Design of Visual Feedback Tracking Algorithm for Nonholonomic Mobile Robots Based on Neural Network

Transcription

1 Sensors & Transducers 2013 by IFSA Desgn of Vsual Feedbac Tracng Algorthm for Nonholonomc Moble Robots Based on Neural Networ 1 Huang Xuan, 2 Wenhua Zeng 1 Cogntve Scence Department, Xamen Unversty, Xamen, Chna, Fuan Key Laboratory of the Bran-le Intellgent Systems, Xamen Unversty, Chna, Zhang Zhou nsttute of technology, Zhang Zhou, Chna, School of Software, Xamen Unversty, Xamen, Chna, Receved: 16 August 2013 /Accepted: 25 September 2013 /Publshed: 31 October 2013 Abstract: Wth the rapd development of the computer and the electronc technque, the applcaton of robots has been wden. The robot vsual servng control system may mmc the human eyes. Then the vson nformaton s used as a feedbac to mprove the ablty of the robot adapton to the envronment. However, tradtonal algorthms whch need the calbraton of vsual parameters spend much tme and become techncal bottlenecs. Ths paper presents the development bacground of the robot and the concept of nonholonomc moble robots wth vsual servong feedbac. Second, the defcency exsts n tradtonal algorthms and fuzzy controller. Thrd, BP neural networ PID s proposed to desgn controller. Combnng BP neutral networ wth PID controller s used to manpulate moble robots frstly. The complex deduces of common tracng controllers s smplfed and tracng control problem wth non calbrated vrtual parameters s solved. Fnally, we program the smulaton code. The smulaton results show that the method s effectve. Copyrght 2013 IFSA. Keywords: Moble Robot, Uncalbrated Vsual Servong, Stablzaton, BP neural networ PID. 1. Introducton Wth the advancement of computer and camera equpment, nonholonomc moble robot vsual servo tracng has become very practcal. Cao Yang and other people desgned a robust speed tracng controller [1] accordng to the concept of servo n the study of traectory tracng control of the global vson of the moble robot. But t s n the case of nown vsual parameters, for the case of unnown parameters, t generally needs to be calbrated vsual parameters. The tradtonal methods are DLT method [2], Tsa's method [3], Weng's teratve method [4]. However, the operaton of these methods s not easy, feasblty of these methods s not hgh, so the research for the vsual parameters calbraton became the hot spot of ths feld. Zhangxue and other people Combned vson algorthms and Lyapunov and constructed an adaptve vsual controllers, although the controller s the vsual parameters calbraton, the mathematcal dervaton of the desgn process s tme-consumng, the structure of the controller s complex [5]. Nonholonomc moble robot tself s non-lnear, and t contans unnown parameters, t s dffcult to use the tradtonal control methods to control t [6]. The fast parallel computng, strong adaptve and nonlnear mappng of the neural networ s applcable to 42 Artcle number P_1384

2 nonlnear systems tracng control of the nonholonomc moble that s parameter uncertanty. Therefore, the choce of the neural networ PID control can acheve a good tracng n the case of vsual calbraton parameters and can avod the tedous formula dervaton [7]. For such nonholonomc moble robot, t s center of mass concdes wth the geometrc center. It s Nonholonomc constrants: x sn θ y cosθ 0 (1) Based on (1), we can ntroduce the nematc model of nonholonomc moble robot n the mage coordnate system: artfcal neural networ actual output value and the desred output value, t s a learnng process whch spreads bac whle corrects the weghts. BP algorthm as the typcal networ s the most mature n the artfcal neural networ [9] Prncple of BP Algorthm The structure of BP networ s shown n Fg. 1, t comprses nput layers, ntermedate layers and output layers. The ntermedate layer s also nown as a hdden layer, t may be a sngle layer or a multlayer. x vα1cos( θ θ0) y vα 2sn( θ θ0) θ w (2) Ths model can be dscretzed nto: x( + 1) x + v α1tcos( θ θ0) y( + 1) y + v α2tsn( θ θ0) θ( + 1) θ + w T The reference traectory of the controlled system s: xr( + 1) xr + vr α1tcos( θr θ0) yr( + 1) yr + vr α2tsn( θr θ0) θr( + 1) θr + wr T (4) y In ths formula, r, x r, θ r, respectvely are the gven deal poston and orentaton of the v nonholonomc moble robot. r, wr respectvely are the gven deal lnear and angular veloctes of the moble robot. So the tracng error nonholonomc moble robot at tme s: ex xr x ey yr y eθ θr θ 2. The BP Neural Networ (5) In 1986, Rumellhart put forward a error bacpropagaton of multlayer feedforward neural networ, that s BP (Bac Propagaton) neural networ, also nown as mult-layer perceptron (multlayer Perceptrons) [8]. BP neural networ uses gradent search, n order to mnmze the error of Fg. 1. Forward neural networ. Assumed that the number of layer of an artfcal neural networ s m, the sample of the nput layer s U X [10]. s the sum of the nputs of the neuron n X the layer, s the output of the neuron n the layer. U X f( U ) (6) W X, 1 (7) where W s the weght coeffcent of neurons n the 1 layer, f( ) s the actvaton functon of the neurons. The learnng process of BP neural networ s dvded nto forward propagaton and bac propagaton. Forward propagaton When t s forward propagaton, the samples of networ nput pass through the layers of hdden layer from the nput layer and then flow to the output layer. In ths process, the state of each layer of neurons wll only affect the state of the next layer neuron. Then, compare the output and the desred output n the output layer, f they are not dentcal, proceed to do the reverse propagaton of next step [11]. 43

