Neuro-based adaptive internal model control for robot manipulators

Transcription

1 Neuro-based adaptive internal model control for robot manipulators ** ** Q. Li, A. N. Poi, C. M. Lim, M. Ang' ** Electronic & Computer Engineering Department Ngee Ann Polytechnic, Singapore Clementi Road, Republic of Singapore * Mechanical & Production Engineering Department National University of Singapore, Singapore Kent Ridge Crescent, Republic of Singapore Abstract The application of nternal Model Control for process control has received much attention during the past decade. n this paper, the application of MC for robot control is investigated. Although the MC approach is shown to be more robust as compared to conventional robot control approaches, such as the computed-torque approach, its performance degrades in the presence of large modelling uncertainties and external disturbances. n this work, a neurobased adaptive internal model control scheme is proposed. Within the framework of this control structure, a back-propagation neural network algorithm is incorporated into a fixed structure internal model controller for robot control. Simulation results, based on a two-link robot confirm the effectiveness of the proposed control algorithm even in the presence of large modelling uncertainties and external disturbances. 1. ntroduction The application of the internal model control (MC) algorithm for process control has received much attention during the past decade [ Recently, this method has been applied for the control of robot manipulators [4]. Compared with the performance of the conventional robot control methods for example, the computed-torque control scheme [5], that of the robot MC algorithm has been shown to be more robust. However, the performance of the MC may become unsatisfactory in the presence of large modelling uncertainties or external disturbances. n this paper, a neuro-based adaptive MC scheme for the control of robot arms is proposed to overcome the detrimental effects due to the above conditions and, as a result, the performance of the overall control structure can be further enhanced. 2. MC in Robots 2.1. Preliminaries of MC The structure of a standard MC system is shown in Fig. 1, where G represents the plant, 6 the model of the plant and G, the controller. Note that q,, U, q and d, indicated in the figure, are the vectors of the reference input to the system, the control input to the Plant, the system output and the external disturbance respectively. Figure 1. Block Diagram of the Standard MC The important properties of the MC structure can be summarised as follows: Property 1. Dual Stability Criterion: When the model of the is perfect, that is = G, the stability of both the controller and the plant is sufficient for the stability of the overall system. d

2 ~ Property 2. Perfect Controller: Assume that the MC system is closed-loop stable and G, = e-' is realisable. Then perfect reference tracking control ( q = q, ) can be achieved for all t > 0, despite any disturbance d. Property 3. Zero Offset: Assume that the steady-state gain of the controller is equal to the inverse of the model gain ( G, (0) = (?'(O) ) and that the closed-loop system in Fig. 1 is stable. Then the control error e = q - q, will vanish asymptotically for all constant inputs and disturbances. t should be noted that in the case of an open-loop unstable plant, pre-stabilisation of the plant by a conventional feedback loop is necessary before the standard MC can be applied [ Robot Dynamics The dynamic equations of a robot manipulator modelled as a set of n rigid bodies connected in a serial chain can be derived as [5]: M(qN + H (q4 + z, = T (1) where q is the n x 1 vector of joint positions, T is the n x 1 vector of applied joint forces/torques, M(q) is the n x n symmetric positive-definite manipulator inertia matrix, H(q, q) is the n x 1 vector of torques arising from centripetal, Coriolis, gravity and frictional forces and zd is the nxl vector of unknown signals due to modelling uncertainties and external disturbances Pre-Compensation of Robot Dynamics The robot dynamics is highly nonlinear and highly coupled [S. Before the standard MC scheme can be applied to control a robot, compensations of the robot dynamics must be accomplished. This procedure consists of two stages, viz. pre-linearisation and prestabilisation, as shown in Fig. 2. n the pre-linearisation stage, the lineariser is chosen to be: T = M(q)u + H(q, 4) (2) where U, in the form of acceleration, is the input vector to the lineariser. The quantities with hats "*" are computed from the estimates of the real manipulator parameters. n the absence of disturbances, if the robot dynamic model is perfectly obtained, then Eqns. (1) and (2) give u=q (3) This is a linearised decoupled system with its transfer function given by: q(s) - 1 (4) u(s) s2 The above decoupled system is critically stable. However, Property 1 of MC requires that the plant should be a stable system. n the pre-stabilisation stage, a conventional PD feedback loop is therefore appended, as shown in Fig. 2, to stabilise the linearised robot. The transfer function of the stabilised robot system is given by: q(s) - -- (5) u'(s) sz + Kvs + KP with K, = diag CKv,, K,... Kvn 1 (6) K, = dlag [KP,,KP2,...,K pnl (7) and K, and K p, are positive feedback gains suitably selected to ensure that the robot system is stable.... lrnearisrd plant... plonl... K,st K,......, pre-conzpensuted plnnr Fig. 2. Pre-compensation of the Robot G, From the foregoing procedures, it can be seen that the original highly nonlinear and coupled robot dynamics have been linearised and decoupled to yield an openloop stable system. The MC can now be applied to the control of the compensated robot Design of the Robot MC The internal model d is simply selected according to the compensated robot dynamics 111: a= s2+k,s+kp For perfect trajectory tracking and complete disturbance rejection, the controller is chosen to be &, as indicated by Property 2. As such, the controller is given as: G, =s2 +K,s+K~ (9) n order to ensure that G' is physically realisable, a pre-filter is added and the modified controller Gun is of the form [3]: s2 + K"S+ K, G, = (T,s + 1)"

