Neural Networks for Advanced Control of Robot Manipulators

Transcription

1 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 13, NO. 2, MARCH Neural Networks for Advanced Control of Robot Manipulators H. Daniel Patiño, Member, IEEE, Ricardo Carelli, Senior Member, IEEE, and Benjamín R. Kuchen, Member, IEEE Abstract This paper presents an approach and a systematic design methodology to adaptive motion control based on neural networks (NNs) for high-performance robot manipulators, for which stability conditions and performance evaluation are given. The neurocontroller includes a linear combination of a set of off-line trained NNs (bank of fixed neural networks), and an update law of the linear combination coefficients to adjust robot dynamics and payload uncertain parameters. A procedure is presented to select the learning conditions for each NN in the bank. The proposed scheme, based on fixed NNs, is computationally more efficient than the case of using the learning capabilities of the neural network to be adapted, as that used in feedback architectures that need to propagate back control errors through the model (or network model) to adjust the neurocontroller. A practical stability result for the neurocontrol system is given. That is, we prove that the control error converges asymptotically to a neighborhood of zero, whose size is evaluated and depends on the approximation error of the NN bank and the design parameters of the controller. In addition, a robust adaptive controller to NN learning errors is proposed, using a sign or saturation switching function in the control law, which leads to global asymptotic stability and zero convergence of control errors. Simulation results showing the practical feasibility and performance of the proposed approach to robotics are given. Index Terms Adaptive control, feedforward neural nets, neurocontrollers, nonlinear systems, robot manipulators, stability. I. INTRODUCTION IN recent years, much attention has been paid to neural-network (NN)-based controllers. The nonlinear mapping and learning properties of NNs are key factors for their use in the control field. These types of controllers take advantage of the capability of an NN for learning nonlinear functions and of the massive parallel computation, required in the implementation of advanced control algorithms. This learning capability of NNs is used to make the controller learn a certain function, highly nonlinear, representing the direct dynamics, inverse dynamics or any other characteristics of the system. This is usually performed during a normally long training period when commissioning the controller in a supervised or unsupervised manner [30]. If the learning capability of the NN is not switched off after the training phase, once the controller is commissioned, the NN-based controller works as an adaptive controller. Manuscript received December 7, 1999; revised January 10, This work was supported in part by ANPCyT (National Agency for Promotion of Science and Technology), CONICET (National Research Council), and Universidad Nacional de San Juan, Argentina. The authors are the Instituto de Automática, Facultad de Ingneniería, Universidad Nacional de San Juan, 5400 San Juan, Argentina ( dpatino@inaut.edu.ar). Publisher Item Identifier S (02)01809-X. This paper deals a neural network-based controller for motion dynamic control of robot manipulators. The dynamical behavior of a rigid manipulator can be characterized by a system of highly coupled and nonlinear differential equations. The nonlinear effects are emphasized for robots working at high speeds direct drive motors or low ratio gear transmissions. Advanced control strategies [4], [36], and [38], generally based on an exact cancellation of the nonlinear dynamics have to be used for these types of robots. The uncertainties on the robot dynamic parameters, such as inertia and payload conditions, have motivated the design of adaptive controllers [17], [25], [37]. This type of controller is designed by assuming an exact knowledge about the structure of the robot dynamic model out including dynamic effects, such as complex nonlinear frictions, elasticity of joints and links, backlash and so on, excepting in cases as in [39] in which the control scheme is robust to arbitrary uncertain inertia parameters of the manipulator, and electrical parameters of the actuators. As mentioned previously, NNs have the capability of learning a nonlinear model out an a priori knowledge of their parameters and structure [22], [24], through experimental data. These attributes have been used to design nonadaptive and adaptive controllers for robot manipulators [11], [15], [16], [26], particularly in feedback adaptive structures in [7], [19], [23], and [43]. However, most of proposed schemes in the literature have been presented out stability analysis and out a rigorous analysis that evaluates the influence of NN learning errors on control errors as an explicit function in terms of the NN learning error and design parameters. Excepting in certain cases as in [2], [3], [27], and [28] in the robotic and [29] and [34] in the control fields. The book [10] provides a good review of neural networks for robot control, which includes proofs of stability properties. This paper presents a feedback adaptive neurocontroller for robots which combines feedforward neural networks adaptive and robust control techniques. An inverse-dynamics modelbased control structure is considered [27]. The neurocontroller is based on a set of fixed NNs, that is, a bank of NNs trained off-line, each NN representing the robot inverse model for a specified and adequate payload condition. This manner of representing the inverse model of robot manipulators has been previously proposed by the authors in [2], [3], and [27]. In the present paper we also propose a procedure to select the learning conditions for the set of NNs. A stable controller-parameter adjustment mechanism, which is determined using the Lyapunov theory, is constructed to adjust the coefficients of a linear combination of NN outputs to uncertain dynamic parameters, such as link inertia or payloads. This approach allows to minimize the on-line computing time because the number of parameters /02$ IEEE

