PID - Gain Scheduling Controller for a Robot Manipulator

Copyright (£) IFAC Digital Control: Past, Present and Future of PID Control, Terrassa, Spain, 2000 PID - GAIN SCHEDULING CONTROLLER FOR A ROBOT MANIPULATOR CarIos Perez, CarIos Vivas, F. R. Rubio Dept. Ingenieria de Sistemas yAutomatica Escuela Superior de Ingenieros. Universidad de Sevilla. Camino de los Descubrimientos sin. 41092-Sevilla. SPAIN Tell + 94 954 48 79 50, Fax: + 94 954 48 79 40 e-mail: {cp!.vivas.rubio}@cartuja.us.es Abstract: This paper presents the implementation of an adaptive PD -Gain Scheduling controller for asix-degree-of-freedom robot manipulator. Uncoupled PD controllers are proposed for each joint where each single manipulator joint is controlled independently of the others. The gain scheduling technique enables the controller to adapt on-line the PD parameters depending on the operation conditions (essentially the robot arm position). The nonlinear dynamic model of the robot arm is then linearized discarding non-linear terms which are considered as model uncertainties, and acontinuous adaptation law for the PD controller parameters is proposed. The overall control strategy can be regarded as composed by anon-linear transformation (adaptation law), followed by alinear PD regulator. This simple technique offers promising results with respect to the traditional worst-case PD design. Copyright @2000 [FAC Keywords: Robotics; PID Thning; Gain Scheduling; Adaptive Control 1. INTRODUCTION The rapidly growing interdisciplinary field of robotics, where systems must operate satisfactorily with different levels of imprecise information, represents one area where adaptive control is finding extensive application. The use ofadaptive controllers requires very little justification in robot manipulators. Avariety of factors including parametric and structural uncertainties, and nonlinearities that usually are neglected in the modeling process, make adaptive control mandatory. In addition avariety ofdisturbances are also present, and accurate trajectory tracking may become ahard goal to meet. Parametric uncertainties are usually present due to imprecise knowledge of the manipulator mass properties, unknown loads, or uncertainty in the 505 load position in the end-effector. Structural uncertainties arise from the presence of high-frequency unmodeled dynamics, actuator dynamics, flexibilities in the links or joint plays. In the presence of such uncertainties, linear feedback controllers can not provide good performance and, hence, adaptive solutions are required. If Gain Scheduling is to be used, it is usually necessary to have agood insight into the dynamics of the process, in order to simplify the controller design. Thus, gain scheduling seems to be apromising and simple control strategy for a robot manipulator, since the nonlinear dynamics of these systems are well known and modeled. According to this, following is presented aPD -Gain Scheduling adaptive controller for asix degrees of freedom robot arm manipulator. This paper is organized as follows: In section 2, adescription of the robot arm manipulator employed is given, as well as some aspects of the robot dynamic model. Section 3 is dedicated to present the adopted control structure. Section 4 presents some simulation results with the proposed controller and finally, conclusions are given in section 6. 2. SYSTEM DESCRIPTION The Gain Scheduling controller proposed in this work has been designed for the RMlO Robot Manipulator. The RMlO, shown in figure 1, is a six-degree-of-freedom revolute joint manipulator arm. Fig. 1. The RMlO Robot Manipulator All the six joints are driven by DC-brushless lowinertia electric motors which provide an uniform torque for all joint positions, and enables high control torque peaks. Torque is delivered to the joint axis through gear reductions, thus RMI0 is an indirect-drive manipulator. The joints also provide an electric brake to block the manipulator arm in any position, (Rob6tica Industriale, 1991). Coupled to every motor axis is atwo-pole resolver device which provides accurate measure of the correspondent joint position. These measures will permit, as usual, the closed-loop control of the system. The RMlO system employs an VME bus based architecture, providing independent control boards for every joint. Particularly, the real-time DSll03 control board, dSPACE trade, was employed (dSPACE,1998), (MOOG GmbH, 1990), (dSPACE, 1999) . The control board were plugged into the expansion bus of commercial PC, holding a333MHz PowerPC as main processor and an additional DSP as input/output processor. The design and simulations of controllers in this paper has been supported by the popular 506 SIMULINK 2.0 toolbox under MATLAB 5.2 environment. For the implementation of control algorithms the dSPACE Real Time Workshop Toolbox has been also used. Before accomplishing the design of acontroller it is necessary to obtain adynamic model of the robot manipulator. Following the Euler-Lagrange formulation, (Craig, 1989), the dynamic model of ageneral n-link rigid-body robot is asecond