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Hybrid Position/Force Controlof the SCORBOT-ER 4pc Manipulatorwith Neural Compensation of NonlinearitiesPiotr GierlakRzeszow University of Technology,Department of Applied Mechanics and Robotics8 Powsta nc ow Warszawy St., 35-959 Rzesz ow, P.plAbstract. The problem of the manipulator hybrid position/force con-trol is not trivial because the manipulator is a nonlinear object, whoseparameters may be unknown, variable and the working conditions arechangeable. The neural control system enables the manipulator to be-have correctly, even if the mathematical model of the control object isunknown. In this paper, the hybrid position/force controller with a neu-ral compensation of nonlinearities for the SCORBOT-ER 4pc roboticmanipulator is presented. The presented control law and adaptive lawguarantee practical stability of the closed-loop system in the sense ofLyapunov. The results of a numerical simulation are presented.Keywords: Neural Networks, Robotic Manipulator, Tracking Control,Force Control.1IntroductionRobotic manipulators are devices which find different applications in many do-mains of the economy. The requirements in relation to precision of motion andautonomy of manipulators are increasing as well as the tasks performed by themare more and more complex. In contemporary industrial applications it is desiredfor the manipulator to exert specified forces and move along a prescribed path.Manipulators are objects with nonlinear and uncertain dynamics, with unknownand variable parameters (masses, mass moments of inertia, friction coefficients),which operate in changeable conditions. Control of such complex systems is veryproblematic. The control system has to generate such control signals that willguarantee the execution of movement along a path with a suitable force andwith 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 approachesrequire precise knowledge about the mathematical model (the structure of mo-tion equations with coefficients) of the control object. Moreover, in such anapproach, parameters in the compensator have nominal values so the controlL. Rutkowski et al. (Eds.): ICAISC 2012, Part II, LNCS 7268, pp. 433441, 2012.c? Springer-Verlag Berlin Heidelberg 2012434P. Gierlaksystem acts without taking into account the changeable operating conditions. Inthe literature exists many variation of algorithms, in which parameters of themathematical model of manipulator are adapted 1,2. However these approachesdo not eliminate the problem with structural uncertainty of the model.In connection with the present difficulties, neural control techniques were de-veloped 3,4,5,6. In these methods the mathematical model is unnecessary. Thesetechniques are used in hybrid position/force controller. In works 7,8 such con-trollers have been presented. But in the first of the works only force normal to thecontact surface is taking into account, and in the second work some assumptionis hard to satisfy in practical applications, namely some stiffness matrix whichcharacterizes features of environment and allows to calculate contact forces, mustbe known.In previous author s paper only position controllers have been considered. Inpresent paper hybrid position/force neural controller is shown. This approachtakes into account all forces/moments which acts on the end-effector. Theseforces/moments are measured by sensor located in the end-effector.2Description of the SCORBOT-ER 4pc RoboticManipulatorThe SCORBOT-ER 4pc robotic manipulator is presented in Fig. 1. It is drivenby direct-current motors with gears and optical encoders. The manipulator has 5rotational kinematic pairs: the arm of the manipulator has 3 degrees of freedomwhereas the gripper has 2 degrees.a)A1q3yzxOBCOOO=d1OA=lAB=lBC=lCD=d1235q1q223u2u1u3q4D4u4q5u5b)contactsurface?FEFig.1. a) SCORBOT-ER 4pc robotic manipulator, b) schemeThe transformation from joint space to Cartesian space is given by the fol-lowing equationy = k(q) ,(1)Hybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator435where q Rnis a vector of generalized coordinates (angles of rotation of links),k(q) is a kinematics function, y Rmis a vector of a position/orientation of theend-effector (point D). Dynamical equations of motion of the analysed modelare in the following form 7, 9:M(q) q + C(q, q) q + F( q) + G(q) + d(t) = u + JTh(q) + F,(2)where M(q) Rnxnis an inertia matrix, C(q, q) Rnis a vector of centrifugaland Coriolis forces/moments, F( q) Rnis a friction