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Hybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator with Neural Compensation of Nonlinearities Piotr Gierlak Rzeszow University of Technology, Department of Applied Mechanics and Robotics 8Powstancow Warszawy St., 35-959 Rzeszow, Poland pgierlakprz.edu.pl Abstract. The problem of the manipulator hybrid position/force con- trol is not trivial because the manipulator is a nonlinear object, whose parameters may be unknown, variable and the working conditions are changeable. The neural control system enables the manipulator to be- have correctly, even if the mathematical model of the control object is unknown. In this paper, the hybrid position/force controller with a neu- ral compensation of nonlinearities for the SCORBOT-ER 4pc robotic manipulator is presented. The presented control law and adaptive law guarantee practical stability of the closed-loop system in the sense of Lyapunov. The results of a numerical simulation are presented. Keywords: Neural Networks, Robotic Manipulator, Tracking Control, Force Control. 1 Introduction Robotic manipulators are devices which find dierent applications in many do- mains of the economy. The requirements in relation to precision of motion and autonomy of manipulators are increasing as well as the tasks performed by them aremoreand more complex.In contemporaryindustrial applicationsit is desired for the manipulator to exert specified forces and move along a prescribed path. Manipulators are objects with nonlinear and uncertain dynamics, with unknown and variable parameters (masses, mass moments of inertia, friction coecients), which operate in changeable conditions. Control of such complex systems is very problematic. The control system has to generate such control signals that will guarantee the execution of movement along a path with a suitable force and with desired precision in spite of the changeable operating conditions. In the control systems of industrial manipulators, the computed torque me- thod 1,2 for non-linearity compensation is used. However, these approaches require precise knowledge about the mathematical model (the structure of mo- tion equations with coecients) of the control object. Moreover, in such an approach, parameters in the compensator have nominal values so the control L. Rutkowski et al. (Eds.): ICAISC 2012, Part II, LNCS 7268, pp. 433441, 2012. c Springer-Verlag Berlin Heidelberg 2012 434 P. Gierlak system acts without taking into account the changeable operating conditions. In the literature exists many variation of algorithms, in which parameters of the mathematicalmodelofmanipulatorareadapted1,2. Howevertheseapproaches do not eliminate the problem with structural uncertainty of the model. In connection with the present diculties, neural control techniques were de- veloped3,4,5,6.Inthesemethodsthemathematicalmodelisunnecessary.These techniques are used in hybrid position/force controller. In works 7,8 such con- trollershavebeen presented.But inthefirstoftheworksonlyforcenormaltothe contact surface is taking into account, and in the second work some assumption is hard to satisfy in practical applications, namely some stiness matrix which characterizesfeaturesofenvironmentandallowstocalculatecontactforces,must be known. In previous authors paper only position controllers have been considered. In present paper hybrid position/force neural controller is shown. This approach takes into account all forces/moments which acts on the end-eector. These forces/moments are measured by sensor located in the end-eector. 