝑋 = 𝑓 𝑋 + 𝑔 𝑋 𝑔 −1 𝑋 [−𝑓 𝑋 + 𝑋 𝑑 + 𝐾𝑒]structure. = 𝑊 𝑇 ∅ − 𝑊 𝑇 ∅ + 𝑋 𝑑 + 𝐾𝑒 (13) Defining 𝑊 = 𝑊 − 𝑊 then equation 13 will beV. NEURAL CONTROLLER 𝑋 = 𝑊 𝑇 ∅ + 𝑋 𝑑 + 𝐾𝑒(14) Feedback linearization is a useful controldesign technique in control systems literature where 𝑋 𝑑 − 𝑋 = 𝑒 = −𝑊 𝑇 ∅ − 𝐾𝑒 (15)a large class of nonlinear systems can be made linear Consider a Lyapunov function candidate 1 1by nonlinear state feedback. The controller can be 𝑉 = 𝑒 2 + 𝑊 𝑇 𝐹 −1 𝑊(16) 2 2proposed in such a way that the closed loop error Where F is a positive definite matrix.dynamics become linear as well as stable. The main Differentiating equation (16),problem with this control scheme is that cancellation 𝑉 = 𝑒𝑒 + 𝑊 𝑇 𝐹 −1 𝑊 (17)of the nonlinear dynamics depends upon the exact Substituting 𝑒 from equation (15) into equationknowledge of system nonlinearities. When system (17) 1379 | P a g e
  5. 5. D Narendra Kumar, S LalithaKumari , Veeravasantarao D / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.1376-1382 𝑉 = 𝑒 −𝑊 𝑇 ∅ − 𝐾𝑒 + 𝑊 𝑇 𝐹 −1 𝑊 (18) kinematic control are𝑘 𝑠 = 0.21, 𝑘 𝜃 = 0.6 𝑎𝑛𝑑 𝑘 𝑠𝑑 = 𝑘 𝜃𝑑 = 0.01. We used 6 hidden neurons and set the Since W is constant, we can write𝑊 = 𝑊 − 𝑊 = gain matrix as 𝐾 = 1.5 0; 0 1.5 . The initial −𝑊. Thus, values of learning rate, weights, centers and sigma 𝑉 = −𝐾𝑒 2 − 𝑊 𝑇 ∅𝑒 − 𝑊 𝑇 𝐹 −1 𝑊 are tuned such a way that it provides good tracking = −𝐾𝑒 2 − 𝑊 𝑇 ∅𝑒 + 𝐹 −1 𝑊 (19) performance. Equating the second term of equation (19) to 0, we get ∅𝑒 + 𝐹 −1 𝑊 = 0 𝑇 Or, 𝑊 = 𝐹∅𝑒 (20) Using update law (equation 20), equation 19 becomes, 𝑉 = −𝐾𝑒 2 (21) Since 𝑉 > 0and 𝑉 ≤ 0, this shows the stability in the sense of Lyapunov so that 𝑒 and 𝑊(hence𝑊) are bounded. So the weight update law is 𝑊𝑛𝑒𝑤 = 𝑊 𝑜𝑙𝑑 + 𝐹∅𝑒 𝑇 (22) VI. MAIN BLOCK DIAGRAM The block diagram of overall controller [7] Fig. 9Tracking the lemniscate trajectory structure is shown in Fig. 8. The errors determined The simulation results obtained by neural between desired trajectory positions and robot actual networkcontroller are shown in Figs. 9-11. Results positions are used to determine the desired velocities achieved in Figs. 9-10 demonstrate the good position using kinematic control discussed in section III. tracking performance. Fig. 11 shows that the error in These desired velocities are compared with actual velocities is almost zero whereas a slight error wheel velocities and use the errors to generate left observed in displacement. It clearly shows that the and right wheel torques for the two motors using the PD kinematic controller performance affects the control law discussed in section V. Here the overall tracking performance. 𝑣 𝜏𝑅 The velocities generated from torque state 𝑋 = , control input is 𝑈 = 𝜏 and the control are exactly matched with the values obtained 𝜔 𝐿 nonlinearities are𝑓 𝑋 = 𝐴 𝑋 𝑋 𝑎𝑛𝑑 𝑔 𝑋 = 𝐵 𝑈 . And from the kinematic control such that it tracks the 𝑣𝑑 − 𝑣 trajectory (Fig. 10). Theproposed neural controller the error is 𝑒 = 𝜔 − 𝜔 . 𝑑 also ensures small values of the controlinput torques Neural for obtaining the reference position trajectories (Fig.Controller (U= 10). Our simulations proved that motor torque of𝑔−1 (𝑋) −𝑓 (𝑋) 1Nm/sec is sufficient to drive the robot motion. This+ 𝑋 𝑑 + 𝐾𝑒 mean that smaller power ofDC motors is requested. PD Controller 𝑣 𝑑 = 𝑘 𝑠 𝑒 𝑠 + 𝑘 𝑠𝑑 𝑒 𝑠 𝜔 𝑑 = 𝑘 𝜃 𝑒 𝜃 + 𝑘 𝜃𝑑 𝑒 𝜃 Fig. 8 Main block diagram of mobile robot A. Trajectory tracking The effectiveness of the neural network controller is demonstrated in the case of tracking of a lamniscate curve. The trajectory tracking problem for a mobile robot is based on a virtual reference robot that has to be tracked. The overall system is designed and implemented within Matlab environment. The geometric parameters of mobile Fig. 10Inputs to the robot robot are assumed as r = 0.08m, D = 0.4m, d = 0.1m. M=5kg, Ia = 0.05, m0=0.1kg and I0=0.0005. The initial position of robot is 𝑥0 𝑦0 𝜃0 = 1 3 300 and the initial robot velocities are 𝑣, 𝜔 = 0.1, 1 . PD controller gains for 1380 | P a g e
