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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 8 Powsta nc ow Warszawy St 35 959 Rzesz ow Poland pgierlak prz 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 1Introduction Robotic manipulators are devices which fi nd diff erent 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 are more and more complex In contemporary industrial applications it is desired for the manipulator to exert specifi ed 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 coeffi cients 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 coeffi cients 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 433 441 2012 c Springer Verlag Berlin Heidelberg 2012 434P Gierlak system acts without taking into account the changeable operating conditions In the literature exists many variation of algorithms in which parameters of the mathematical model of manipulator are adapted 1 2 However these approaches do not eliminate the problem with structural uncertainty of the model In connection with the present diffi culties neural control techniques were de veloped 3 4 5 6 In these methods the mathematical model is unnecessary These techniques are used in hybrid position force controller In works 7 8 such con trollers have been presented But in the fi rst of the works only force normal to the contact surface is taking into account and in the second work some assumption is hard to satisfy in practical applications namely some stiff ness matrix which characterizes features of environment and allows to calculate contact forces must be known In previous author s 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 eff ector These forces moments are measured by sensor located in the end eff ector 2Description of the SCORBOT ER 4pc Robotic Manipulator The SCORBOT ER 4pc robotic manipulator is presented in Fig 1 It is driven by direct current motors with gears and optical encoders 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 q3 y z x O B C O OO d1 O A l AB l BC l CD d 1 2 3 5 q1 q2 2 3 u2 u1 u3 q4 D 4 u4 q5 u5 b contact surface FE 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 Manipulator435 where 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 the end eff ector 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 JT h q F 2 where M q Rnxnis an inertia matrix C q q Rnis a vector of centrifugal and Coriolis forces moments F q Rnis a friction vector G q Rnis a gravity vector d t Rnis a vector of disturbances bounded by d 0 u Rnis a control input vector Jh q Rm1xnis a Jacobian matrix associated with the contact surface geometry Rm1is a vector of constraining forces exerted normally on the contact surface Lagrange multiplier F Rnis a vector of forces moments in joints which come from forces moments FE Rm applied to the end eff ector except the constraining forces The vector Fis given by F JbT q FE 3 where Jb q Rmxnis a geometric Jacobian in body 2 The Jacobian matrix Jh q can be calculated in the following way Jh 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 n1 n m1 so the analysed system can be described by the reduced position variable 1 Rn1 7 The remainder of variables depend on 1in the following way 2 1 5 where 2 Rm1 and arise from the holonomic constraint The vector of generalized coordinates may be written as q T 1 T2 T Let defi ne the extended Jacobian 7 L 1 In1 1 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 JT h 1 J bT 1 FE 9 where V1 1 1 M 1 L 1 C 1 1 L 1 Pre multiplying eq 9 by LT 1 and taking into account that Jh 1 L 1 0 the reduced order dyna mics is given by M 1 V1 1 F G d LTu 10 where M LTML V1 LTV1 F LTF G LTG d LT d JbTFE 436P Gierlak 3Neural Network Hybrid Control The aim of a hybrid position force control is to follow a desired trajectory of motion 1d Rn1 and exert desired contact force d Rm1normally to the surface By defi ning a motion error e a fi ltered motion error s a force error and 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 be written in terms of the fi ltered motion error as M s V1s LTf x LT d JbTFE LTu 15 with a nonlinear function f x ML 1 V1 1 F G 16 where x eT eT T 1dv T 1d T 1d T The mathematical structure of hybrid posi tion force controller has a form of 7 u f x KDLs JT h d KF 17 where KDand KF are positive defi nite 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 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 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 is given by f x WT x 19 Hybrid Position Force Control of the SCORBOT ER 4pc Manipulator437 S 1 S N S 2 S j y1 y2 yr x1 x2 xn D T W T hidden layer inputsoutputs u 1 e s KD MANIPULATOR environment I f x 1d NEURAL NETWORK CONTROLLER L KF d Jh T J bT FE uF 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 fi ltered motion error M s LTKDLs V1s LT WT x LT d JbTFE 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 Defi ne a Lyapunov function candidate which is a quadratic form of the fi ltered motion error and the weight estimation error 4 V 1 2sTMs 1 2tr WT 1 W W 21 where Wis a diagonal design matrix tr denotes trace of matrix The time derivative of the function V along the solutions to 20 is V sTLTKDLs tr WT x sTLT 1 W W sTLT d JbTFE 22 where a skew symmetric matrix property of M 2V1 was used Defi ning an adaptive law of the weight estimation as 7 W W x sTLT k W Ls W 23 with k 0 and choosing robustifying term in the form JbTFE 24 function 22 may be written as V sTLTKDLs k Ls tr WT W sTLT d 25 438P Gierlak Function V 0 if at least one of two the following conditions will be satisfi ed s s Ls b N kW2 max 4 KDmin bs 26 W W W F Wmax 2 b N k W2 max 4 bW 27 where KDminis the minimum singular value of KD W F Wmax Fde notes Frobenius norm This result means that the function V is negative outside a compact set defi ned by 26 and 27 According to a standard Lyapunov the orem extension 10 both Ls and W Fare uniformly ultimately bounded to sets sand Wwith practical limits bsand bW 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 fi ltered motion error taking into account 17 18 19 and 24 After conversion we obtained JT h KF I ML s V1s KDLs WT 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 I 1 JhJT h 1JhB s s x W 29 where JhJT h is nonsingular This result means that the force error is bounded and may be decreased by increasing the force gain KF 4Results of the Simulation In order to confi rm the behaviour of the proposed hybrid control system a simu lation was performed We assumed that the contact surface was fl at rough and parallel to xy plane The end eff ector 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 fi ve simple tasks For control of each link a separate neural network with a single output was 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 diag 1 1 1 1 KD diag 1 1 1 1 1 W 4I where I is an identity matrix with suitable dimension and moreover KF 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 compensatory signal f2 x2 Fig 4b generated by the compensator was not accurate because the initial weight estimates were set to zero The signal uPD2 Fig 4b generated by the Hybrid Position Force Control of the SCORBOT ER 4pc Manipulator439 Fig 3 a the desired patch of the end eff ector b the desired force c the desired trajectory in a joint space Fig 4 Control inputs for the second link a u2 the total control signal 2 the robustifying term uF2 the second element of the term JT h d KF b uPD2 the second element of the PD term KDLs f2 x2 the compensatory signal 440P Gierlak PD controller takes majority meaning at the beginning and then the infl uence of the PD signal decreases during the movement because the weight estimates adaptation and the meaning of the compensatory signal increases The signal uF2 Fig 4a which results from force control take an important part in the total control signal u2 Fig 4a The robustifying term 2 Fig 4a is associated with the presence of a dry friction force T Fig 5a where 0 2 is a friction coeffi cient 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 has the highest values Afterwards it is decreased during the adaptation 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 Manipulator441 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 defi ned as s 1 n n k 1 Ls 2 k 0 0363 rad s and 1 n n k 1 2 k 0 0667 N 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 brid controller was tested in the same work conditions Such controller is based on the mathematical model of the manipulator For testing of the adaptive controller in case of modelling errors model of dry friction in joints is omit in the controller structure In this case we obtain s 0 0439 rad s and 0 0671 N These indices show that the neural hybrid controller is better than the adaptive hybrid controller if the model of control object is not well known Acknowledgments This