2 344 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 13, NO. 2, MARCH 2002 to be adapted is small. Hence, the adaptation to changes in robot parameters is faster than for the case the learning capability of full NN is used for the adaptation task. Due to the NN learning error, a practical stability result for the neurocontrol systems is given. That is, the control error converges asymptotically to a neighborhood of zero, whose size is evaluated and depends on the approximation error degree of the NN bank.the analysis also allows for a control error evaluation as an explicit function of the NN learning errors and design parameters. In all design and analysis of NN-based control systems, it is important to take into account the NN learning error and its influence on the control error of the plant. In addition, the neurocontroller is modified to build a robust adaptive controller to NN learning errors, which leads to reach a global asymptotic stability and zero convergence of control errors. In order to verify the stability properties and performance of proposed control schemes, simulation studies were carried out using a PUMA-560 model robot. Simulation results showing the practical feasibility and performance of the proposed approach to robotics are given. This paper is organized as follows. Section II presents the modeling of the inverse robot dynamics by a bank of NNs and a procedure for selecting the learning conditions. In Section III, the problem of motion adaptive control of rigid robot manipulators is stated and formulated, and the proposed neural network-based feedback adaptive and feedback robust adaptive controllers are given. Section IV describes the used NN structure, and a method for training is proposed. Section V presents the stability results for NN-based controllers, including the NN learning errors. Finally, Section VI shows the simulation results for the tracking adaptive control problem. Conclusions are given in Section VI. II. MODELING THE INVERSE ROBOT DYNAMICS BY A SET OF NNs A. Dynamic of Rigid Robots The joint-space (robot coordinate) inverse dynamics of an -link rigid robot manipulator can be written as denotes the vector of joint displacements in robot coordinates, is the vector of applied joint torques or forces (vector of control inputs in robot coordinates), is the symmetric and positive definite manipulator inertia matrix, is the vector of centripetal and Coriolis torques, is the vector of gravitational torques and is a vector representing the dynamic effects as nonlinear frictions, small joint and link elasticities, backlash and bounded torque disturbances. This dynamic model can be expressed as a linear function [18] of a suitably selected set of robot and load parameters is a function matrix, is a function vector of robot parameters which may be unknown or uncertain, is a vector representing dynamics independent of unknown parameters, is the regressor, and (1) (2) is an extension of the unknown parameter vector. The manipulator model described by (1) and (2) is considered to be nonredundant. It is also assumed that the robot is equipped joint position and velocity sensors at their joints. Some fundamental properties of the robot dynamic model used in the stability analysis can be seen in [25] and [36]. B. NN-Based Inverse Robot Dynamics In [5], [9], [12] and other works it is shown that any wellbehaved nonlinear function can be approximated to any desired accuracy by a two-layer NN a sufficiently large number of neurons, over a compact domain of a finite dimensionally normed space. In [24], it is shown that neural networks can be used for both identification and control of nonlinear dynamical systems. In this work, the nonlinear mapping capability of NNs is used for inverse robot model in order to develop an adaptive control scheme for robot manipulators. Consider a set of feedforward neural networks, each one representing the inverse robot dynamics for a determined payload condition characterized by a value of the parameters. Taking into account the linear parameterization property of the robot model and assuming that each element of the NNs represents the inverse robot dynamics exactly for each payload condition, it can be written and. Now, let us consider a particular robot payload condition, characterized by a value of an unknown parameter vector, and assume it is expressed as a linear combination of the values Hence, as the last component of vector is equal to one it must be verified that. Then, the inverse robot dynamics for a particular payload condition characterized by can be expressed as Now, by substituting (3) into (5), the inverse robot dynamics can be interpolated for any payload condition by a NN bank as Equation (4) can be written in matricial form as (3) (4) (5) (6) (7)

