Adaptive Control for Dual-Arm Robotic System Based on Radial Basis Function Neural Network

Nội dung text: Adaptive Control for Dual-Arm Robotic System Based on Radial Basis Function Neural Network

1. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 050-058 Adaptive Control for Dual-Arm Robotic System Based on Radial Basis Function Neural Network Luu Thi Hue2, Nguyen Pham Thuc Anh1* 1Hanoi University of Science and Technology, Hanoi, Vietnam 2Electrical Power University, Hanoi, Vietnam *Email: anh.nguyenphamthuc@hust.edu.vn Abstract The paper has developed an adaptive control using neural network for controlling a dual-arm robotic system in moving a rectangle object to the desired trajectories. Firstly, the overall dynamics of the manipulators and the object have been derived based on Euler-Lagrangian principle. And then based on the dynamics, a controller has been proposed to achieve the desired trajectories of the grasping object. A radial basis function neural network has been applied to compensate uncertainties of dynamic parameters. The adaptive algorithm has been derived owning to the Lyapunov stability principle to guarantee asymptotical convergence of the closed dynamic system. Finally, simulation work on MatLab has been carried out to reconfirm the accuracy and the effectiveness of the proposed controller. Keywords: Adaptive control, dynamics, Radial Basis Function Neural Network, dual-arm robotic. 1. Introduction* Nowadays neural networks (NNs) have created drastic changes in the development of controllers, Researches on multi-manipulator control have especially in robotics. The ability of NNs to received increasing attention due to their advantages approximate nonlinear uncertainties leads to the idea over single-manipulator in industrial applications of using a neural network directly in a model-based such as assembling, transporting heavy objects, etc. control strategy. The idea traced here is based on the The development of multi-manipulator systems possibility of training networks to compensate the creates an opportunity to replace humans in dynamic parameters of multi-manipulators. A dual dangerous environments. However, control of multi- neural network has been used to control the manipulator systems is always a challenge due to the coordination of two redundant robots in real-time [7]. high nonlinearity and complication of their dynamics. An adaptive hybrid force/position has been developed A coordination scheme for cooperative manipulation for cooperative multiple-manipulators carrying and with two-arm systems was introduced by Yun and manipulating a common rigid object by Panwar et al Kumar and a nonlinear-feedback control algorithm [8]. A framework for NNs based consensus control has proposed [1]. A robust algorithm for cooperative has been proposed for multiple robotic manipulators control of closed-chain manipulators has been under leader-follower communication topology. Two proposed with uncertain dynamics [2]. Adaptive situations: fixed and switching communication control of multiple-robot manipulation in a dynamical topologies, were studied by using adaptive and robust environment has been proposed [3]. An adaptive control principles [9]. A synchronized NN approach controller combined with a sliding mode controller has been proposed for controlling multiple robotic has been introduced [4]. A robust adaptive hybrid manipulators based on the leader-follower network force/position control scheme for two planar- communication topology [10]. An adaptive robust manipulators coordinating to move an object without control (SOSMC) algorithm has been considered for knowing its parameters, but knowning the parameters dual-arm manipulators using the combination of of the robots has been proposed [5]. A robust second-order sliding mode control and neural adaptive algorithm has been proposed for controlling networks. The SOSMC deals with the system dual planar-manipulators in cooperative manipulation robustness when faced with external disturbances and of an object under uncertainty of dynamic parameters parametric uncertainties [11]. The problem of self- [6]. Almost the aforementioned controllers were tuning control with a two-manipulator system holding based on inverse dynamics to create adaptive update a rigid object in the presence of inaccurate algorithms, then they were complicated in translational base frame parameters is addressed. An mathematic formulae and the practical applicability adaptive robust neural controller is proposed to cope was limited. with inaccurate translational base frame parameters, internal force, modeling uncertainties, joint friction, and external disturbances. A radial basis function neural network is adopted for all kinds of dynamical ISSN: 2734-9373 estimation, including undesired internal force. Specialized robust compensation is established for Received: 28 April 2020; accepted: 02 October 2020 50
2. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 050-058 global stability [12]. NN-based adaptive controllers 2.2 Kinematic Relations have proved the effectiveness in compensating dynamic uncertainties. However, they are based on Refer to the Fig.1, the position ov (, xy )of the inverse dynamics and complicated in mathematical mass center of the object in the reference frame formulae. {OXY} can be calculated as: The aforementioned adaptive NN-based i L xx= 0i −−( 1) cosθθ+Yi sin ; controllers are effective in compensation of 2 (1) uncertainties of dynamic parameters of systems. L yy= −−( 1)i sinθθ−Ycos , However, they are based on inverse dynamics of the 0i 2 i systems, then the proposed controllers are complicated and do not ensure for keeping contact where Exiii(,00 y ) is the position of the end-effector i between end-effectors and the object. in the frame {OXY}. In the paper, we propose Radial Basis Function Differentiating (1) Neural Network (RBFNN) in controlling a dual-arm i L  robotic system manipulating a rectangle object. The xx=0ii +−( 1) sinθθ . +Y cos θθ . rest of the paper is organized as follows: section 2 2 addresses to formulating dynamics of dual-arm robot i L yy= −−( 1) cosθθ . +Y sin θθ . . and object system, section 3 aims to build up adaptive 0ii2 RBFNN-based controller, section 4 introduces simulation results and section 5 is for conclusions. then 2. Formulation of Dynamics of Overall Dual x  Robot-Object System θθ−= θθ [cos sin Yi] y[cos sin] Jq0ii . , (1a)  2.1. System Description θ The model under study consists of a dual-arm where J0i is the Jacobian matrix. robotic system in grasping a rigid object and is T depicted in Fig.1. The dual-arm robotic system has ∂∂xy two 3-DOF planar robots in antagonistic J = 00ii, 0i ∂∂ arrangement. The left robot is numbered the first and qqii the right one is the second. The object is rigid and It is possible to express (1a) into rectangle, then its surfaces are flat. Coordinated frames, main parameters and variables of the x objective system have been defined in detail in the  + y= D.[ cosθθ sin] .Jq . previous work [13]. The dual-robot system is  0ii responsible for stable grasping the object and then θ manipulating it dexterously. It is assumed that the +−TT1 end-effectors and the object are rigid and point- with DY= [cosθθ sin − i ] and D= D(. DD ) contacted. When the end-effector i contacts with the is pseudo-inverse matrix of D matrix. corresponding surface, there exists a force fi that T occurs perpendicular and a force λi that occurs Define vector z=[ xy θ] then tangential to the contact surface. The whole system works in the vertical plane, then it is affected by  z= Aqii.; z = Aq i .  + Aq i  , gravity. That means yv Y xv q= Az.; q = Az .  + Bz  , (2) E2 θ f 23 E 1 y 1 f2 01 λ d23 where λ Y1 Y2 2 d 1 θ θ 13 13 y ov θ −−11T TT θ 22 AA= [( ) ,( A ) ] , d22 12 d12 L −−11T − 1 − 1TT B= [(- A1 AA 11 ) , (− A2 AA 2 2 ) ] . d11 Mg d21 θ12 θ11 θ21 X 2.3. Dynamics O x01 x a The dynamics of the system has been formulated First arm-robot Second arm-robot based on the Euler-Lagrange principle. The Lagrangian function is defined as: Fig. 1. Description of the dual-arm robotic and object = − system LKP 51
3. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 050-058 where K is the kinematic energy and P is the potential + in which Hp= HqA() + JEHBz; energy. + The dynamic equation of the whole system can Cp=++ HqB( ) Cqq ( , ). A JBz EC ( zz , ); be expressed as: G= Gq() + J E+ g . a) For the ith robot p Bz It is clear that Hp is an inertia matrix, Cp is a τ i=++Hqq iii() Cqq iii (,)  Gq ii () (3) centrifugal/Coriolis matrix, and Gp is a gravity force iT θθT in the dynamic equation of the whole system (8). It is −−( 1) Jcos fJ+ sin λ , 00isinθθ ii −cos  i known that dynamic parameters of system dynamics such as mass and inertia term of links and the object, where qi is joint angle vector, Hii (q )is inertia friction coefficients are uncertain. It is reasonable to consider Hp, Cp, Gp including two terms: known H0, matrix, Cqqiii(, )is Coriolis and centrifugal matrix, C0, G0, and unknown ∆∆∆H,C,G. That means: and Gi(qi) is gravity vector, for i=1,2. ppp b) For the object: HHpp=0 +∆ H;  CC= +∆ C; cosθθsin 0 pp0 22  H z+−( 1)i f sin θλ − −cos θ +Mg . = 0 , (4) zi∑∑ i GGpp=0 +∆ G. ii=11= −YL0 i −−( 1)i .  2 Therefore the dynamic equation (8) can be expressed in the following form: where Hz is the inertia matrix of the object. Hz000+ Cz  + G +∆ f(,) zz =τ , (9) 3. Design Control Law where ∆f(,) zz includes unknown terms 3.1. Controller Design The dynamic equation of the dual-arm system ∆f(,) zz =∆ Hzppp  +∆ Cz  +∆ G . can be rewritten in a general form as follows: The error between desired trajectory and the Hqq()+ Cqqq (,)   ++ Gq () JB . F =τ , (5) actual trajectory of the object can be defined as: TTT where q= [] qq12, ; ezzp =d − . (10) T τ= [, ττ12 ]; Define: T se=pp +Λ. e; ξ ()t= As . . (11) Ff= [ 112λλ f 2] ; where Λ is diagonal positive matrix. If s → 0 then H( q )= blockdiag [H11 ( q ), H 2 ( q 2 )]; zz→ d when t →∞ C( q , q )= blockdiag [C111 ( q , q  ), C 2 ( q 2 , q 2 )]; T Substituting (10), (11) into (9) leads to Gq( )= [ G11 ( q ), G 2 ( q 2 )] . And the dynamic equation of the object also can be Hsp= fee0 (,) pp +∆ fee (,) pp  − Cs p . −τ , (12) rewritten in the following form: where Hzz+ C z(,) zzz   += g zz F. (6) feeH00(ppdp , )= (z  +Λ . eC  ) + 0 .(z  dp +Λ . eG ) + 0 Forces and moments that apply to the object are: is a nonlinear and known function and Fz = EF. . ∆fe(pp , e ) =∆ Hpp (z  dp +Λ . e  ) +∆ C (z  dp +Λ . e ) (13) + +∆ So F= EF. z , (7) Gp where E+−= ETT(. EE )1 is pseudo-inverse matrix of is the unknown function. E. Combining (2), (5), (6), (7) leads to the dynamic In the ideal case, ∆=fe(,)0pp e , the control equation of the whole system can be expressed as input is proposed as: follows: τξ=fx0 () + Ks , (14) Hzppp+ Cz  += G τ , (8) where Ks is positive matrix. 52
4. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 050-058 Weight update Wˆ Estimation of ∆fe(,)pp e (13) ∆fˆ z= [,,] xyθ T T + z= [, xy ,θ ]+ Control Dynamics of T TT d d dd + + τ q= [,] qq Λ algorithm the robot-object 12 Desired trajectory + (9) T TT - (14) q= [,] qq 12 d dt Fig. 2. Scheme of overall dual-arm robot and object system under RBRNN-adaptive controller The closed dynamics work stably according to neuron. The Gaussian-type function can be the Lyapunov principle. It is proved that expressed as: zz→  d h1  when t →∞ x w1 zz→ 1  d ∆f (x) However, due to the uncertainties of dynamic w2 x h2 ∑ ∆ 2 parameters, the term fe(,)pp e always exists that . . . . Output breaks the stability of the whole system. In order to . . wm compensate for the uncertainties of the system xn dynamics, adaptive control based on Radial Basis h Function Network is proposed. Fig.2 illustrates the i m control model using RBFNN to compensate for the Inputs j uncertainties of the system Hidden The control algorithms are: Fig. 3. Typical structure of RBF Neural Networks τξ=fee(,) +∆ feeˆ (,)K + , (15) 2 0 pp pp s xc− j hj =−=exp jm1,2 , , (16) where ∆feˆ(,) e is an approximated function of 2b2 pp j ∆fe(,)pp e . where x is the input signal of RBFNN, m is the 3.2 Design of Radial Basis Function Neural number of nodes in the hidden layer, cj is the center th Network and bj is the variance of j basic Gaussian function. The general RBF Neural Network has a The output of the linear layer RBFNN is structure as depicted in Fig.3 [15]. Its simplest form m is a three-layer feedforward network. The first layer ∆=fx()∑ Wjj hx (), corresponds to the inputs of the network, the second j=1 one is a hidden layer consisting of nonlinear where Wj is the gradient matrix associated with each activation units and the last one is the output layer node j of RBFNN. By selecting the appropriate corresponding to the final output of the network. weight vector, RBF network can approximate a Each of n components of the input vector x feeds continuous function with arbitrary precision. forward to m basis functions whose output are linearly combined with weighs into the network ∆=f() x WT hx () +ε , (17) output. Neurons in the hidden layer have Gaussian transfer functions for which outputs are inversely where W* is the optimal weight vector and ε is the proportional to the distance from the center of the error that arises from the approximation procedure of 53
5. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 050-058 RBF NN. Given an infinitely small constant ε 1 N V= sTT Hs + s Hs  +Γ trW (  T .−1 W ). there exists ε satisfies a condition εε< N . The 2 * ≤  T optimal weights are bounded WWmax , therefore, Due to H− 2 ACp is skew-symmetric matrix then an estimate of ∆fx() can be written as follows: TT s( H−= 2 ACp )s 0. ∆=fxˆ() Whˆ T , (18) Combining with (22) leads to: ˆ TTTTTT−1   where W is the estimated weight of the optimal V=− sAW Ks As +tr(. W Γ+ W)s A h(x) weight of the NN, which is generated by the online Due to sTT. A= ( As .) T weight update algorithm. V =−(A.s)T K( As .) + trW (T . Γ+−1 W ) ( As .)T W T h . According to the theory of RBFN, component s ∆fe(,)pp e of the dynamics of the system (12) is According to the Lyapunov stability principle, approximated by using RBFNN. Base on equation the condition for stability of closed dynamics is  (17), ∆fe(,)pp e is represented as follows: V ≤ 0 then − T ⇒=tr( WTT .Γ+1 W ) ( A .) s W0 T h ∆=+f( epp , e ) W hx () ε . (19) tr( WTT .(Γ+−1 W ) h ( A . s ) ) = 0 ˆ ⇒ Γ+−1  T = ∆fe(,)pp e is the estimation of the function Wh(As .) 0 ∆fe(,)pp e , base on (18) is determined: So the updated law for the weights of the neural network may have the form: ˆ ˆ T ∆=f( epp , e ) W hx (). (20) Wˆ = Γ h( As .)T . (24) The weight estimation error as It is possible to prove that the dynamic system is WW =* − Wˆ . stable under the control input (15) combining with the updated law (24). Then, the error of the function ∆fe(,)pp e and 4. Simulation ˆ ∆fe(,)pp e ) is determined: In order to confirm the effectiveness of the proposed RBFNN control, we carry out the ∆fe(,) e =∆ fe (,) e  −∆ feˆ (,) e pp pp pp (21) simulation work in MatLab/Simulink. The desired T ˆ  =W hx() +−εε Whx ()= Whx ()+ position (,xydd ) and rotational angle θd of the grasping object are planned in 5 order-polynomial = T The inputs of RBFNN are x[] eepp . trajectories as follows: From (12), with the control in (15) and using 345 xd =+−+0,54 0,4845 ttt 0,2907 0,0465 ; (21), the closed dynamics becomes 34 5 yd =+−+1,4 0,3841 tt 0,2304 0,0369 t ; ˆ 345 H ps=∆−∆−− fe(,) pp e fe (,) pp e  C p . s KAs s . θ =−+ 0,513ttt 0,1508 0,0241 .  d =∆−−f(,) epp e C p . s K s As . (22)  - The control parameters are: =W()h x −−Cps. s K As Ks = diag(15,15,15,15,15,15); 3.3 Update Law for NN Weighs Λ=diag(350,350,350) . The candidate of Lyapunov function is chosen as: - RBFNN has 6 inputs, 6 outputs, and 50 neural 11− nodes in hidden layer. V= sTT H s +Γ tr ( W .1 W ), (23) s 2 Initiating value of weights W0 = 6. T Center cj= [-2, 2]50. where H= AHp . Width bj =10; Γ=2 . TT TT + If define AJB = E then AJB E= I, and The simulation is carried out under some cases as follows: H= AHTT, N = AC then HN − 2 is skew- pp 4.1. Case 1: The differences between the estimated symmetric matrix. terms and the actual term are 10% of the actual Differentiation of (23) leads to term as follows: ∆=Hp10% H0 ; ∆= C pp 10% CG 00 ; ∆= 10% G. 54
6. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 050-058 We investigated 2 cases: the desired position of the mass center (,xydd ) in (Fig.4b), (Fig.5b) and rotate it to the desired a) Without compensation ∆fe(,)pp e orientation in Fig.6b. The time needed for exact We apply the control input (14) to the dynamics tracking to the trajectories are very short, only 0.5 of the dual arm robot – object (9). second for translational move in x- and y- directions, and 1 second for rotation about z- axis. It is possible to see the errors between the real trajectories and the desired trajectories in x- (Fig.4a), The simulation results display the effectiveness y-axis (Fig.5a) and rotation around z-axis (Fig.6a) are of the RBFNN controller in tracking trajectories of bigger in time. the object and its rotational angle to the desired trajectories in the case of uncertain level of dynamic b) With compensation ∆fe(,)pp e parameters is not so much, 10%. For the case, we increase the uncertain level to 20%. We apply the control input (15) to the dynamics of the dual arm robot – object (9). It is possible to realize that the dual-arm robot can move the object to 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ˆ ˆ (a) When ∆=fe(,pp e )0 (b) When ∆≠fe(,)0pp e Fig. 4. Response of move in x-axis of the object 2 2 1.9 1.9 1.8 1.8 1.7 1.7 1.6 1.6 1.5 1.5 1.4 1.4 1.3 1.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ∆=ˆ ˆ (a) When fe(,pp e )0 (b) When ∆≠fe(,)0pp e Fig. 5. Response of move in y-axis of the object 55
7. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 050-058 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ∆=ˆ ˆ (a) When fe(,pp e )0 (b) When ∆≠fe(,)0pp e Fig. 6. Response of rotational angle of the object 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ˆ ˆ (a) When ∆=fe(,pp e )0 (b) When ∆≠fe(,)0pp e Fig. 7. Response of move in x-axis of the object 2.1 2 2 1.9 1.9 1.8 1.8 1.7 1.7 1.6 1.6 1.5 1.5 1.4 1.4 1.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ˆ ∆≠ˆ (a) When ∆=fe(,pp e )0 (b) When fe(,)0pp e Fig. 8. Response of move in y-axis of the object 56
8. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 050-058 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 a) When ∆=ˆ fe(, e )0 ˆ pp (b) When ∆≠fe(,)0pp e Fig. 9. Response of rotational angle of the object 4.2. Case 2: The differences between the estimated been carried out to reconfirm the accuracy and the terms and the actual term are 20% of the actual effectiveness of the proposed controller. term as follows: Acknowledgements ∆=H20% H ; ∆= C 20% CG ; ∆= 20% G. p0 pp 00This research is funded by Hanoi University of We also investigated 2 cases: Science and Technology (HUST) under project number T2020-PC-021 a) Without compensation ∆fe(,) e . pp References It is easy to realize the divergence between the [1]. V. K. X. Yun, An approach to simultaneous control desired trajectories and actual trajectories of object of trajectory and interaction forces in dual-arm movement in x- axis (Fig.7a), in y-axis (Fig.8a) and configurations, IEEE Transactions on Robotics and rotation around z-axis (Fig.9a). Automation.7 (5) (1991) 618–625. b) With compensation ∆fe(,) e . pp [2]. S. S. W. Gueaieba, M. Bolica, Robust computationally efficient control of cooperative The desired trajectories and actual trajectories of closed-chain manipulators with uncertain dynamics, object in x- axis, y-axis and rotation around z-axis Automatica. 43 (2007) 842–851. respectively are shown in Fig.7b, Fig.8b and Fig.9b The simulation results show that the adaptive [3]. V. M. Tuneski AI, Dimirovski GM, Adaptive control controller that based on RBFNN works effectively in of multiple robots manipulation on a dynamical compensation the dynamic uncertainties of the whole environment, Proc Inst Mech Eng I: J Syst Control system. It is possible to conclude that when the Eng (2001) 385–404 uncertain level of dynamic parameters is in a small range, the responses of systems in tracking the [4]. B. R. R. Uzmay I I, Sarikaya H H, Application of desired trajectories are reasonable. robust and adaptive control techniques to cooperative manipulation, Control Eng Pract. (2004). 5. Conclusion In this paper, the adaptive control using Radial [5]. M. R. R. Mohajerpoor, A. Talebi, M. Noorhosseini, Basis Function Neural Network for controlling dual- R. Monfaredi, A robust adaptive hybrid force/position arm robotic system in manipulating a rectangle object control scheme of two planar manipulators handling tracking to the desired trajectories has been applied. an unknown object interacting with an environment, The overall dynamics of the manipulators-object Proceedings of the Institution of Mechanical system have been formulated based on the Euler- Engineers, Part I: Journal of Systems and Control Lagrangian principle. Radial basis function neural Engineering. (2011). network has been applied to compensate uncertainties of dynamic parameters. The adaptive algorithm has [6]. B. R. Uzmay I, and Sarikaya H, Application of robust been derived owning to Lyapunov stability principle and adaptive control techniques to cooperative to guarantee asymptotical convergence of the closed manipulation, Control Eng Pract. (2004) 139–148. dynamic system. Simulation work on MatLab has 57
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