Yang, R., Yang, C., Chen, M., & Na, J. (16). RBFNN base aaptive control of uncertain robot manipulators in iscrete time. In 16 UKACC 11th International Conference on Control (CONTROL 16): Proceeings of a meeting hel 31 August - September 16, Belfast, Unite Kingom [7737617] Institute of Electrical an Electronics Engineers (IEEE). DOI: 1.119/CONTROL.16.7737617 Peer reviee version Lin to publishe version (if available): 1.119/CONTROL.16.7737617 Lin to publication recor in Explore Bristol Research PDF-ocument This is the author accepte manuscript (AAM). The final publishe version (version of recor) is available online via IEEE at http://ieeexplore.ieee.org/ocument/7737617/. Please refer to any applicable terms of use of the publisher. University of Bristol - Explore Bristol Research General rights This ocument is mae available in accorance ith publisher policies. Please cite only the publishe version using the reference above. Full terms of use are available: http://.bristol.ac.u/pure/about/ebr-terms
RBFNN Base Aaptive Control of Uncertain Robot Manipulators in Discrete Time Runxian Yang 1,,3, Chenguang Yang, Mou Chen 1 an Jing Na 4 1 College of Automation Engineering, Nanjing University of Aeronautics an Astronautics, Nanjing, China Zienieicz Centre for Computational Engineering, Sansea University, Sansea, UK 3 College of Electric an IT, Yangzhou Polytechnic Institute, Yangzhou, China 4 Department of Mechanical Engineering, University of Bristol, Bristol, UK Abstract The trajectory tracing control problem for a class of n-egree-of-freeom (n-dof) rigi robot manipulators is stuie in this paper. A novel aaptive raial basis function neural netor (RBFNN) control is propose in iscrete time for multiple-input multiple-output (MIMO) robot manipulators ith nonlinearity an time-varying uncertainty. The high orer iscrete-time robot moel is transforme to facilitate igital implementation of controller, an the output-feebac form is erive to avoi potential noncausal problem in iscrete time. Furthermore, the esire controller base on RBFNN is esigne to compensate for effect of uncertainties, an the RBFNN is traine using tracing error, such that the stability of closeloop robot system has been ell guarantee, the high-quality control performance has been ell satisfie. The RBFNN eight aaptive la is esigne an the semi-global uniformly ultimate bouneness (SGUUB) is achieve by Lyapunov base on control synthesis. Comparative simulation stuies sho the propose control scheme results in supreme performance than conventional control methos. I. INTRODUCTION WITH avances of technologies, robot applications in inustry an our aily life become increasingly popular, the relevant research ors have been an attractive topic. Hoever, most robot manipulators are usually subject to unmoelle ynamics an various uncertainties in practice [] [4], an ieal control esign for a class of robot manipulators is challenging. Various approaches for trajectory tracing control of robot manipulators have been propose. Feebac linearization methos [5], [6], sliing moe an other robust control methos [1], [7] [9] have all been extensively investigate for robot control, an global tracing error convergence are able to be guarantee. Furthermore, the avance intelligent methos an relevant research results have been ell applie to robot control, e.g., aaptive control [1], [11], aaptivefuzzy control [1], aaptive-sliing control [13], an