3 PATIÑO et al.: NEURAL NETWORKS FOR ADVANCED CONTROL OF ROBOT MANIPULATORS 345 or (8) When analyzing (8), if and is nonsingular, there is a unique solution for, given by.now,ifthe columns of do not form a basis, because or the training conditions have been chosen in such a way as to make some columns of be linearly dependent on the rest, then only an approximative value can be found for any using any fitting algorithm such as least squares. Finally, it is the case when the columns of form a basis for which the whole space can be generated and a set of parameters can be found for all. That is, the inverse robot dynamics can be approximated by the set of NN (bank of NNs) for any uncertainty value, such as payload and link inertia, parameterized in. Therefore, this bank of NNs, each one under off-line training for an adequate load state, models the inverse robot dynamics in a sort of look-up table approach, which lets us make interpolation between the models corresponding to particular load conditions at any payload condition. The following examples illustrate the above concepts. Consider a case for the second and third joints (shoulder and elbow) of a PUMA-560 robot. Assuming uncertainty for the payload grasping-center-point (gcp), a robot parametrization can be accomplished a three-component vector,,, the position of the mass center of load on the -axis of the last link-fixed system,. Fig. 1(a) shows the set of vectors in the space, assuming admissible values of between 0.05 m and 0.2 m. For two learned conditions and,any vector can be approximated as on the plane generated by. For three learned conditions any vector can be exactly represented. Now, assuming payload uncertainty, a parametrization of the robot can be achieved a two-component vector,,, the payload mass. Fig. 1(b) shows the set of vectors in space, assuming admissible values for ranging from 0 Kg to 20 Kg. In this case, for two learned conditions and,any vector can be exactly represented. In order to select the learning conditions, the following criterium associated to the geometrical surface of vectors is proposed. 1) Choose as the maximum length vector on the parameters surface. 2) Choose as the point on the surface having maximum distance to the subspace generated by. 3) Choose as the point on the surface having maximum distance to the subspace generated by. 4) Continue likewise for choosing. This criterium maximizes the angle between vectors, optimizing the linear independency of vectors. Remark 1: To model the inverse robot dynamics by neural networks, it is needed to train a bank of NNs, each one for a different payload condition. The learning error is defined as (9) Fig. 1. (a) Representation of ( ; ; ) space. (b) Representation of ( ; ) space. (a) (b). Then, the approximation error can be bounded by, being the learning error of each NN of the bank which depends on how well and accurately the NNs were trained and on the absolute values of parameters. III. NN-BASED MOTION ADAPTIVE CONTROL A. Statement of the Robot Motion Adaptive Control Problem The motion adaptive control problem of robot manipulators can be formulated as follows. For a desired motion trajectory, specified in joint-space and characterized by, let us consider the robot model as given in (1) or (2), the dynamic parameter vector of the manipulator, the payload is constant but unknown, and the regressor known. Then, under conditions of uncertainty in these parameters, it should be found: 1) an adaptive control law, thas is, the vector of torques or forces to be applied at each articulation; and 2) a parameter update law, which estimates the vector of parameters uncertainty or unknown, in such a way that the tracking motion objective is asymptotically attained, i.e., as. If this objective is verified at all times, it means that the

4 346 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 13, NO. 2, MARCH 2002 manipulator is displaced asymptotically, thus accomplishing the desired motion trajectory. As a solution to the above problem, an NN-based motion adaptive controller is presented, which was originally inspired on the controller proposed in [17]. It consists of a computed torque plus compensation control law structure that avoids acceleration measurement. In this work, we present two feedback adaptive controllers for robot motion control, which combine a bank of NNs trained off-line adaptive and robust control techniques. The control objective is accomplished by reaching a sufficiently small tracking motion error, keeping bounded all others internal signals of the controller. In addition, for the other NN-based motion robust adaptive controllers proposed, the tracking of the desired motion function is attained asymptotically all other signals kept bounded. B. Neural Network-Based Feedback Adaptive Controllers 1) Feedback Adaptive Controller: Considering the inverse robot dynamics described in the Section II, and the control law (10) have the same functional form as, respectively, estimated parameters, and is a feedback PD control action plus given by diagonal positive definite gain ma- is an auxiliary signal defined by trices; and (11) From a practical point of view, can be implemented as (12) which does not require to measure the joint acceleration. Now, using the robot parametrization property of (2), (10) can be written as Considering the inverse robot dynamics modeling of (5), the NN-based feedback control law can be expressed by (13). An adequate structure for the NN bank,, as well as the learning method used during off-line training are explained in detail in Section V. In order to obtain the vector in (13), the following parameter update law is used, which is a -modification-type law [14] (14) is the same vector as defined in (13), and are positive definite gain matrices. Considering (9) and, (13) can be written as (15) is the approximation error-nn bank learning errorwhose th component is which is bounded by. 2) NN-Based Feedback Robust Adaptive Control: Here a robust adaptive controller for the NN learning error is proposed, including a sign or a saturation switching function included in the control law (9), which leads to a globally asymptotic stability and zero convergence of the control