  6. 6. D Narendra Kumar, S LalithaKumari , Veeravasantarao D / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.1376-1382 Sudden forcesFig. 11 displacement, angular and velocity errors Fig. 13Left and right wheel motor torquesB. Neural controller performance with C. Neural controller performance with disturbances convergence This test is performed to analyze control As the neural control structure is adaptive,performance when any disturbance occurred on the the weights are automatically adjusted using updaterobot. We have chosen sinusoidal trajectory for this law such that it tracks the trajectory though anypurpose as to prove neural controller performance changes happen to dynamics. So the velocity errorimproves when time increases. We applied sudden keeps on reducing with the time and hence theforces on robot at two different time instants and tracking performance improves. If the controlobserved robot come back to the desired trajectory. structure uses previous saturated weights as initialThe neural controller performance is compared with weights for the next time reboot of robotmakes thethe classical PD controller. The dynamic PD error further decreases to lower values. Whereas this is not possible in case of PD controller as the gainscontroller 𝜏 𝑅 = 𝑘 𝑤𝑟 𝑒 𝑣 + 𝑘 𝑤𝑟𝑑 𝑒 𝑣 , 𝜏 𝐿 = are fixed for a particular dynamics and external 𝑘𝑤𝑙 𝑒𝑤 + 𝑘𝑤𝑙𝑑𝑒𝑤gains which are used to generate environments. Fig. 14 shows that in case of neuraltorques from the velocity errorsare𝑘 𝑤𝑟 = 0.8, 𝑘 𝑤𝑙 = controller, the RMS error in X, Y coordinates0.2 𝑎𝑛𝑑 𝑘 𝑤𝑟𝑑 = 0.53, 𝑘 𝑤𝑙𝑑 = 0.01 . The total run decreases faster with time than a PD controller.time is 150sec and two high forces (equal to 10 and 2 215 Nm/sec) are appliedat 75sec and 50sec. Fig. 12 𝑖 𝑖 𝑁 𝑒 𝑥 +𝑒 𝑦shows that the neural controller is able to stabilize 𝑅𝑀𝑆 𝐸𝑟𝑟𝑜𝑟 = 𝑖=1 , where N is 𝑁the robot quickly and makes the robot move in the number of iterations.𝑒 𝑖𝑥 , 𝑒 𝑖𝑦 are𝑖 𝑡𝑕 iteration errors indesired path smoothly compared to PDcontroller.From Fig. 13, we can say that the neural 𝑥, 𝑦 coordinates.controller generated torques is smooth and low. Disturbances Fig. 14 RMS error with number of iterations (or w.r.tFig. 12Tracking performance when sudden forces time)applied CONCLUSIONS In this paper, we presented a simple method of controlling velocities to achieve desired trajectory by converting 𝑥, 𝑦, 𝜃 into linear displacement ( 𝑆) 1381 | P a g e
  7. 7. D Narendra Kumar, S LalithaKumari , Veeravasantarao D / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.1376-1382and 𝜃which takes care of nonholonomic constraints.We also proposed direct adaptive control methodusing RBF networks to generate motor torques suchthat the velocities generated from kinematic controlare achieved. We observed that neural controllerperformance is better than better than PD controllerwhen disturbances occurred. It also converges fasterthan PD.REFERENCES [1] I. kolmanovsky and N. H. McClamroch, “Development in Nonholonomic Control Problems,” IEEE Control Systems, pp. 20– 36, December 1995. [2] T. Fukao, H. Nakagawa, and N. Adachi, “Adaptive Tracking Controlof a Nonholonomic Mobile Robot,” IEEE transactions on Roboticsand Automation, vol. 16, pp. 609–615, October 2000. [3] R. Fierro and F. L. Lewis, “Control of a Nonholonomic MobileRobot Using Neural Networks,” IEEE Transactions on neural networks,vol. 9, pp. 589–600, July 1998. [4] Luca, A., Oriolo, G., Samson, C., and Laumond, J. P. (1998). Robot Motion Planning and Control, chapter Feedback Control of a Nonholomic Car-like Robot. [5] Frederico C. VIEIRA, Adelardo A. D. MEDEIROS, Pablo J. ALSINIA, Antonio P. ARAUJO Jr. “Position And Orientation Control of A Two Wheeled Differentailly Driven Nonholonomic Mobile Robot” [6] IndraniKar, LaxmidharBehera, “Direct adaptive neural control for affine nonlinear systems,” Applied Soft Computing., vol. 9, pp. 756–764, Oct. 2008. [7] Ali Gholipour, M.J. Yazdanpanah “Dynamic Tracking Control of Nonholonomic Mobile Robot with Model Reference Adaption for Uncertain Parameters”Control and Intelligent Processing center for Excellence, University of Tehran [8] IndraniKar, “Intelligent Control Schemes for Nonlinear Systems,” Ph.D. thesis, Indian Institute ofTechnology, Kanpur, India, Jan. 2008. [9] JasminVelagic, Nedim Osmic, and BakirLacevic, “Neural Network Controller for Mobile Robot Motion Control,” World Acadamy of Science, Engineering and Technology, 47, 2008. [10] JasminVelagic, BakirLacevic and Nedim Osmic “Nonlinear Motion Control of Mobile Robot Dynamic Model”University of Sarajevo Bosnia and Herzegovina 1382 | P a g e