complex aaptive control base on fuzzy an sliing-moe theories for robot manipulators [14], function approximators have also been utilize. In orer to compensate for uncertainties of This or as supporte in part by Engineering an Physical Sciences Research Council (EPSRC) uner Grant EP/L6856/, an Marie Curie Intra-European Felloships Project AECE uner Grant FP7-PEOPLE-13- IEF-6553174. *Corresponing author. Email: cyang@theiet.org robot manipulators, aaptive neural netor (ANN) techniques have been popular in recent years [15] [17], ANNs have universal approximation capability for nonlinear functions. An aaptive RBFNN algorithm guaranteeing close-loop stability has been propose for robot manipulator systems in [18]. A novel RBFNN estimator has been esigne to compensate for uncertainties in [19], []. These approaches are able to guarantee UUB of close-loop system of robot manipulators. But the igital implementation of robot controllers an netor communication of high-spee computers are becoming increasingly popular an poerful. Thus, the increasing research ors for robot manipulators have no been carrie out in iscrete time. Discrete-time robot manipulator moels an iscrete-time control methos are use in [18], [1], the iscrete-time controllers applie to on-line robot control provies convenience for implementation. In [], a ANN controller has been propose base on combining one-step-ahea control ith ANN control for a class of MIMO iscrete-time systems ith nonaffine nonlinearity. In [8], a class of MIMO nonlinears systems ith bloc triangular structure can be ecompose in iscrete time, by applying pure-feebac metho, an ANN control has been presente base on all subsystems ith couplings an unnon irections. These iscrete-time approaches perform ell to guarantee robust stability of nonlinear robot systems. Hoever, these research ors only guarantee stability of close-loop robot manipulator systems, hile realizing trajectory tracing control is selom in iscrete time. Thus, a novel control scheme propose for a class of robot manipulators ith uncertainty in iscrete time is the main research objectives of this paper. Aiming to aress satisfie trajectory tracing performance base on stable close-loop robot system, e evelop a iscrete-time novel RBFNN base aaptive control for uncertain robot manipulators. The folloing notations are employe in this paper. represents the Eucliean norm of vectors an inuce norm of matrices. b := a enotes b is efine as a. [] T represents the transpose of a vector or a matrix. [] 1 represents the inverse of a n-orer reversible matrix.
[p] enotes the imension of zero vector is p-imension. I [m] stans for m-imension unit matrix. W represents the iea neural net eight matrix. Ŵ represents the estimate value matrix of neural net iea eight W at the -th step. W = Ŵ W enotes the eight estimate error. II. DISCRETIZING FOR ROBOT MODEL The ynamic moel of general n-dof nonlinear rigi robot manipulators can be escribe using orinary ifferential equation as M(q) q + C(q, q) q + G(q) = + (1) here q R n is the joint position, an q R n is the joint velocity, q R n is the joint acceleration, M(q) R n n is the symmetric an positive efinite inertia matrix, C(q, q) R n n is the Coriolis-Centrifugal torque matrix, G(q) R n enotes the gravity torque vector, R n is the control input torque vector, R n is the external force torque vector. Accoring to [], the folloing properties hol for rigi robot