errors. This NN-based motion robust adaptive controller can be classified into the group of variable structure robust controllers [13]. The proposed feedback robust adaptive control is made up of the original control law plus a nonlinear term, a sign or a saturation switching function or (16), a constant diagonal matrix; and, the sign function. By considering the inverse robot dynamics modeling as given by (5), the NN-based feedback robust adaptive control law can be expressed as (17), and. To find in (17), the following parameter update law is used, which is a gradient-type law: (18) is the same function as defined for (13), and is a diagonal positive definite gain matrix. Likewise, the control law for this controller can be expressed in terms of the NN-learning error as (19) is the approximation error of the NN bank, whose th component is given by, and bounded by. The structure of the NN-based feedback adaptive control system is shown in Fig. 2. IV. NN STRUCTURES AND TRAINING STRATEGY This section describes the NN structure that allows to implement functions of the control law of (10) and (17), as well as to find a procedure for training each NN of the bank. Each net in the NN bank of Fig. 2 has the structure shown in Fig. 3 both blocks represent NNs. The upper block models the nonlinear elements of matrix, and the lower one models

5 PATIÑO et al.: NEURAL NETWORKS FOR ADVANCED CONTROL OF ROBOT MANIPULATORS 347 Fig. 2. Feedback robust adaptive control system. Fig. 5. Learning phase II: Feedback error learning. Fig. 6. Learning phase III: Specialized learning structure. ture, by equating the robot inverse dynamics to the there results output, Fig. 3. Neural-network structure. Then, through the definition for the signal as stated in (11) Fig. 4. Learning phase I: Generalized learning. Here, when the learning error tends to zero, tend to zero too, and also does as well. This means that is an appropriate training signal for this learning structure. The backpropagation learning rule is used to adjust any parameter of the NN. defines the connection weight from neuron th to neuron th of the th layer. The backpropagation rule is based on the gradient descendent for some performance function respect to such as the function (see (10) and (17)). The outputs of each NNs are the components of matrix of (10) and (17). The proposed learning strategy for training one NN of the bank, is made up of three phases. The first learning phase the general learning structure given in [8] and shown in Fig. 4 is used to obtain on-line (or off-line) first approximation to the inverse dynamic model (or theoretical model) of the robot. In the second phase, the feedback error learning structure [16] is applied (as shown in Fig. 5) the signals are fed to the inputs and, respectively. The following analysis justifies this learning procedure. The control action is stated by,. Then,. Now, when tends to tends to zero, and the approximates the robot inverse dynamics. This means that is an appropriate training signal. In the third phase, the specialized learning architecture [8] is considered, which allows a fine tuning input signals in the operative domain, Fig. 6. Here, the signals and are fed to the inputs and, respectively. For this struc-, the step size, is a suitably chosen constant, and denotes the nominal value of for which the gradient is computed. In this work, the performance criterion is defined as, for both structures described above. Then, the gradient of in the parameter space is obtained by is the state of the th processing element, is the th element output of the

6 348 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 13, NO. 2, MARCH 2002 layer, and is the sigmoid function for neurons in the hidden layers as well as the linear function for the output layer. For the particular NN structure of Fig. 3, is evaluated as have the same functional form as, respectively, as regards the estimated parameters. Then, the closed-loop equation in terms of the error model is and can be computed at every instant using conventional static backpropagation. V. STABILITY ANALYSIS In this section, a practical stability analysis for the neurocontrol systems is given, considering [29] and [3] as main references. Theorem 1 (Stability With Controller III-B1): Suppose that the control law is given by (13) and the parameter updating law is given by (14). Then, the entire system denoted by (24) By substituting from (24) into (21), and considering the robot model property, (see [25]) as well as the parameter update law (14), it can be obtained Then, substituting becomes (25) into (25) and assuming ( is a positive constant value) (25) (20) has strong practical stability (see [20]) under a certain perturbation as. Proof: System (20) under perturbation (closed-loop equation) is described by Recalling that bilinear terms can be expressed as (26) (21) Consider the following positive definite function: (22) in which is a diagonal positive definite matrix and, is the optimal parameter vector as defined before. We need to show that the derivative of along the solutions of the system (21) satisfies the condition of the Lemma given in Appendix I. Define, denotes the minimum singular value. can be bounded both below and upper by for some, (26) can be rewritten as and. Equivalently, (27) can be expressed as (27) (28). The time derivative of the system (21) and (23), evaluated along the trajectories of consider. Let us choose, i.e.,, and (29) for some. Equation (28) turns to be for all and external to, since By considering the robot model structure (1), the control law (10) can be rewritten as and for all and all outside. Then, applying the Lemma presented in Appendix I, (28) implies that (21) possesses strong practical stability.