3 Bac-propagaton Bac-propagaton s a way to reverse transfer the error sgnal from the output layer to the hdden layer and then to the nput layer, n the process of passng the weghts of each neuron n the hdden layer wll be fxed, and ultmately mnmze the error [12] Descrpton of BP Algorthm Mathematcal The nature of BP algorthm s to use the negatve gradent descent method to modfy the weght value, so that the error functon eventually reaches a mnmum [13]. Frst defne the error functon e, to represent sad the sum of squared dfferences between the actual output and the desred output. 1 e X Y 2 ( m 2 ) (8) Assume that the m layer n the BP networ s the output layer, n the formula X represents the actual output value of the networ, Y means the desred output value of the networ. m (9) where W s the modfcaton amount of the weght value coeffcent W, η s the step length, generally taes a value between 0-1. U W U W Due to the followng equaton: U W 1 ( WX ) X W 1 We can get another formula as follow: (10) (11) d (14) U So the modfcaton amount s: d (15) And the n the above formula can be expressed as follow: d X (1 X ) W d + 1 (16) Summarzed from the above mathematcal relatonshp we can get that neural networ BP algorthm frst does the forward propagaton, adds the nput sample nto the networ, forward transfers through layers of networ to get equaton as follow: X f( U ) And then compare the actual output X and the desred output U, the error generated e modfes the value of weghts through the bac-propagaton: (17) In order to speed up the convergence rate, we can add a weght coeffcent to the modfed formula. (18) In the formula, α s the weght coeffcent correcton constant. Weghts have been modfed, and fnally maes the error tends to be mnmum. The more layers of the neural networ, the more number of neurons, the greater the amount of computaton s, the speed of convergence s slower The Implementaton Steps of BP Neural Networ The BP algorthm flow chart s shown as Fg. 2: W X U 1 (12) 3. Moble Robot Tracng Controller Based on Neural Networ So we can get that: 3.1. The Selecton of the BP Neural Networ PID n Ths Paper (13) PID ncremental control algorthm: And we mae: (19) 44

4 M (2) (2) (1) (2) net w o θ, 1,2, Q 1 (22) (2) (2) o f[ net ] f ( ) n ths formula taes Sgmod functon that s postve and negatve symmetrcal. tanh f x x e e x x e + e x x (23) The nput-output relatonshp of output layer s: Q (2) netl wl o θ l, l 1,2,3 1 ol g[ netl ] (24) g [ ] can be wrtten as follow: 1 g[ x] [1+ tanh( x)] (25) 2 Fg. 2. Flowchart of BP algorthm. e( ) s the error between system's desred output and the actual output at tme, that s: () () e y y (20) r The output of the neural networ s: o1 p o2 o3 d (26) The BP networ selected n ths paper s a threelayer structure for M Q 3, there are M nodes n the nput layer of the networ, Q nodes n the hdden layer, three nodes n the output layer (Fg. 3). The nput sample s: o ( ) x 1, 2, M (21) (1), The functon of the performance ndex s: E ( yr y) e (27) 2 2 The modfed weght coeffcent s: x1 p (28) x2 E And n the formula w l s: xm nput layer Hdden layer Output layer Fg. 3. Model of BP neural networ. The nput-output relatonshp of hdden layer s: d E E y u ol netl wl y u ol netl wl (29) But net w l l o (2) ( ) (30) 45