manipulators in (1): Property 1: M(q) is uniformly boune, an satisfies the folloing inequality m M(q) m () Property : The matrix C(q, q) an the vector G(q) are boune by C(q, q) c q, an G(q) g, respectively, here c an g are positive constants. It is very important an meaningful to esign robot controller in iscrete time. For a class of n-dof rigi nonlinear robot manipulators ith uncertainty in (1), hich can be iscretize by using iscretization theory ith a small sampling time interval T. The sample joint angle is q = q t, the sample joint angle velocity is q = q t, the control torque is = t an the external isturbance torque is = t at the sampling time instant t = T, respectively. Define p = q R n an v = q R n, then, the equivalent ynamics form in iscrete time can be obtaine [3], [6], [7] as (M(ξ )/T )(v +1 v )=(M(ξ ) M(p ))v f(p,v )+ + (3) here M(ξ ) R n n is also the inertia matrix ith ξ = p + Tv R n, f(p,v )=C(p,v )v + G(p ) R n, C(p,v ) R n n is Coriolis-Centrifugal torque matrix an G(p ) R n is gravitational synthetic torque vector in iscrete time, respectively. Accoring to Property 1, M(ξ ) is also symmetric, positive efinite an boune, satisfying m M(ξ ) m ith non constants m > an m >. III. TRANSFERING TO FEEDBACK SYSTEM To avoi possible noncausal problem in control esign, e exten our previous research ors [4], [9], [3] to a class of nonlinear time-varying MIMO robot manipulators ith uncertainty in iscrete time. The iscrete-time ynamics in (3) can be transferre into the output-feebac control system [31] as p +1 = p + Tv v +1 =[(1+T )I [n] TM 1 (ξ )M(p ) TM 1 (ξ )C(p,v )]v TM 1 (ξ )G(p (4) ) + TM 1 (ξ ) + TM 1 (ξ ) here is boune as ith an non constant It is easily non that M 1 (ξ ) is also boune, satisfying m M 1 (ξ ) m ith non constants m > an m >. The control objective is to synthesize an aaptive RBFNN control input for robot system (4), not only all signals of close-loop robot system are boune, but also the joint position signal p is able to ell trac the ieal trajectory signal of robot manipulators p Ω p, finally, the satisfie control performance is able to be obtaine, here Ω p is a compact set. It is note that v +1 epens on control output, hile p +1 is associate ith p an v at the ( +1)-th step in (4). We can rerite the first equation of the system (4) as p +1 p Tv = [n], an v is esigne as v = 1 T (p+1 p ). To preict the ( +)th step of robot manipulators, e have p + = p +1 + Tv +1 =[(+T )I [n] TM 1 (ξ )M(p ) TM 1 (ξ )C(p,v )]p +1 [(1 + T )I [n] TM 1 (ξ )M(p ) TM 1 (ξ )C(p,v )]p T M 1 (ξ )G(p ) + T M 1 (ξ ) + T M 1 (ξ ) Furthermore, e nee to move (5) bac to the ( +1)-th step, the output-feebac metho is applie to get the p +1 as p +1 =[(+T )I [n] TM 1 (ξ 1 )M(p 1 ) TM 1 (ξ 1 )C(p 1,v 1 )]p [(1 + T )I [n] TM 1 (ξ 1 )M(p 1 ) TM 1 (ξ 1 )C(p 1,v 1 )]p 1 T M 1 (ξ 1 )G(p 1 ) + T M 1 (ξ 1 ) 1 + T M 1 (ξ 1 ) 1 Substituting (6) to (5), e note that there is no more explicit future outputs an input signals. For convenience, let us efine L =(+T )I [n] TM 1 (ξ )M(p ) TM 1 (ξ )C(p,v ) R =(1+T )I [n] TM 1 (ξ )M(p ) TM 1 (ξ )C(p,v ) M = T M 1 (ξ ), G = G(p ) Consiering equation (6), e no that future state at the ( +1)-th step is able to be obtaine by getting values of the (5) (6)