7 PATIÑO et al.: NEURAL NETWORKS FOR ADVANCED CONTROL OF ROBOT MANIPULATORS 349 Remark 2: Theorem 1 implies that all trajectories of system (21) will be ultimately in, which implies that and for all. If the learning error is zero, is zero as well. Then, as is definite positive, from (28) there results that,as. Remark 3: From a practical point of view it is important to calculate the boundaries for the motion control errors in terms of the NN learning errors,, and the design parameters. Integration of (28) on can be bounded by Proof: Consider the control law equation (16), written as (17). By substituting for and combining it (1), the closed-loop system equation can be obtained as (33) on as can be determined as, and the supremum of the bound Now, let us consider the following nonnegative function: (34) (29) Now, by taking into account (23), both and can be evaluated as explicit functions of the NN learning error and design parameter as In addition, from (12) considering the following transfer function matrices: The time derivative of evaluated along the trajectories of (33) can be substituted from (33), considering the robot model property for an appropriate definition of (see [25]), and the parameter update law equation, (18), it results (35) The second and third terms of (35) can be expressed as (30) and evaluating the induced norm of linear mappings and as (see [6]), can be bounded by (31) By choosing the gain diagonal matrix have its elements verify large enough so as to Then, taking into account (29) and (31), the motion control errors can be evaluated as an explicit function of the NN learning error and design parameters as is a safety factor. Then, (34) can be rewritten as (32) Note that the control errors,, are bounded by an explicit function of the learning error and the design parameters, as. Theorem 2 (Stability With Controller III-B2): Consider the control law equation (16), the parameter updating law (18), in closed loop the robot manipulator equation (1). Then, the following holds: a b c d as (36) From (34) and (36), it results that (see [20]). Also, by time-integrating the (36), it is concluded that. Now, from (11) (37) Since, and (35) representing an exponentially stable and strictly proper linear system, from Theorem 6 [6, pp. 59] it is concluded that and as. Assuming that, and since are bounded, then and are bounded too. Besides, there exists a positive constant such that, so that exists (see [25]). Then, from (35), hence. Now, by applying Barbalat s Lemma

8 350 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 13, NO. 2, MARCH 2002 Fig. 8. Adaptive controller. Evolution of q ;q. Fig. 7. PUMA-560 Robot. (see [35]), one can be concluded that the tracking motion errors asymptotically converge to zero, that is, as. Remark 4: As, from (12) it results. Following the Barbalat s lemma, as. Then, as is bounded, and. From (18) it results that converges to a constant value. Remark 5: The sign switching function in the control law (16) may cause an undesirable chattering. To prevent it, a saturation function can be used, as proposed in [42]. In this case, the ultimate boundedness for control errors can be proved. Fig. 9. Evolution of the control errors q ;q for the adaptive controller. VI. SIMULATION STUDIES To show the feasibility and performance of the proposed neural network-based feedback adaptive controllers, as well as the stability properties obtained in the preceding theoretical development, a simulation study has been carried out for the PUMA-560 robot. The mathematical model used is given in [40] a gear ratio of 30 (Fig. 7). The simulation setup was as follows. Controllers were designed for controlling the dynamic motion of two joints (2 and 3). It was assumed a payload uncertainty, so the robot model parametrization, (2), is accomplished a vector of two components, is the payload mass. The component 1 is included to consider a part of robot dynamics which is independent of the parametric uncertainty. Admissible values are also considered for, ranging from 0 Kg to 20 Kg. The geometrical surface, representing possible values of, is shown in Fig. 1(b). A set of NNs is used to compensate the inverse dynamics of the robot. Each NN has eight inputs (equal to the dimension of the vector), two hidden layers of 90 and 45 neurons, and one output layer of two neurons corresponding to the dimension of vector. The NNs were trained for the following conditions of payload : 20 Kg and 0 Kg, following the criterium proposed in Section II-B. The reference signal used and the desired motion trajectory for the training process consisted on the random position, velocity and acceleration levels linked by fifth-order polynomial interpolation. In all phases, the NNs were trained using the backpropagation learning rule, fuzzy set theory-based variable learning rate, similarly as is proposed in [8], which allows to accelerate the learning time and to avoid falling into a local minimum. Upon completing the training process, [V], [V] are obtained for the learning error. Then, through (32), the bounds [rad] and [rad/s] are obtained. The feedback PD gains were selected as:, and. Update gains were fixed to and. The payload, set initially to 0 kg, was changed to 15 kg at 15 s and finally to 0 kg, again, at s. The evolution of the desired and output signals of the system, and, is presented in Fig. 8. The control errors,, can be seen in Fig. 9, and the estimated parameters in Fig. 10. It can be observed that, due to the adaptation, the parameters are adjusted to the new payload pattern, thus allowing to maintain a good performance in spite of changes in payload (see Fig. 10). It can be observed as well that the motion control error remains in calculated bounds, as is shown in Fig. 11. To show the advantage of the adaptive controller, the adaptive coefficients were given the values corresponding to the 0 kg payload condition and kept constant while the payload was changed