5 The converson of y u quotes partal dervatves of formula put forward by Psalts and some other people. y y y ( 1) (31) u u u ( 1) y can approxmately be replaced by a sgn u y functon sgn[ ]. u Learnng algorthm of the coeffcent of output layer n the neural networ can be derved from the above dervaton: δ ' l l u () ol () (32) y () u () e sgn g( net ),( l 1,2,3) (33) deferent from the conventonal method, t s based on the error between actual coordnates (x (), y (), θ ()) and the coordnates of the tracng targets (x r (), y r (), θ r ()), obtans the amount of control (u x (), u y (), u θ ()) of each step, through the PID controller of BP neural networ. Model for poston and orentaton of the controlled obects s: x( + 1) ux + x y( + 1) uy + y θ( + 1) uθ + θ (39) And then substtute (u x (), u y (), u θ ()) nto the nematc model wth unnown parameters, through recurrence, get the correspondng amount of control (v (), w ()). Choose three BP neural networs PID control system to respectvely control the x, y, θ n order to prevent mutual nterference. The structure of PID tracng controller of BP neural networ of the moble robot s shown n Fg. 4: Use the same toen, learnng algorthm of the coeffcent of hdden layer n the neural networ can be derved from the above dervaton: δ 3 (2) ' (2) δl l l 1 (34) f ( net ) w,( 1,2,, Q) (35) In the formula, θr u θ θ θ g ' ( ) g( x)[1 g( x)] (36) 1 (37) ' 2 f f x We can obtan from equatons (21) (37) that: u e e( 1) o1 u e o2 u e 2 e( 1) + e( 2) o Moble Robot Tracng Controller Based on Neural Networ PID (38) Tradtonal controller of the moble robot usually desgns drectly the lnear and angular veloctes of the nonholonomc moble robot [14]. The nonlnear structure of the nonholonomc moble robot causes the desgn process of the nonholonomc moble robot cumbersome [15]. The method n ths paper s Fg. 4. PID controller of neural networ. The specfc algorthm s as follows. As the formula (39) show that: ux v α1tcos( θ θ0) uy v α2tsn( θ θ0) uθ w T (40) and Assumes that θ 0 s unnown, α1 α2 α are unnown. It s physcal meanng s that the magnfcaton of the longtudnal drecton and a transverse pxel of a CCD camera s the same. What can be calculated by the formula (40) s: uy uy v Tα uxcos(arctan ) + uysn(arctan ) ux ux uθ w T (41) 46

6 uy In the formula arctan ux and then: s denoted as ϕ ( ), 1. Assume that learnng rate s unchanged and η0.25, and then change the nerta coeffcent α. When η 0.25, α 0.1, the tracng error curve s shown as Fg. 5. v Tα u cos( ϕ) + u sn( ϕ) (42) x y v( + 1) Tα ux( + 1)cos( ϕ( + 1)) + uy( + 1)sn( ϕ( + 1)) (43) The left and rght ends of formula (42) and (43) are respectvely dvded, and then we can get that: v ( 1) ( 1)cos( ( 1)) ( 1)sn( ( 1)) + ux + ϕ + + uy + ϕ + v () u x()cos(()) ϕ + uy()sn(()) ϕ (44) The rght sde of the equaton of the formula (44) s nown, and then the followng equaton can be recursve: v( 1) ( 1) cos( ( 1)) ( 1) sn( ( 1)) + ux + ϕ + + uy + ϕ + v(1) ux(1) cos( ϕ(1)) + uy(1) sn( ϕ(1)) (45) Fg. 5. Error n the condton that η 0.25, α 0.1. When η 0.25, α 0.2, the tracng error curve s shown as Fg. 6. The gven lne speed of nonholonomc moble robot gven at the frst tme s v (1), then accordng to equaton (45) we can get that: ux( + 1) cos( ϕ( + 1)) + uy( + 1) sn( ϕ( + 1)) v ( + 1) v (1) ux(1)cos( ϕ(1)) + uy(1)sn( ϕ(1)) (46) The actual needs of lnear veloctes v and angular veloctes w of every step of the nonholonomc moble robot can be obtaned as follow: uxcos( ϕ) + uysn( ϕ) v v(1) ux(1) cos( ϕ(1)) + uy(1) sn( ϕ(1)) uθ w T (47) Fg. 6. Error n the condton that η 0.25, α 0.2. When η 0.25, α 0.5, the tracng error curve s shown as Fg. 7. Thus n the case that the vsual parameters of nonholonomc moble robot are unnown, and at last the tracng control of the moble car s acheved. 4. Results and Analyss In the structure dagram, the PID controllers of the three BP neural networ use the BP networ structure, and then accordng to the above algorthm to do the smulaton. In the formula, v r 1.5, w r 2.3, v (1) the parameter selecton of BP neural networ: Fg. 7. Error n the condton that η 0.25, α 0.5. When η 0.25, α 0.52, the tracng error curve s shown as Fg