current -th step an the past ( 1)-th step. Then, the output p + is obtaine as p + =(L L 1 R )p L R 1 p 1 L M 1 G 1 M G + L M 1 + M + L M 1 1 + M 1 an e further efine L p =(L L 1 R )p L R 1 p 1 + L M 1 1 L G = L M 1 G 1 + M G L = L M 1 1 + M Thus, equation (7) can be reritten as p + = L p L G + M + L = ψ(p 1,p,, 1, 1, ) It is easily non that the function ψ(,,,,, ) in (8) is continuous for all the arguments an continuously ifferentiable. Lemma 1: M is symmetric positive efinite matrix, an is boune as m M m ith m = T m an m = T m. Accoring to Lemma 1, e no that L L (3 + T + T m c ) m :=. (7) (8) is boune an IV. ADAPTIVE RBFNN CONTROLLER DESIGN A. RBFNN Approximation The RBFNN can approximate any nonlinear function F (z), hich can be expresse as [5]: F (z) =W T S(z), W R Ns No, S(z) R Ns (9) here z = [z 1,z,,z N n ] R Nn in Ω z is the input vector of RBFNN, N s is neuron noe number, N o is output imension of RBFNN, W is eight matrix, S(z) = [s 1 (z),s (z),..., s N s (z)] T is hien layer output function of RBFNN, an s i (z) is the i-th neuron output function, the Gaussian RBFNN function is chosen as follos s i (z) =e zi cij /b i (1) here i =1,,,N n, j =1,,,N s, c ij is the center of the j-th neuron noe for the i-th input signal, b i is the ith of the j-th neuron. A number of research results have shon that for any continuous smooth function ϕ(z) : Ω z R over a compact set Ω z R Nn [3], [33], e can apply RBFNN (9) to approximate ϕ(z). In particular, if N s is chosen a sufficiently large value, such that the ieal boune eight W exists, e have ϕ(z) =W T S(z)+μ(z) (11) here μ(z) is the approximation error, hich is boune as μ(z) <μ ith a given small constant μ. RBFNN in (9) or in (11) has the folloing property, hich ill be use in the control esign: S(z) T S(z) <N s (1) Noting the ieal RBFNN eight W is unnon in practice, e often use Ŵ as estimate eight of ieal eight W to approximate the unnon nonliear function ϕ(z). By esigning an appropriate learning rule, the estimate Ŵ can be renee. Then, equation (11) can be reritten as ϕ(z) Ŵ T S(z) (13) B. Desire Control The ieal system tracing output is p +. The ynamics of tracing error e + R n can be obtaine as e + = p + p + = L p L G + M + L p + (14) here p + is efine in the (8). There exists a continuous ieal control input n [9], such that L p L G + M n p + = (15) Lemma : There are positive constants m =1/ m an m =1/m, an M 1 is boune as m M 1 m. Thus, the preictor for to-step trajectory error e + can be constraine as e + = L (16) It is note that the esire control n is not obtaine ith the unon M 1, L p an L G. We apply the aaptive RBFNN to learn an to approximate the esire input n, such that tracing error e + =after steps can be achieve, if = an 1 =in (14). C. RBFNN Base Control From Section IV-A, an ieal eight matrix W exists, e apply RBFNN Gaussian function S ( z ) to approximate the ieal control input n as follos n( z )=W T S ( z )+ɛ ( z ) (17) here S ( z ) R N is the regression matrix, N is neuron noe number, ɛ ( z ) ɛ ith ɛ > is the approximation error, the ieal eight matrix W = R N n, an the RBFNN input vector z is esigne as z =[p T,p 1T,v T,v 1T, 1T,p +T ] T Ω z here Ω z is a sufficient large compact set corresponing to Ω p. It is easy to verify the ieal control n( z ) is boune. Accoring to (14) an (15), e apply RBFNN to approximate the ieal control input n( z ), an introuce PD metho to improve control performance, the system control input is esigne as: = p e + e 1 +ˆ n ( z ) (18) ˆ n ( z )=Ŵ T S ( z ) here p = p + >, an p >, > are scaling factors, Ŵ R N n is use to approximate the ieal control input n( z ) in (17) ith the compact set Ω z.