9 PATIÑO et al.: NEURAL NETWORKS FOR ADVANCED CONTROL OF ROBOT MANIPULATORS 351 Fig. 10. Evolution of parameters estimates ; for the adaptive controller. Fig. 13. Evolution of tracking errors q ;q for the nonadaptive case. Fig. 11. Boundedness of the motion errors. Fig. 14. Evolution of q and q for the robust adaptive controller. Fig. 12. Evolution of q ;q and q ;q for the nonadaptive case. as the previous case. Fig. 12 shows the evolution of and, and Fig. 13 shows the tracking motion errors. As it can be seen, the control performance is degraded when a fixed controller is used. With the aim of testing the tracking performances of the NN-based motion robust adaptive controller, the same simulation setup previously planed was developed. Design parameters for this controller were selected as follows: Feedback PD gains, ; also,, and adaptation gains. The payload change sequence in the motion Fig. 15. Evolution of the control errors for the robust adaptive controller. desired trajectory was set initially to 0 kg, and changed to 15 kg at 10 s, and finally to 0 kg, again at s. Fig. 14 shows the evolution of desired values of and. Fig. 15 presents the control errors, and Fig. 16 the estimated parameters. In this case, two facts can be observed: 1) due to adaptation of parameters, the controller keeps good behavior in spite of changes in payload and 2) due to the algorithm robustness the motion control errors converge to zero. However, as shown in Fig. 17, an undesirable chattering

10 352 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 13, NO. 2, MARCH 2002 Fig. 16. Evolution of parameters estimates for the robust adaptive controller. Fig. 18. Control action for the robust adaptive controller saturation switching function. Fig. 17. function. Control action for the robust adaptive controller sign switching Fig. 19. Evolution of control errors for the robust adaptive controller saturation switching function. control input may occur due to the sign switching function. From an implementation point of view, it is practically impossible to achieve a high switching control. To prevent this, the following saturation switching function can be used:, when, when and, when,. Fig. 18 shows the evolution of the control action, and Fig. 19 the control errors, which do not converge to zero, but remain ultimately bounded. VII. CONCLUSION We have presented an approach and a systematic design methodology to a motion adaptive control based on NNs for high-performance robot manipulators, for which stability conditions and performance evaluation have been given. The neurocontroller includes a linear combination of a set of off-line trained NNs (bank of fixed neural networks), and an update law of the linear combination coefficients to adjust robot dynamics and payload uncertain parameters. A procedure has been proposed to select the learning conditions for each NN in the bank. The stable controller-parameter adjustment mechanism, which is determined using the Lyapunov theory, is constructed using a -modification-type updating law. This particular scheme, based on fixed NNs, is computationally more efficient than the case of using the learning capabilities of the neural network to be adapted, as that used in feedback structures that need to propagate back control errors through the model (or network model) to adjust the neurocontroller. A practical stability result for the neurocontrol system was given. That is, we proved that the control error converges asymptotically to a neighborhood of zero, whose size is evaluated and depends on the approximation error of the NN bank. An explicit evaluation of control error in terms of the NN learning error and design parameters was performed. In addition, a robust adaptive controller to NN learning errors was proposed, using a sign or saturation switching function in the control law, which leads to global asymptotic stability and zero convergence of control errors. To show the practical feasibility and performance of the NN-based adaptive control algorithms as well as stability properties obtained in the present work, a simulation study was carried out for motion control of a PUMA-560 robot. The results show the practical feasibility and good performance of the proposed approach to robotics.