vector, G(q) Rnisa gravity vector, d(t) Rnis a vector of disturbances bounded by | | d| | 0, u Rnis a control input vector, Jh(q) Rm1xnis a Jacobian matrixassociated with the contact surface geometry, Rm1is a vector of constrainingforces exerted normally on the contact surface (Lagrange multiplier), F Rnisa vector of forces/moments in joints, which come from forces/moments FE Rmapplied to the end-effector (except the constraining forces). The vector Fisgiven byF= JbT(q)FE,(3)where Jb(q) Rmxnis a geometric Jacobian in body 2. The Jacobian matrixJh(q) can be calculated in the following wayJh(q) =h(q)q,(4)where h(q) = 0 is an equation of the holonomic constraint, which describesthe contact surface. This equation reduces the number of degrees of freedom ton1= n m1, so the analysed system can be described by the reduced positionvariable 1 Rn17. The remainder of variables depend on 1in the followingway2= (1) ,(5)where 2 Rm1, and arise from the holonomic constraint. The vector ofgeneralized coordinates may be written as q = T1T2T. Let define the extendedJacobian 7L(1) =?In11?,(6)where In1 Rn1xn1is 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+V1(1,1)1+F(1)+G(1)+d(t) = u+JTh(1)+JbT(1)FE,(9)where V1(1,1) = M(1)L(1) + C(1,1)L(1). Pre-multiplying eq. (9) byLT(1) and taking into account that Jh(1)L(1) = 0, the reduced order dyna-mics is given by:M1+ V11+ F + G + d= LTu ,(10)where M = LTML, V1= LTV1, F = LTF, G = LTG, d= LT?d JbTFE?.436P. Gierlak3Neural Network Hybrid ControlThe aim of a hybrid position/force control is to follow a desired trajectory ofmotion 1d Rn1, and exert desired contact force d Rm1normally to thesurface. By defining a motion error e, a filtered motion error s, a force errorand an auxiliary signal 1as: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 bewritten in terms of the filtered motion error asM s = V1s + LTf(x) + LT?d JbTFE? LTu ,(15)with a nonlinear functionf(x) = ML 1+ V11+ F + G ,(16)where x =?eT eTT1dvT1dT1d?T. The mathematical structure of hybrid posi-tion/force controller has a form of 7u =f(x) + KDLs JTh?d+ KF? ,(17)where KDand KFare positive definite matrixes of position and force gain, is a robustifying term,f(x) approximates the function (16). This functionmay be approximated by the neural network. In this work a typical feedforwardneural network (Fig. 2b) with one hidden layer is assumed. The hidden layerwith sigmoidal neurons, is connected with an input layer by weights collectedin 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 weightsinitially are equal zero, and will be tuned during adaptation process. Such neuralnetwork is linear in the weights, and has the following description 3,4:f(x) = WT(x) + ,(18)with output from hidden layer (x) = S(DTx), where x is an input vector,S(. ) is a vector of neuron activation functions, is an estimation error boundedby | | | | 0. The matrix W is unknown, so an estimationW is used, and a mathematical description of a real neural network, whichapproximates function f(x) is given byf(x) =WT(x) .(19)Hybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator437S(.)1S (.)NS(.)2S(.)jy1y2yrx1x2xnDTWThiddenlayerinputsoutputs+u?1e?sKDMANIPULATOR+environment?If(x)?1dNEURALNETWORKCONTROLLERLKF?dJhTJbTFEuFvNEURALNETWORKCONTROLLERa)b)Fig.2. a) scheme of closed-loop system, b) neural networkSubstituting equations (18), (18) and (19) into (15), we obtained a descriptionof the closed-loop system (Fig. 2a) in terms of the filtered motion errorM s = LTKDLs V1s + LTWT(x) + LT?d+ JbTFE+ ?,(20)whereW = W W is an error of weight estimation.In order to derive an adaptation law of the weights and the robustifyingterm , the Lyapunov stability theory is applied. Define a Lyapunov functioncandidate, which is a quadratic form of the filtered motion error and the weightestimation error 4V = 1/2sTMs + 1/2tr?WT1WW?,(21)where Wis a diagonal design matrix, tr(. ) denotes trace of matrix. The timederivative of the function V along the solutions to (20) isV = sTLTKDLs + tr?WT?(x)sTLT+ 1WW?+sTLT?d+ JbTFE+ ?,(22)where a skew symmetric matrix property ofM 2V1was used. Defining anadaptive law of the weight estimation as 7W = W(x)sTLT kW| | Ls| |W ,(23)with k 0, and choosing robustifying term in the form = JbTFE,(24)function (22) may be written asV = sTLTKDLs + k| | Ls| | tr?WTW?+ sTLTd+ .(25)438P. GierlakFunctionV 0 if at least one of two the following conditions will be satisfieds= s : | | Ls| | (b + N+ kW2max/4)/KDmin= bs ,(26)W= W : | |W| |F Wmax/2 +?