3 ~~~ where T, is the filter time constant and N, its order, is selected to be N 2 2 in this case. 3. Neuro-Based Adaptive MC for Robots The above described MC for robots has been shown to have satisfactory performance via simulation and experimental studies 141. However, to achieve better control performance, accurate knowledge of the robot dynamics is required. f the estimation of the robot dynamics is not precise, then errors will emergence in the result of the linearisation of the robot dynamics, which will then cause nonideal stabilisation result. The model mismatch between the ideal internal model and the actual compensated robot dynamics may break the fundamental properties of the MC algorithm and cause high gain control and possible loop instability. operates independently based on the following relationship: f, (4, q, q) = T,, i = 1 to n (11) where T, is a scalar, indicating the specific torque actuating on the i-th link, and f,(.) is a scalar function, describing the dynamic model of link i. The parallel combination of the n subnets gives the following relationship: F(q, q, ii) = T (12) where F(.) and T are nxl vectors. This expression is equivalent to the inverse robot dynamic model. Robot nputsn, Subnetworh Robot Fig. 4. Structure of the Subnets. Fig. 3. The Neuro-Based Lineariser n this paper, incorporation of an adaptive algorithm, based on the neural network technology, into the fixed structure MC scheme is considered. The motivation of using the adaptive approach is that the modelling accuracy of the robot manipulator with unknown or varying dynamics can be improved with time. This is because the neuro-based adaptation mechanism in the control structure can continuously extract information from the tracking errors and then update the estimated dynamics along the way. Since an accurate linearisation result is sufficient to guarantee a precise internal model 6, only the prelinearisation structure in the MC scheme is modified by using a back-propagation neural network adaptation mechanism. The structure of the neurobased lineariser is shown in Fig. 3. n order to make full use of the advantages of neural networks, the on-line re-learning algorithm of the network during real-time control operation is required. To achieve a fast and reliable training, the lineariser in the MC is implemented using an n sub-network structure as shown in Fig. 4. Each sub-network The operation of the proposed neuro-based lineariser comprises of two stages: a control stage and an on-line re-learning stage. n the first stage, the lineariser takes in its input signals, u(k,), q(k1) and q(k,), to generate its output torque T(k1). The signal flows of this stage are indicated by the solid lines shown in Fig. 3. The torque T(k1) then actuates the robot to move to a new state q(kz) and q(k,), from which q(k,) is computed using the finite difference approximation. n the second operation stage, relearning is performed using q(kz), q(k,) and q(k,) as set of new input data. The network generates a new output T(k,) and the error between T(k1) and T(k,) is used to adjust the weights of the network so that it can match the actual inverse dynamics of the robot. 4. SMULATON STUDY 4.1. Specification of the MC n this simulation study, the feedback gains of the prestabiliser in the internal model controller are designed to be Kv = 2 and K, = 1 for the roots of the characteristic Eqn. (5) to be shifted to -1. The time constant of the pre-filter in the control structure are