11 PATIÑO et al.: NEURAL NETWORKS FOR ADVANCED CONTROL OF ROBOT MANIPULATORS 353 APPENDIX I Definition: Let a system be given by (20), which has the equilibrium state at the origin for all. Let the perturbed system be given by (21). Let be a set which is closed and bounded containing the origin and let be a subset of. Let be the solution of (21) satisfying. Let be the set of perturbations satisfying for all and for all,. If for each in, each in, and each is in for all, then, the equilibrium of (20) at the origin is said to be practically stable. In other words, practical stability of (20) implies that the solution of (21) which starts initially in remains thereafter in. In analogy to the classic asymptotic stability, we can consider a strong practical stability. For this case, given, if the origin is practically stable and if, in addition, we require that every solution of the system (21) for each be ultimately in, then we say that the system (20) has a strong practical stability. The following result is from [20]. Lemma: ]Let be a scalar function which has continuous first partial derivative for all and the property that as. Let denote the time derivative of along the solutions of system (21). If for all in and all outside, then system (20) possesses strong practical stability. REFERENCES [1] I. Akio, T. Furuhashi, and S. Okuma, A neural-network compensator for uncertainties of robotics manipulators, IEEE Trans. Ind. Electron., vol. 39, pp , [2] R. Carelli, E. F. Camacho, and D. Patiño, Neural-network-based adaptive control for robots, in Proc. Europ. Contr. Conf. ECC 93, 1993, pp [3], A neural-network-based feedforward adaptive controller for robots, IEEE Trans. Syst., Man, Cybern., vol. 25, [4] J. J. Craig, Adaptive Control of Mechanical Manipulators. Reading, MA: Addison-Wesley, [5] G. Cybenko, Approximation by Superpositions of a Sigmoidal Function, Dept. Elect. Comput. Eng., Univ. Illinois, Urbana-Champaign, Tech. Rep. 856, [6] C. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. New York: Academic, [7] M. Ertugrul and O. Kagnak, Neural network adaptive sliding mode control and its application to scara type robot manipulator, Proc. IEEE Int. Conf. Robot. Automat., pp , [8] T. Fukuda, T. Shibata, M. Tokita, and T. Mitsuoka, Adaptation and learning for robot manipulator by neural network, Proc. IEEE Conf. Decision. Contr., pp , [9] K. Funahashi, On the approximate realization of continuous mapping by neural networks, Neural Networks, vol. 2, no. 3, pp , [10] S. Z. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulator, Singapore: World Scientific, [11] A. Guez and I. Bar-Kana, Two-degree-of-freedom robot neurocontroller, in Proc. 29th Conf. Decision Contr., 1990, pp [12] K. Hornik, M. Stinchcombe, and H. Wite, Multilayered feedforward network are universal approximators, Neural Network, vol. 2, no. 5, pp , [13] J. Y. Hung, W. Gao, and J. C. Hung, Variable structure control: A survey, IEEE Trans. Ind. Electron., vol. 40, pp. 2 22, [14] P. A. Ioannou and P. V. Kokotovic, Adaptive Systems Reduced Models. New York: Springer-Verlag, [15] A. Ishiguro, T. Furuhashi, and Y. Uchikawa, A neural-network compensator for uncertainties of robotics manipulators, IEEE Trans. Ind. Electron., vol. 39, pp , [16] M. Kawato, Y. Uno, M. Isobe, and R. Suzuki, A hierarchical model for voluntary movement and its application to robotics, IEEE Contr. Syst. Mag., vol. 8, no. 2, pp. 8 16, [17] R. Kelly, R. Carelli, and R. Ortega, Adaptive motion control design of robot manipulators: An input-output approach, Int. J. Contr., vol. 50, no. 6, pp , [18] P. Khosla and T. Kanade, Parameter identification of robot dynamics, Proc. IEEE Conf. Decision Contr., [19] Y. H. Kim and F. L. Lewis, Output feedback control of rigid robots using dynamic neural networks, Proc. IEEE Int. Conf. Robot. Automation, pp , [20] J. La Salle and S. Lefschetz, Stability by Liapunov: Direct Method Applications. New York: Academic, [21] W. Li and J. J. Slotine, Indirect adaptive robot control, Proc. IEEE Int. Conf. Robot. Automat., pp , [22] G. Lightbody and G. Irwin, Neural networks for nonlinear adaptive control, in Proc. IFAC Workshop Algorithms Architectures Real-Time Contr., 1992, pp [23] J. B. Mbede, X. Huang, and M. Wang, Robot fuzzy and recurrent neural-network motion control among dynamic obstacles, Proc. IEEE Int. Conf. Robot. Automat., pp [24] K. S. Narendra and K. Parthasarathy, Identification and control of dynamical systems using neural networks, IEEE Trans. Neural Networks, vol. 1, pp. 4 27, [25] R. Ortega and M. Spong, Adaptive motion control of rigid robots: A tutorial, Automatica, vol. 25, no. 6, pp , [26] T. Ozaki, T. Suzuki, T. Furuhashi, S. Okuma, and Y. Ochikawa, Trajectory control of robotic manipulators using neural networks, IEEE Trans. Ind. Electron., vol. 38, pp , [27] D. Patiño, R. Carelli, and B. Kuchen, Stability analysis of neural-networks-based adaptive controllers for robot manipulators, in Proc. Amer. Contr. Conf., vol. 1, 1994, pp [28] D. Patiño, Dynamic Control of Robot Manipulators Using Neural Networks, Doctoralal dissertation, Ed. Fundación Univ. Nacional de San Juan, Argentina, [29] D. Patiño and D. Liu, Neural network-based model reference adaptive control system, IEEE Trans. Syst., Man, Cybern., vol. 30, pp , Feb [30] D. Psaltis, A. Sideris, and A. A. Yamura, A multilayered neural-network controller, IEEE Contr. Syst. Mag., vol. 8, pp , [31] D. E. Rumelhart and J. L. McClelland, Parallel Distributed Precessing. Cambridge, MA: MIT Press, [32] M. Saad, L. A. Dessaint, J. F. Basset, and Y. Dube, Adaptive robot controllers: A comparative study, in Symp. Design Methods Cont. Syst., IFAC, 1991, pp [33] N. Sadeg and R. Horowitz, Stability and robustness analysis of a class of adaptive controllers for robotic manipulators, Int. J. Robot. Res., vol. 9, pp , [34] M. R. Sanner and J. J. E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neural Networks, vol. 3, pp , [35] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness. Englewood Cliffs, NJ: Prentice-Hall, [36] L. Sciavicco and B. Siciliano, Modeling and Control of Robot Manipulators: Springer-Verlag, [37] J. J. Slotine and W. Li, Adaptive manipulator control: A case study, IEEE Trans. Automat. Contr., vol. 33, pp , [38] M. Spong and M. Vidyasagar, Robot Dynamics and Control. New York: Wiley, [39] C.