Integrated Direct / Indirect Adaptive Robust Control of Multi - Dof

In a general DIARC framework [13], the emphasis is always on the guaranteed transient performance and accurate trajectory tracking in the presence of uncertain nonlinearity and parametric uncertainties along with accurate parameter estimation for secondary purpose such as system health monitoring and prognosis. Need for accurate parameter estimation calls for the use of Least Square Estimation (LSE) type of algorithms for such a seamless integration of good tracking performance and accurate parameter estimation. This paper presents a physical model based integrated direct/indirect adaptive robust control (DIARC) strategy for a hydraulically actuated 3-DOF robotic arm. To avoid the need of acceleration feedback for DIARC backstepping design, the property, that the adjoint matrix and the determinant of the inertial matrix could be linearly parameterized by certain suitably selected parameters is utilized. Unlike gradient-type parameter estimation law, which used overparamterization, there is no multiple estimation of the single parameter. Theoretically, the resulting controller is able to take into account not only the effect of parametric uncertainties coming from the payload and various hydraulic parameters but also the effect of uncertain nonlinearities. Furthermore, the proposed DIARC controller guarantees a prescribed output tracking transient performance and final tracking accuracy while achieving asymptotic output tracking in the presence of parametric uncertainties only. Simulation results based on a three degree-of-freedom (DOF) hy∗THE WORK IS SUPPORTED IN PART BY THE US NATIONAL SC ENCE FOUNDATION GRANT NO. CMS-0600516. Idraulic robot arm (a scaled down version of an industrial backhoe/excavator arm) are presented to illustrate the proposed control algorithm. INTRODUCTION Robotic manipulator driven by hydraulic actuators has been widely used in the industry for the tasks such as material handling and earth moving due to its high power density. These types of tasks typically require that the end-effectors of the manipulators follow certain prescribed desired trajectories in the working space. In order to meet the increasing requirement of productivity and performance of modern industry, the development of high speed and high accuracy trajectory tracking controllers for the coordinated motion of robot manipulator driven by hydraulic actuators is of practical importance. Compared to the the conventional robotic manipulator driven by electrical motors, the controller design for the robotic manipulator driven by hydraulic actuators is more difficult both theoretically and experimentally due to the following several reasons. First of all, unlike the electrical motors, the hydraulic cylinders are linear actuators and complicated mechanical mechanisms are needed to drive revolute joints. Such a configuration results in additional nonlinearities and stronger couplings among the dynamics of various joints. Secondly, in addition to the coupled MIMO nonlinear dynamics of the rigid robot arm, the dynamics of the hydraulic actuators must be considered in the control of a hydraulic arm, which substantially in1 Copyright c © 2007 by ASME creases the controller design difficulties. It is well known that a robot arm including actuator dynamics has a relative degree more than three [17]. Synthesizing a controller for such a system usually requires joint acceleration feedback for a complete state feedback, which may not be a practical solution. Furthermore, the single-rod hydraulic actuator studied here has a much more complicated dynamics than electrical motors. The dynamics of a hydraulic cylinder is highly nonlinear [9] and may be subjected to non-smooth and discontinuous nonlinearities due to directional change of valve opening and frictions. The dynamic equations describing the pressure changes in the two chambers of a single-rod hydraulic actuator cannot be combined into a single load pressure equation, which not only increases the dimension of the system to be dealt with but also brings in the stability issue of the added internal dynamics. Finally, a hydraulic arm normally experiences large extent of model uncertainties including the large changes in load seen by the system in industrial use, the large variations in the hydraulic parameters (e.g., bulk modulus), leakages, the external disturbances, and frictions. Partly due to these difficulties, so far, the model-based robust control of a hydraulic arm has not been well studied and fewer results are available. In [4] the singular perturbation was used to synthesize a controller for a 6 axis hydraulically actuated robot. In [8] a variable structure controller was developed to control a Caterpillar 325 excavator without considering parametric uncertainties and uncertain nonlinearities associated with the system simultaneously. Theoretically, none of above schemes could address all the difficulties mentioned above well. In [13] an integrated direct/indirect ARC (DIARC) framework is presented for the high performance trajectory tracking control of SISO nonlinear systems in a semi-strict feedback form taking into account both the parametric uncertainties and uncertain nonlinearities. The resulting DIARC controller not only achieves a guaranteed robust tracking performance but also as good on-line parameter estimation as possible. In [3] a physical model based ARC controller, which uses overparameterization for a 3 DOF hydraulic robot arm to avoid need of acceleration feedback was proposed. The same problem was solved by the duo in [2] using an acceleration observer. In both the cases, however, the presented ARC controllers are of direct type, which only admit gradient-type of parameter adaptation law, leading to poor on-line parameter estimation in practice. This paper continues the work done in [3] and [2] but will extend the design to include LSE-type of parameter estimation laws for accurate on-line parameter estimates and reduce the order of control by avoiding overparameterization. The proposed method makes full use of the property of the inertial matrix that the adjoint matrix and the determinant of the inertial matrix could be linearly parametrized by certain suitably selected parameters. Theoretically, the proposed DIARC approach achieves a guaranteed transient and final tracking accuracy for output trajectory tracking, which overcomes the drawbacks of traditional robust 2 adaptive control designs. At the same time, asymptotic output tracking is achieved in the presence of parametric uncertainties only, which overcomes the performance limitation of traditional robust control designs. Simulation results based on a 3 DOF hydraulic arm will be presented to illustrate the effectiveness of the proposed control algorithm. Experimental verification is being carried out and comparative experimental results will be presented at the conference when available. PROBLEM FORMULATION AND DYNAMIC MODEL