2 Description of the SCORBOT-ER 4pc Robotic Manipulator The SCORBOT-ER 4pc robotic manipulator is presented in Fig. 1. It is driven by direct-currentmotorswith gearsandopticalencoders.The manipulator has 5 rotational kinematic pairs: the arm of the manipulator has 3 degrees of freedom whereas the gripper has 2 degrees. a) A 1 q 3 y z x O B C O OO=d 1 OA=l AB=l BC=l CD=d 1 2 3 5 q 1 q 2 2 3 u 2 u 1 u 3 q 4 D 4 u 4 q 5 u 5 b) contact surface c108 F E Fig.1. a) SCORBOT-ER 4pc robotic manipulator, b) scheme The transformation from joint space to Cartesian space is given by the fol- lowing equation y = k(q) , (1) Hybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator 435 where q R n is a vector of generalized coordinates (angles of rotation of links), k(q) is a kinematics function, y R m is a vector of a position/orientation of the end-eector (point D). Dynamical equations of motion of the analysed model are in the following form 7, 9: M(q)q +C(q, q)q +F(q)+G(q)+ d (t)=u+J T h (q)+ F , (2) where M(q) R nxn is an inertia matrix, C(q, q) R n is a vector of centrifugal and Coriolis forces/moments, F(q) R n is a friction vector, G(q) R n is a gravity vector, d (t) R n is a vector of disturbances bounded by | d | 0, u R n is a control input vector, J h (q) R m 1 xn is a Jacobian matrix associatedwith the contactsurface geometry, R m 1 is a vectorofconstraining forces exerted normally on the contact surface (Lagrangemultiplier), F R n is a vectorofforces/momentsinjoints, which come fromforces/momentsF E R m applied to the end-eector (except the constraining forces). The vector F is given by F = J bT (q)F E , (3) where J b (q) R mxn is a geometric Jacobian in body 2. The Jacobian matrix J h (q) can be calculated in the following way J h (q)= h(q) q , (4) where h(q) = 0 is an equation of the holonomic constraint, which describes the contact surface. This equation reduces the number of degrees of freedom to n 1 = nm 1 , so the analysed system can be described by the reduced position variable 1 R n 1 7. The remainder of variables depend on 1 in the following way 2 = ( 1 ) , (5) where 2 R m 1 ,and arise from the holonomic constraint. The vector of generalized coordinates may be written as q = T 1 T 2 T . Let define the extended Jacobian 7 L( 1 )= bracketleftbigg I n 1 1 bracketrightbigg , (6) where I n 1 R n 1 xn 1 is an identity matrix. This allows to write the relations: q = L( 1 ) 1 , (7) q = L( 1 ) 1 + L( 1 ) 1 , (8) and write a reduced order dynamics in terms of 1 ,as: M( 1 )L( 1 ) 1 +V 1 ( 1 , 1 ) 1 +F( 1 )+G( 1 )+ d (t)=u+J T h ( 1 )+J bT ( 1 )F E , (9) where V 1 ( 1 , 1 )=M( 1 ) L( 1 )+C( 1 , 1 )L( 1 ). Pre-multiplying eq. (9) by L T ( 1 ) and taking into account that J h ( 1 )L( 1 ) = 0, the reduced order dyna- mics is given by: M 1 +V 1 1 +F +G+ d = L T u, (10) where M = L T ML, V 1 = L T V 1 , F = L T F, G = L T G, d = L T bracketleftbig d J bT F E bracketrightbig . 436 P. Gierlak 3 Neural Network Hybrid Control The aim of a hybrid position/force control is to follow a desired trajectory of motion 1d R n 1 , and exert desired contact force d R m 1 normally to the surface. By defining a motion error e , a filtered motion error s, a force error and an auxiliary signal 1 as: e = 1d 1 , (11) s =e +e , (12) = d , (13) 1 = 1d +e , (14) where is a positive diagonal design matrix, the dynamic equation (10) may be written in terms of the filtered motion error as M s = V 1 s+L T f(x)+L T bracketleftbig d J bT F E bracketrightbig L T u, (15) with a nonlinear function f(x)=ML 1 +V 1 1 +F +G, (16) where x = bracketleftBig e T e T T 1d v T 1d T 1d bracketrightBig T . The mathematical structure of hybrid posi- tion/force controller has a form of 7 u = f(x)+K D LsJ T h bracketleftBig d +K F bracketrightBig , (17) where K D and K F are positive definite matrixes of position and force gain, is a robustifying term, f(x) approximates the function (16). This function may be approximated by the neural network. In this work a typical feedforward neural network (Fig. 2b) with one hidden layer is assumed. The hidden layer with sigmoidal neurons, is connected with an input layer by weights collected in a matrix D, and with an output layer by weights collected in a matrix W. The input weights are randomly chosen and constant, but the output weights initially are equal zero, and will be tuned during adaptation process. Such neural network is linear in the weights, and has the following description 3,4: f(x)=W T (x)+, (18) with output from hidden layer (x)=S(D T x), where x is an input vector, S(.) is a vector of neuron activation functions, is an estimation error bounded by | 0. The matrix W is unknown, so an estimation W is used, and a mathematical description of a real neural network, which approximates function f(x)isgivenby f(x)= W T (x) . (19) Hybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator 437 S(.) 