-Y. Su and Y. Stepanenko, Hybrid adaptive/robust motion control of rigid-link electrically-driven robot manipulators, IEEE Trans. Robot. Automation, vol. 11, no. 3, pp , June [40] K. Troch and K. Desoyer, Benchmark Problems for Control System Design,, Rep. IFAC Theory Committee, E. J. Davinson, Ed., [41] T. Yabuta and T. Yamada, Possibility of neural networks controller for robot manipulators, in Proc. IEEE Int. Conf. Robot. Automat., 1990, pp [42] K. S. Yeng and Y. P. Chen, A new controller design for manipulators using the theory of variable structure systems, IEEE Trans. Automat. Contr., vol. 33, pp , [43] W. Yu, A. S. Poznyak, and E. N. Sanchez, Neural adaptive control of two-link manipulator sliding mode compensation, in Proc. IEEE Int. Conf. Robot. Automat., 1999, pp

12 354 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 13, NO. 2, MARCH 2002 H. Daniel Patiño (M 87 S 92 M 95) was born in San Juan, Argentina. He received the electronic engineering degree and the doctorate degree in engineering from the Universidad Nacional de San Juan (UNSJ), Argentina, in 1986 and 1995, respectively. From 1987 to 1993, he was fellowship holder of the National Council for Scientific and Technical Research (CONICET) of Argentina. Dr. Patiño received the National Dr. José A. Balseiro Award for the Best Research Group in the technological area and the technology transfer to industry in 1994, as a Member of the Research Staff of the Instituto de Automática. From 1997 to 1998, he was a Postdoctoral Research Associate and taught in the Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ. He is an Associate Professor in the Instituto de Automática and the Electronics and Automatics Department, Faculty of Engineering, UNSJ. He is also currently Secretary of Research in this Faculty. His current research areas cover computational intelligence, operations research, systems and control theory, data mining, computational finance, and robotics. Specific research and teaching topics include artificial neural networks, hybrid intelligent systems, adaptive critic designs, neurodynamic programming and evolutionary computation for optimization and control, intelligent agents, and autonomous machines and vehicles. He is a member of the IEEE and he has collaborated and served as a technical reviewer in the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, and in numerous IEEE conferences. Benjamín R. Kuchen (M 90) was born on October 1, 1941 in Santa Maria, Santa Fe, Argentina. He received the Bachelor s degree in electronics engineering at the Universidad Católica de Córdoba, Córdoba, Spain, in During 1968, he did a specialization course at the Philips International Institute, Eindhoven, The Netherlands, on power electronics. From 1971 to 1974, he worked on his doctorate degree at the Rheinisch-Westfälischen Technischen Hochschule Aachen (RWTH Aachen), Germany; a thesis on the adaptive model of man acting as a controller of processes. He is currently Professor at the Universidad Nacional de San Juan (UNSJ) and Director of the Instituto de Automática (INAUT) since 1983 developing research and graduate activities. He is also an Independent Researcher of the National Council for Scientific and Technical Research of Argentina (CONICET). He was Visiting Professor in the RWTH Aachen, Germany in 1983, in the National Council for Scientific Research of Spain, Madrid, during 1992, and in the Universidad de Salamanca, Spain, from October 2000 to March He has conducted several national and international research projects and participated in organizing and program committees of many national and international conferences. Currently he is the Coordinator of the evaluation process of the National Agency for Science and Technology in the area of Information Technology, Electronics and Communication. His interest areas include robot control, manufacturing systems and control of power electronic systems. Dr. Kuchen is member of AADECA-IFAC. As director of the INAUTs Research Group, he received the National Dr. José A. Balseiro Award for his activities in the technological area and the technology transfer to industry. Ricardo Carelli (M 76 SM 98) was born in San Juan, Argentina. He received the Bachelor s degree in engineering from the National University of San Juan, Argentina, and the Ph.D. degree in electrical engineering from the National University of Mexico (UNAM). He is presently full Professor at the National University of San Juan and Senior Researcher of the National Council for Scientific and Technical Research (CONICET, Argentina). He is Adjunct Director of the Instituto de Automática, National University of San Juan. His research interests include robotics, manufacturing systems, adaptive control, and artificial intelligence applied to automatic control. Dr. Carelli is a member of AADECA-IFAC.