(b + N)/k + W2max/4 = bW ,(27)where KDminis the minimum singular value of KD, | | W| |F Wmax, | | . | |Fde-notes Frobenius norm. This result means, that the functionV is negative outsidea compact set defined by (26) and (27). According to a standard Lyapunov the-orem extension 10, both | | Ls| |and | |W| |Fare uniformly ultimately bounded tosets sand Wwith practical limits bsand bW. Adaptive law (23) guaranteesthat 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) interms of the filtered motion error, taking into account (17), (18), (19) and (24).After conversion, we obtainedJThKF+ I = ML s+V1s+KDLsWT(x)d= B?s, s,x,W?, (28)where all quantities on the right hand side are bounded. Pre-multiplying eq. (28)by Jhand computing the force error, we obtain: = KF+ I1JhJTh1JhB?s, s,x,W?,(29)where JhJThis nonsingular. This result means, that the force error is boundedand may be decreased by increasing the force gain KF.4Results of the SimulationIn 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 andparallel to xy plane. The end-effector was normal to the contact surface, movedon 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 bysolving the inverse kinematics problem.Problem of nonlinearities compensation have been decomposed on five simpletasks. For control of each link, a separate neural network with a single output wasused. Neural networks have correspondingly 11, 10, 10, 12 and 4 inputs. Neuralnetworks for links 1-4 had 15 neurons, and for link 5 had 9 neurons in the hiddenlayer. The input weights are randomly chosen from range . Thedesign matrixes were chosen as: = diag1,1,1,1, KD= diag1,1,1,1,1,W= 4I, where I is an identity matrix with suitable dimension, and moreoverKF= 3, k = 0. 1.In relation to the controller, only the results for the second link are presentedin this paper. At the beginning of the movement, the compensatory signalf2(x2)(Fig. 4b) generated by the compensator was not accurate, because the initialweight estimates were set to zero. The signal uPD2(Fig. 4b) generated by theHybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator439Fig.3. a) the desired patch of the end-effector, b) the desired force, c) the desiredtrajectory in a joint spaceFig.4. Control inputs for the second link: a) u2- the total control signal, 2- therobustifying term, uF2- the second element of the term JTh?d+ KF?, b) uPD2- thesecond element of the PD term KDLs,f2(x2) - the compensatory signal440P. GierlakPD controller takes majority meaning at the beginning, and then the influenceof the PD signal decreases during the movement, because the weight estimatesadaptation, and the meaning of the compensatory signal increases. The signaluF2(Fig. 4a), which results from “force” control, take an important part in thetotal control signal u2(Fig. 4a). The robustifying term 2(Fig. 4a) is associatedwith the presence of a dry friction force T = (Fig. 5a), where = 0. 2 is afriction coefficient. The force error (Fig. 5b) was bounded.Sometimes, the friction forces are neglected in theoretical considerations, andin practical applications are treated as disturbances. But in such approachescontrol quality is worse.In the initial movement phase motion errors have the highest values, so | | Ls| |(Fig. 6a) has the highest values. Afterwards, it is decreased during the adaptationof weight estimates (Fig. 6b). In accordance with the theory presented in thepaper, the weight estimates were bounded.Fig.5. a) exerted force normal and T = tangential to the contact surface, b) theforce errorFig.6. a) |Ls|, b) the weight estimates of neural network associated with the secondlinkHybrid Position/Force Control of the SCORBOT-ER 4pc Manipulator4415ConclusionAll 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=?1/nnk=1| | Ls| |2k= 0. 0363rad/s and =?1/nnk=12k=0. 0667N, where k is an index of sample, n is a number of sample. In order tocomparison neural hybrid controller with other control technique, adaptive hy-brid controller was tested in the same work conditions. Such controller is based onthe mathematical model of the manipulator. For testing of the adaptive controllerin case of modelling errors, model of dry friction in joints is omit in the controllerstructure. In this case we obtain s= 0. 0439rad/s and = 0. 0671N. Theseindices show, that the neural hybrid controller is better than the adaptive hybridcontroller if the model of control object is not well known.Acknowledgments. This research was realized within a framework of researchproject No. U-8313/DS/M
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