1 S (.) N S(.) 2 S(.) j y 1 y 2 y r x 1 x 2 x n D T W T hidden layer inputs outputs + + + u c113 1 e c113 s K D MANIPULATOR +environment c76I f(x) c113 1d NEURAL NETWORK CONTROLLER L K F c108 c108 d J h T J bT F E u F v NEURAL NETWORK CONTROLLER a) b) Fig.2. a) scheme of closed-loop system, b) neural network Substituting equations (18), (18) and (19) into (15), we obtained a description of the closed-loop system (Fig. 2a) in terms of the filtered motion error M s = L T K D LsV 1 s+L T W T (x)+L T bracketleftbig d +J bT F E + bracketrightbig , (20) where W = W W is an error of weight estimation. In order to derive an adaptation law of the weights and the robustifying term , the Lyapunov stability theory is applied. Define a Lyapunov function candidate, which is a quadratic form of the filtered motion error and the weight estimation error 4 V =1/2s T Ms+1/2tr parenleftBig W T 1 W W parenrightBig , (21) where W is a diagonal design matrix, tr(.) denotes trace of matrix. The time derivative of the function V along the solutions to (20) is V = s T L T K D Ls+tr bracketleftBig W T parenleftBig (x)s T L T + 1 W W parenrightBigbracketrightBig + +s T L T bracketleftbig d +J bT F E + bracketrightbig , (22) where a skew symmetric matrix property of M 2V 1 was used. Defining an adaptive law of the weight estimation as 7 W = W (x)s T L T k W |Ls| W, (23) with k0, and choosing robustifying term in the form = J bT F E , (24) function (22) may be written as V = s T L T K D Ls+k|Ls|tr parenleftBig W T W parenrightBig +s T L T d + . (25) 438 P. Gierlak Function V 0 if at least one of two the following conditions will be satisfied s = s : |Ls| (b+ N +kW 2 max /4)/K Dmin = b s , (26) W = W : | W| F W max /2+ radicalbig (b+ N )/k+W 2 max /4=b W , (27) where K Dmin is the minimum singular value of K D , |W| F W max , |.| F de- notes Frobenius norm. This result means, that the function V is negative outside a compact set defined by (26) and (27). According to a standard Lyapunov the- orem extension 10, both |Ls| and | W| F are uniformly ultimately bounded to sets s and W with practical limits b s and b W . Adaptive law (23) guarantees that weight estimates will be bound without persistency of the excitation con- dition. In order to prove, that force error is bounded, we write equation (9) in terms of the filtered motion error, taking into account (17), (18), (19) and (24). After conversion, we obtained J T h K F +I = MLs+V 1 s+K D Ls W T (x) d = B parenleftBig s, s,x, W parenrightBig , (28) where all quantities on the right hand side are bounded. Pre-multiplying eq. (28) by J h and computing the force error, we obtain: =K F +I 1 J h J T h 1 J h B parenleftBig s, s,x, W parenrightBig , (29) where J h J T h is nonsingular. This result means, that the force error is bounded and may be decreased by increasing the force gain K F . 4 Results of the Simulation In order to confirm the behaviour of the proposed hybrid control system, a simu- lation was performed. We assumed, that the contact surface was flat, rough and parallel to xy plane. The end-eector was normal to the contact surface, moved on that surface on a desired circular path (Fig. 3a) and exerted prescribed force (Fig. 3b). The desired trajectory in a joint space (Fig. 3c) was obtained by solving the inverse kinematics problem. Problem of nonlinearities compensation have been decomposed on five simple tasks.Forcontrolofeachlink,aseparateneuralnetworkwithasingleoutputwas used. Neural networks have correspondingly 11, 10, 10, 12 and 4 inputs. Neural networks for links 1-4 had 15 