Accoring to equation (15), e have p + = L p L G + M n. Then, equation (14) is reritten as follos e + = L p p + = M ( n )+L (19) For convenience, e efine: S = S ( z ), ɛ = ɛ ( z ) From Lemma 1, it is obvious that M is boune ith m an m. Noting W = Ŵ W, e substitute (17) an (18) into (19), then, e + = M ( p e + e 1 )+M T W S + p () here p = M ɛ + L. It is easy to sho that p M ɛ + L m ɛ + := p. Then, the error equation in () can be converte as: e + + M p e M e 1 = M T W S + p (1) We efine a ne error function as follos e + 1 = e + + M p e M e 1 () Substitute () into (), the error function e + 1 is reritten as e + 1 = M T W S + p (3) It is note that the error function base on the aaptive RBFNN algorithms (3) is the ( +)th step error for robot system, then, e can obtain the th step system error by efining = e 1 = M W T S + p (4) here m M m accoring to Lemma 1, an e 1 = e + M p e M e 1 Base on system tracing error e 1, RBFNN upate rule ΔŴ for (18) is given by ΔŴ Ŵ +1 = Γ S e T 1 = Ŵ +ΔŴ (5) N N here Γ = γ I [N ] R is a iagonal action system learning rate matrix ith γ >. D. Stability Analysis It has been shon that an ieal control input n( z ) exists an can guarantee e + =, if the unnon isturbance p =. Base on above all assumptions are only vali in compact set Ω z, the system all outputs an inputs signal must be prove remain in corresponing compact sets. A positive efinite Lyapunov function V for the system (8) is chosen as V = n tr[ j= +it W Γ 1 W +i ] (6) here W = Ŵ W. Note the error function in (4), it is obvious that the Lyapunov function V contains system tracing error, strategic signal error an parameter aaptation for RBFNN eights. The ifference of (6) is given by here b = S T Defining A (7), e have ΔV = e T 1 Γ S. = M W T S + be T 1 e 1 (7) W T S ΔV = (A + an substituting (4) into p )T M 1 A + b(a + p )T (A + p ) (A + p )T (M 1 (A + p ) AT + J bi [n] ) M 1 A (8) here J = T p M 1 p J = p m. Accoring to Lemma, it is easy to no that M 1 is symmetric positive efinite matrix, an it can be boune ith m M 1, i =1,,,n, it is obvious that λ λ i m. If the eigenvalues of M 1 i >. We further efine λ max = max(λ i ) an λ min = min(λ i ), then, nλ 1 min M = n i=1 λ T i(m 1 M 1 ) nλ max, i =1,,,n. For convenience, e efine P = M 1 bi [n]. The matrix P being symmetric positive efinite can be satisfie uner folloing conition: 1 b m > n Accor to the property in (1), it is obvious that b = S T Γ S = γ S T S <γ N. Analyse the ifference of Lyapunov function in (8), the esign parameter of controller are selecte as n <γ < N m (9) Furthermore, the folloing theorem is presente to analyze stability of the system in (8), such that the close-loop system stability an the trajectory tracing performance can be guarantee by choosing appropriate parameters an aaptive eight gain of the controller. Theorem 1: Assume that the conitions set above are satisfie, an efine B = A +, then, e have ΔV B T P B + J (3) Proof. There exists an invertible matrix Q so that P = Q T Q. Accoringly, ΔV can be satisfie uner folloing conitions: B > J Q 1 (31) A iscrete-time elay factor z 1 is introuce in (3), e have p are e =(I [n] + M p z M z 3 ) 1 e 1 (3)
Accoring to (31) an (3), e no there exists a finite running step K, hich maes B J Q ( 1), then, e e 1 uner (I [n] +M p z M z 3 ) 1 being Huritz-stable for all >K. Consier the bouneness of M an p, B = A + p, such that the error e 1 is boune as or, e can get e 1 = e T 1 e 1 A T e e 1 < the proof is complete. A + T < 4 J Q ( 1) +6 p p p (33) 4 J Q ( 1) +6 p (34) V. SIMULATION STUDIES To verify the above evelope aaptive RBFNN control approach, a testing example, -DOF robot manipulator interacting is use in this section. A. Robot Manipulator Dynamics Moel The folloing parameters of robot manipulator are specifie. The mass are m 1 = m =1.g, the length are l 1 = l =.m, the inertia are I 1 = I =.3gm, the istance are l c1 = l c =.1m. Then, ynamics of the robot manipulator ith G(q) = [] is given as M(q) =[M 11 M 1 ; M 1 M ] (35) C(q) =[C 11 C 1 ; C 1 