Direct Fuzzy Adaptive Control of a Manipulator with Elastic Joints

T HE benefits of lightweight space robotic manipulators are gained at the expense of higher elasticity in the joints. This leads to a more complex dynamic behavior. The additional flexible dynamics introduce two more state variables for each joint, which implies that the description of the complete dynamics model (including joint elasticity) requires four states for each joint: position and velocity of the motor rotor and position and velocity of the link. As a result, accelerating and stopping the arm produces large vibrations, making positioning of the end-effector very difficult. Achieving accurate motion and dampening elastic vibrations thus requires advanced control techniques. Conventional model-based control strategies for flexible-joint manipulators, such as nonlinear feedback control [1] and feedback linearization [2], require good knowledge of the plant in the form of a mathematical model and its parameters. Consequently, if significant or unpredictable plant parameter variations arise, or if there are modeling errors due to complex flexible dynamics behaviors, model-based control approaches may perform inadequately. Although indirect adaptive control techniques may be used to estimate unknown plant parameters upon which the controller gains are obtained using some design procedure, this class of adaptive control methodologies nevertheless requires good knowledge of the dynamics model [3]. For example, Cao and de Silva [4] use an indirect adaptive control scheme in which neural networks approximate the unknown manipulator inertia and centrifugalmatrices that are used explicitly in their control law. Alternatively, direct adaptive control techniques, with the controller gains updated directly in response to tracking errors, without requiring estimation of unknown plant parameters or mathematical models of the system to be controlled, can be used to address this problem [5]. Ulrich et al. [6] developed an adaptive composite control strategy using the singular perturbation-based (SPB) theory, in which a slow direct adaptive control term is added to a fast control term designed to dampen the elastic vibrations at the joints of a flexible-joint manipulator system. The direct adaptive control systemwas based on the modified simple adaptive control (MSAC) methodology, using the tracking errors between the ideal system and the actual system outputs to adapt the controller gains. In addition to direct adaptive control, fuzzy logic techniques have also been successfully applied to situations where mathematical modeling of the plant was uncertain; with flexible-joint manipulators for example. Goulet et al. [7] applied a complex multilayer approach with a conventional control bottom layer, with a preprocess layer and an intelligent fuzzy logic top layer, to a two-link flexible-joint manipulator, composed of revolute and prismatic joints. Ahmad et al. [8] used a fuzzy logic control system with triangular membership functions, to vary the input voltage of the robot’s actuator, in response to the end-effector position errors and change-of-errors for a linearized state-space model of a single-link flexible-joint manipulator. The performance of this fuzzy control scheme, combined with an input shaping technique, was evaluated in numerical simulations in a set-point control scenario. Park and Cho [9] designed a fuzzy model reference adaptive control strategy and applied it to a single-link elastic-joint manipulator that was mathematically modeled by a Takagi — Sugeno (TS) fuzzy model. The design of this controller was based on the TS dynamics model using a parallel distributed compensation technique. Poor performance were achieved when tracking a sine-wave trajectory described in terms of joint angles, as demonstrated by the 2 deg oscillating tracking error along the desired trajectory.Weiming et al. [10] proposed a fuzzy proportionalintegral controller, with the joint angular position and velocity errors chosen as inputs, and an incremental control torque vector selected as the controller output. The control strategy included an inner-loop system that managed the variation of the incremental control gain, which significantly increased the complexity of the control scheme, largely due to the real-time numerical integration of various signals. The purpose of this work is to address the problem of trajectory tracking by a two-link flexible-joint manipulator which is subject to large parametric uncertainties and modeling errors. The proposed strategy employs automatic tuning of the controller, enabling it to adapt to different plant conditions. To achieve this, a composite controller which uses a direct fuzzy adaptive control law as the slow control term is proposed. Within this context, the main contributions of this work are as follows. First, the strategy uses a novel direct adaptation mechanism based on fuzzy logic that adjusts the gains of the slow control term to yield good trajectory tracking performance in the presence of adverse conditions. This is, to the best of our knowledge, the first time a fuzzy logic approach is employed in a composite control scheme for flexible-joint robot manipulators. Second, as opposed to the previously mentioned existing flexible-joint control schemes, this new direct fuzzy adaptive method does not rely on online identification of plant parameters, is simple to implement, and renders effective trajectory-tracking performance regardless of large parametric uncertainties and dynamics modeling errors. Third, most existing control schemes for flexible-joint manipulators have been validated in numerical simulations using only the classic linear joint stiffness dynamics model [11]. In this work, similar to Ulrich et al. [6], the performance of the proposed control strategy will also be assessed with a comprehensive dynamics model that captures nonlinear effects observed in experiments, such as friction and nonlinear joint stiffness. This nonlinearmodel will be used as amean to validate the robustness of the proposed controller to modeling errors.