order nonlinear equation: J(q)ij +C(q, q)q +F(q, q) +G(q) =T (1) where qE nn is the joint position; T is the generalized torque; J(q) (Inertial forces), C(q, q) E nnxn (Coriolis and centripetal forces terms), F( q, q) E nnxn (Friction model) and G(q) (gravitational components) E nn are nonlinear functions of q and q. The equation of motion is complex and contains anumber of hard to handle nonlinear terms. In order to simplify the controller design, Coriolis, centripetal and gravitational terms in equation 1have been neglected, and included as model uncertainty. Anumber of additional parameters are required to characterize the dynamic model of arobot manipulator, such as link masses and inertias. These parameters have been estimated by geometric measures and dynamic experiments of the robot arm. 3. CONTROL STRUCTURE In this section is described the structure of an adaptive PD-Gain Scheduling controller for the previously described RMI0 robot arm manipulator. Several ways ofdealing with the non-linear control problem in robot manipulators are proposed in classical literature. One of the most simple approaches consists oflinearizing the dynamic model equations of the robot arm, neglecting the coupling between links. In this case, each robot axis is treated as aSISO system that can be modeled as, (Sciavicco and Siciliano, 1996) (2) where qi is the angular displacement of joint i, Ti is the actuator torque and di is adisturbance due to Coriollis and centripetal acceleration, gravitationalload and Coulomb friction Torque. This assumption is done frequently in industrial robots with gearing between the motor and the link. The problem becomes then to control anumber Fig. 2. Diagram of the gain scheduling structure 3.1 Gain Scheduling compensator design Coriollis/Centripetal terms, depends weakly on most of these variables, and the dependency on this terms can be neglected. Thus, the Coriolis and centripetal forces terms can be neglected for moderate speeds in the whole manipulator work space, so operating condition dependence on q is discarded. (3) y Operating 1---;" condition Auxiliary variable Process u T= J{j + BB +Tp Adjustment mechanism The gain scheduling adaptation law is designed as acontinuous PO-parameter compensator. That is, the parameters of the six independent PO controllers are continuously changed in an openloop fashion depending on the robot arm position. Gain scheduling can thus be viewed as afeedback control system in which the feedback gains are adjusted using feedforward compensation. Neglecting Coriolis/centripetal, gravitational and coulomb friction terms and linearizing equation 1 for arobot arm position, the following approximate expression holds for the system dynamics of each independent joint: where Tp is adisturbance torque including the neglected dynamics. The PO control structure is described in figure 3 and includes proportional and derivative terms with constants Kp and Kdrespectively. The disturbance torque, Tp, is included as aexternal disturbance input, up =Tp/Km, where Km is a motor actuator constant. The most important dependencies appear in the principal diagonal components of matrix J(q) in equation 1, the inertial terms. Since the control scheme do not explicitly consider couplings between links ,only the diagonal elements in matrix J(q) are considered to depend on the operating conditions. Furthermore, not all these terms present asignificant dependence on every qparameter. For the RMlO physical configuration, the main dependence in J(q) is associated to the angle displacements in joints 2 and 3, (q2 and q3), and to aminor extent to the rest of joints. of second order plants in the presence of disturbances. This kind of controllers are usually designed employing the worst-case rule, so that the performance of the system is not very precise for operation points far from the linearization point. Asecond category of controllers includes the algorithms based on what is commonly referred to as the computed torque method. The basic idea is to determine the terms J, C, Fand Gin equation 1 at every instant of time, and cancel the nonlinear terms thereby decoupling the plant. Control is then effected by generating acontrol torque that results in alinear time-invariant homogeneous error equation whose equilibrium state is asymptotically stable. One of the main drawbacks of this method is the necessity of computing the terms of equation 1 at every instant of time. This computation may be quite time-consuming, depending on the configuration and complexity of the system to be controlled, and may happen that the controller can not meet the computation time requirements. Most of the solutions proposed for this problem consist ofconsidering some specific terms as aconstant, so that computations can be accomplished meeting the required times. Adifferent approach to this problem is proposed in this paper: To avoid the former requirement of an on-line computation of the non-linear terms in equation 1, no feedback linearization is accomplished as in the computed torque method. Nonlinear terms are instead