C ] here M 11 = m 1 lc1 + m (l1 + lc +l 1 l c cos(q )) + I 1 + I M 1 = M 1 = m (l c + l 1 l c cos(q )+I M = m l c + I C 11 = m l 1 l c sin(q ) q C 1 = m l 1 l c sin(q )( q 1 + q ) C 1 = m l 1 l c sin(q ) q 1, C = The external force torque may be cause by isturbance, a smaller an a larger amplitue force torque s an b are assume as, respectively, s =[.5cos(.1t)cos(q 1 ),.5cos(.1t)cos(q )] T b =[4cos(.1t)cos(q 1 ), 4cos(.1t)cos(q )] T To ifferent types of esire trajectory q an q g are assume as [ ] q =[q 1,q ] T 1.5+.5(sin(.3t)+sin(.t)) = 1.5+.5(cos(.4t)+sin(.3t)) [ ] q g =[q g1,q g ] T.6sign(cos(πt/)) +.4 =.5sign(sin(πt/)) Position-q 1 Position-q 3 1 Desire Trajectory Trajectory Using Aaptive RBFNN Trajectory Using Traitional PD -1 4 - time(secon) Fig. 1. Position trajectory for tracing q using aaptive RBFNN an PD for a small isturbance s Tracing error of q 1 Tracing error of q - - Tracing error curve Using RBFNN Tracing error curve Using Traitional PD -4 time(secon) Fig.. Error curve for tracing ieal q using aaptive RBFNN an PD for a small isturbance s Position-q 1 Position-q 5-5 - Desire Trajectory Trajectory Using Aaptive RBFNN Trajectory Using Traitional PD -4 time(secon) Fig. 3. Position trajectory for tracing q g using aaptive RBFNN an PD for a large isturbance b B. Test Results The initial states of robot manipulator in (35) are q() = [, ] T an q() = [, ] T. We construct the aaptive RBFNN Ŵ T S, hich approximates system tracing error using N = 496 ith all the centres of Gaussian function evenly in [ 1; 1] an all the iths b = 1. The esign parameters are chosen as γ =.1, p =6.5, = 115, The initial eights Ŵ () = [ N ]. Simulation results are presente ith the controller sampling interval T =.1s. To sho the effectiveness, e use the above same esign parameters, an compare the position trajectory accuracy an capability beteen the aaptive RBFNN control an traitional PD control = p e (e e 1 ) for the robot manipulator (4) ith s an b in Figs. 1-3. Fig.1- sho trajectory tracing trajectories an error trajectories of q 1 an q for the esire q ith ae
external isturbance torque s, respectively. Fig.3 shos trajectory tracing trajectories of q 1 an q for the esirable q g ith ae external isturbance torque b. Comparing ith a traitional PD control base on the above simulation results ith a small isturbance signal s, the first joint of the propose iscrete-time aaptive RBFNN control has an initial error an eviates from the esire trajectory for less than 8s, but it can ajust itself quicly to achieve the esire trajectory; an the secon joint using the propose control has also an excellent tracing performance than the traitional PD control. Furthermore, a large isturbance signal b ae to test tracing performance of the propose control for the esire trajectory q g, the simulation results are given in Fig.3, hich shos that to joints of robot manipulator have achieve satisfie tracing trajectories using aaptive RBFNN control. VI. CONCLUSION An iscrete-time aaptive RBFNN has been evelope for a class of uncertain robot manipulators to achieve precise tracing control performance. The aaptive RBFNN controller is esigne to estimate system error, here the control la is aaptively tune online. To ifferent types of given trajectory an to ins of external isturbances are use to test the performance of the propose approach in the simulation. The propose iscrete-time aaptive RBFNN control is able to overcome effects of external isturbances an internal uncertainties. Not only the close-loop system stability is guarantee via Lyapunov stability analysis, but also excellent tracing performance is achieve. REFERENCES [1] C. Pueboon. Lyapunov Optimizing Sliing Moe Control for Robot Manipulators. Applie Mathematical Sciences, 7(63), 313-3139, 13. [] F. L. Leis, D. M. Dason, C. T. Aballah. Marine Control Systems: Guiance, Navigation an Control of Ships, Rigs an Unerater Vehicles. r. Marcel Deer, Inc., Ne Yor, USA, 4. [3] F. L. 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