7 Fg. 8. Error n the condton that η 0.25, α Fg. 11. Error n the condton that α0.05, η0.4. When η 0.25, α 0.6, the tracng error curve s shown as Fg. 9. When α0.05, η0.5, the tracng error curve s shown as Fg. 12. Fg. 9. Error n the condton that η 0.25, α 0.6. Accordng to the above smulaton results, when the neural networ s n the case that η s not changed, when mae the value of α less than 0.5, the control effect s good, when the value of α exceeds 0.5 and the error ncreases as the value ncreases, and even there wll be phenomenon out of control as shown n Fg. 8 and Fg assume that the nerta coeffcent s constant and α 0.05, change the learnng rate η. When α0.05, η0.3, the tracng error curve s shown as Fg. 10. Fg. 12. Error n the condton that α0.05, η0.5. When α0.05, η0.6, the tracng error curve s shown as Fg. 13. Fg. 13. Error n the condton that α0.05, η0.6. Fg. 10. Error n the condton that α0.05, η0.3. When α0.05, η0.4, the tracng error curve s shown as Fg. 11. Accordng to the above smulaton results, when the neural networ s n the case that α s not changed, the value of η has lttle effect on the tracng effect. Thus mae the nerta coeffcent A 0.05, learnng rate B 0.25, and then ths wll allow the controller to meet the tracng requrements. Assume that the start poston and orentaton of the target that the system want to trac s (2,2,0), the actual start poston and orentaton of the movng car s (0,0,0). The tracng results as shown as Fg. 14. Because the tracng target s movng randomly, the PID parameters are changng for adaptve adustment, the 48

8 PID parameters are shown as Fg. 15, the tracng error s shown as Fg. 16. PID method. Moble robot that based on vsual parameters are not calbrated, has been a research focus because of usefulness n the feld of robotcs. However, as far as s concerned traectory tracng problem of the robot that vsual parameters are not calbrated, there s a dfferent shortcomngs of exstng methods. Fg. 14. Spral curve tracng. Fg. 17. Straght-lne tracng. Fg. 15. The PID parameters n the Y drecton. Ths paper analyzes and compared the nadequaces of the exstng methods, and put forward a way that s to apply BP neural networ PID to the robot, and desgn a new tracng control algorthm, n order to acheve the tracng of nonholonomc moble robot. In ths paper we prove the valdty of the BP neural networ PID controller by usng a smulaton method. We put forward a dynamc tracng control of nonholonomc moble robot that vsual parameters are not calbrated, and results of the smulaton show the effectveness of ths method. In connecton wth nonholonomc moble robot model that the vsual parameters are not calbrated, we put forward BP neural networ PID control theory, desgn the tracng controller, and the results of smulaton verfy the tracng results. Fg. 16. Tracng error. Assume that the startng poston and orentaton of the target of the nonholonomc moble robot s (2,1,2.3), the actual startng poston and orentaton of the moble car s (0,0,0), and the tracng results are shown as Fg Conclusons In ths paper n connecton wth nonholonomc moble robot model that the vsual parameters are not calbrated, we put forward a method to desgn the tracng controller that uses the BP neural networ References [1]. P. Montesnos, L. Ceze, J. Torrellas, DeLorean: recordng and determnstcally replayng sharedmemory multprocessor executon effcently, n Proceedngs of the 35 th Annual Internatonal Symposum on Computer Archtecture (ISCA'08), 2008, pp [2]. J. Devett, B. Luca, L. Ceze, Lus Ceze, Mar Osn, DMP: determnstc shared memory multprocessng, n Proceedngs of the 14 th Internatonal Conference on Archtectural Support for Programmng Languages and Operatng Systems, 2009, pp [3]. J. Yu, S. Narayanasamy, A case for an nterleavng constraned shared-memory mult-processor, n Proceedngs of the 36 th Annual Internatonal 49

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