4 defined to be T, = s. The order of the filter N is selected to be The Robot Structure The structure of the robot manipulator, along with its specifications, is illustrated in Fig. 5. For simplicity, the masses of the links are assumed as point masses at the end of the massless links. The dynamic equations of the robot are estimated as [6]: T, = [m,c + m21a cos(q,>lq, +m,l;q, + q;m21,l2 sin(q,) +m,gl, cos(q,+42) The parameters of the model are chosen as m1=2kg, m2=3kg, 1,=lm and 1,=15m respectively. Y Fig. 5. Structure of the 2-link robot The Neural Network Structure Two sub-network structure is applied to represent the robot dynamics, with the first sub-network learning the dynamics of link 1 and the second link 2. Each of the networks employs a 2-hidden-layer backpropagation algorithm. Both of them have 18, 8 and 1 nodes in their first hidden, second hidden and output layer respectively. However, the input layer for subnetwork 1 has 6 nodes whereas that of sub-network 2 has 5. Both sub-networks were pre-trained off-line using Eqn. (13). Specifically, torques T and T2, the desired outputs of the sub-networks are computed according Eqn. (13) with their inputs (q,q,q) randomly generated within a set of pre-selected range of values. n order to achieve effective training, both input and output values need to be scaled before they can be presented to train the neural networks. n this paper, the weight adaptation algorithm used is the normal delta rule [7] with the learning rate and the momentum coefficient selected as 0.4 and 0.2 respectively. The activation function used is the sigmoid function [7]. Off-line training of both networks is terminated after the training errors have dropped within 5% of the desired output. After training, the networks are then embedded in the lineariser of the control structure. Since on-linc weight adaptation is applied, any modelling uncertainties of the robot dynamics and external disturbances can be accounted for Simulation results: A trajectory-tracking control is studied in this simulation. The initial and final velocity and acceleration of the motion are all assumed to be zero. To ensure a smooth torque profile and satisfy the boundary conditions, the following quintic polynomial is selected to be the task trajectory [5]: q,(t) =(10J-15-t+69 t2)t3 4 9 t: t; t: nitially, both links are commanded to move simultaneously from their initial positions qol =qm =O at t= 0 to the final positions q,, = qf2 = 1 rad exactly in the time interval t, = 2 s. Then the motion task requires that the two links remain at their final positions for 1 s. During the tracking path, link 2 is assumed to pick up a heavy payload at t = 0.5 s and release it at t = 1.5 s. The variation of the payload can be considered as a serious parameter modelling error. n addition to this parameter error, other modelling uncertainties, for example the friction effects and the external disturbances, are assumed to interfere the system simultaneously. n this simulation example, the control performances of both MC structures are studied in the case of the combined effect of all these modelling uncertainties. Table 1 gives the assumptions of the magnitudes of the uncertainties. Fig. 6(a) shows the simulation results obtained in the absence of any modelling uncertainties. This is to test how accurate the neural network can learn to map the non-linear robot dynamics. Fig.6(b) and 6(c) show the trajectory errors in the presence of modelling uncertainties described above. For comparison purposes, the results due to the MC scheme without using the neural network are also shown in the figures. The solid lines represent the error curves of the neuro-based MC,

5 while the dotted lines indicate the results for the fixed structure MC scheme. Variation of m, 0.2 kg (10%of ml) degraded with severe errors presenting both during the tracking path and at the final stage. Compared with these, the errors of the neuro-based MC are much smaller in several orders of magnitude. Friction Torque f,, Viscous Friction Torque f,, Disturbance d, Disturbance d, 0.4 q2 (20% of max. T, ) 20 N 1. Assumptions of Uncertainties me in Second Fig. 6(c). Tracking errors of link 2 (with modelling uncertainties) 5. CONCLUSONS Fig. 6(a). Tracking errors of link 1 (perfect model) The proposed neuro-based adaptive internal model control of robot manipulators has been shown to have very good performance even in the presence of large unmodelled dynamics and external disturbances. However, back-propagation learning algorithm requires intensive computation. Further study will focus on developing a neural network algorithm which is simple in its structure yet effective in its performance. 6. REFERENCES Fig. 6(b). Tracking errors of link l(with modelling uncertainties) From the results shown in Fig. 6(a), it can be seen that the tracking performance of the neuro-based MC system is very closed to that of the fixed structure MC system. The difference between the tracking performances is so small that it can be neglected in reality. This means that the neural network has learnt the inverse dynamic model of the robot very well. Fig. 6(b) and (c) show that tracking performance for the fixed structure MC system has been significantly 1. C.E. Garcia and M. Morari, nternal model control. 1. A unifying review and some new results, nd. Eng. Chem. Process Des. Dev. 21(2), M. Morari and E. Zafiriou, Robust Process Control. Englewood Cliffs, NJ, Prentice-Hall, D.G. Fisher, Process control: an overview and personal perspective, The Canadian J. Chem. Eng. 69(1), Q. Li, A.N. Poo, C.M. Lim C M and C.L. Teo, nternal model structure in the control of robot manipulators, Submitted to Mechatronics, J.J. Craig, ntroduction to Robotics: Mechanics and Control, 1st edition, Reading, Mass: Addison- Wesley, F.L. Lewis, C.T. Abdallah and D.M. Dawson, Control of Robot Manipulators, 1st edition, Macmillan Publishing Company, R. Hecht-Nielsen, Neurocomputing. 1st edition, Reading, Mass: Addison-Wesle, 1990.