included as uncertainties in the manipulator model and the system is controlled as six-order uncoupled system. This way equation 2holds for every joint. In order to compensate the neglected non-linear dynamics, an adaptive gain scheduling approach is proposed for the controller design, (Astrom and Wittenmark, 1989). Thus, the nonlinear system dynamic is compensated by means of anon-linear adaptation law (see figure 2) which regulates the parameters of the PO controllers according to the operation conditions of the plant (robot link positions). The regulator thus obtained can be regarded as composed by anonlinear transformation (adaptation law), followed by alinear PO regulator. Amain problem in the design of systems with gain scheduling is to find suitable scheduling variables. As far as robot manipulator control concerns, the variables which determine the robot operating condition correspond to the joint angular displacements q, and its derivatives q. For the problem of concern, asix-degrees-of-freedom robot manipulator, this amounts to 12 parameters. Fortunately, the robot operating condition, say the value of the inertial, Gravitational and 507 The closed-loop dynamic from the desired joint position, Od, to the robot effective joint position 0 is described by: (4) 4. SIMULATION RESULTS The Gain Scheduling design was examined by computer simulation and compared with fixed gain linear controller. The simulation uses an arm model ofthe RMlO manipulator (Fig. 1). Thesim- The controller parameters, Kd and Kp,are computed by forcing the dynamic in equation 4 to acritically damped second order response. Thus, assuming such response as: p(s) = S2 + 2cSw ns+ w~ (5) and identifying terms in both equations, 4 and 5, the following identities hold _ JKmK p. Wn- --J-' .... ,....-----.----.---~-----.-----,----, .... .... ..,··'0'------0-':. '--~-T-i-m""'e-(-s )--0------,."""------' Fig. 3. PO Control Structure 25 ... 0 .... .'" 9- ., -::. '" 0 .. ~- Vb·Otl ••• .- 0• ..1 Time (s) Fig. 5. Trajectory error with Gain Scheduling controller ulation verifies how a PD controller was improved if gain scheduling is applied for Kd gain, especially in trajectories with high inertia variation for axes 1and 2. For acritically damped response an adaptation law for Kdgain is used along the trajectories. Figures 4 y 5 shows tracking errors for axes 1and 2when acontinuous adaptation law was implemented. It can be observed, that for trajectories moving from high inertia to low inertia configuration, aclassical PO controller works with acceptable results, because the increasing damping factor which would yield slower response, is compensated by an increasing wn.Thus, aKd scheduling improve the system performance. Fig. 4. Trajectory error with PO controller (7) K_ Jw;. p- Km' In order to guarantee acritically damped close loop response a PO controller are designed as: From the previous expressions is easily observed that as Jdecreases, the natural frequency response of the system, Wn,increases yielding faster responses. On the other hand, this faster dynamic is counteracted by increasing values of cS as J decreases, yielding overdamped responses. These factors counteract each other in the case of constant parameters controller (classical PO control) inducing system stability for awide range of operating conditions. In order to improve performance of the system with respect to aclassical PO controller, asimple adaptation law for the gain scheduling controller is proposed. The proportional gain parameter, Kp, is fixed for all operating conditions while the derivative parameter, Kd, is adjusted to get a critically damped second orderresponse according to equation 6, that is cS = 1. The matrix J is calculated in every position of the manipulator, so that, the position is scheduling variable in the adaptive controller. 5. CONCLUSIONS An adaptive PD gain scheduling controller is proposed for the six-degree-of-freedom robot manipu- 508 lator RMlO. The controller avoids computational expensive feedback linearization schemes, as the classical computed torque method, considering instead an uncoupled system model. Neglected model couplings and non-linearities are compensated by means of acontinuous adaptation law which tunes the PD controller according to the manipulator operating conditions. This control strategy yields promising results in terms of trajectory error with respect to the classical fixed parameters PD controllers. Acknowledgments. The authors would like to acknowledge CICYT for funding this work under grant TAP98-0540. 6. REFERENCES Astrom, K.J. and B. Wittenmark (1989). Adaptive control. Addison Wesley. Craig, J.J. (1989). Introduction to robotics, mechanics and control. Addison Wesley. dSPACE (1998). DSll03 PPC Controller Board. Instalation and Configuration Guide. dSPACE (1999). Real Time Interface (RTI and RT-MP) Implementation Guide. MOOG GmbH (1990). T158-llX Controller User Manual. Rob6tica Industriale (1991). RMlO Manuale d'uso. System Robot. Sciavicco, L. and B. Siciliano (1996). Modelling and control of robot manipulators. Mc Graw Hill. 509