neurons, and for link 5 had 9 neurons in the hidden layer. The input weights are randomly chosen from range .The design matrixes were chosen as: = diag1,1,1,1, K D = diag1,1,1,1,1, W =4I,whereI is an identity matrix with suitable dimension, and moreover K F =3,k =0.1. In relation to the controller, only the results for the second link are presented in this paper. At the beginning of the movement,the compensatorysignal f 2 (x 2 ) (Fig. 4b) generated by the compensator was not accurate, because the initial weight estimates were set to zero. The signal u PD2 (Fig. 4b) generated by the Hybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator 439 Fig.3. a) the desired patch of the end-eector, b) the desired force, c) the desired trajectory in a joint space Fig.4. Control inputs for the second link: a) u 2 - the total control signal, 2 -the robustifying term, u F2 - the second element of the term J T h bracketleftBig d +K F bracketrightBig ,b)u PD2 -the second element of the PD term K D Ls, f 2 (x 2 ) - the compensatory signal 440 P. Gierlak PD controller takes majority meaning at the beginning, and then the influence of the PD signal decreases during the movement, because the weight estimates adaptation, and the meaning of the compensatory signal increases. The signal u F2 (Fig. 4a), which results from “force” control, take an important part in the total control signal u 2 (Fig. 4a). The robustifying term 2 (Fig. 4a) is associated with the presence of a dry friction force T = (Fig. 5a), where =0.2isa friction coecient. The force error (Fig. 5b) was bounded. Sometimes, the friction forces are neglected in theoretical considerations, and in practical applications are treated as disturbances. But in such approaches control quality is worse. In the initial movement phase motion errors have the highest values, so |Ls| (Fig.6a)hasthehighestvalues.Afterwards,itisdecreasedduringtheadaptation of weight estimates (Fig. 6b). In accordance with the theory presented in the paper, the weight estimates were bounded. Fig.5. a) exerted force normal and T = tangential to the contact surface, b) the force error Fig.6. a) |Ls|, b) the weight estimates of neural network associated with the second link Hybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator 441 5Conclusion All signals in the control system were bounded, so control system was stable. Moreover, the motion errors decreased during movement. For numerical evalua- tion of the hybrid control system quality, we used a root mean square of errors, defined as: s = radicalbig 1/n n k=1 |Ls| 2 k =0.0363rad/sand = radicalBig 1/n n k=1 2 k = 0.0667N, where k is an index of sample, n is a number of sample. In order to comparison neural hybrid controller with other control technique, adaptive hy- bridcontrollerwastestedinthesameworkconditions.Suchcontrollerisbasedon themathematicalmodelofthemanipulator.Fortestingoftheadaptivecontroller in case ofmodelling errors,model ofdry friction in joints is omit in the controller structure. In this case we obtain s =0.0439rad/sand =0.0671N. These indices show,that the neural hybrid controlleris better than the adaptive hybrid controller if the model of control object is not well known. Acknowledgments. This research was realized within a framework of research project No. U-8313/DS/M. Apparatus/equipment purchased in the project No POPW.01.03.00-18-012/ 09 from the structural funds, the Development of Eastern Poland Operational Programme co-financed by the European Union, the European Regional Devel- opment Fund. References 1. Canudas de Wit, C., Siciliano, B., Bastin, G.: Theory of Robot Control. Springer, London (1996) 2. Tchon, K., Mazur, A., Duleba, I., Hossa, R., Muszynski, R.: Manipulators and mobile robots: models, movenent planning and control. AOW PLJ, Warsaw (2000) (in polish) 3. Gierlak, P., Zylski, W.: Tracking control of manipulator. In: Methods and Models in Automation and Robotics, vol. 14, part 1 (2009), ifac- 4. 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