Approximate Feedback Linearization Control for Spatial 6-DOF Hydraulic Parallel Manipulator

Traditional feedback linearization approach (TFL) requires a priori knowledge of plant, which is difficult and the computational efficiency of controller is low due to the complex dynamics of spatial 6-DOF hydraulic parallel manipulator. In order to improve the tracking performance of spatial 6-DOF hydraulic parallel manipulator and to conquer the drawbacks of TFL, a novel approximate feedback linearization approach, non-model based method, is proposed in this paper. The mathematical model of spatial hydraulic parallel manipulator is established. The approximate feedback linearization control is designed for the parallel manipulator in joint space, with position and stored force in the previous time step are employed, as a learning tool to yield improved performance. Under Lyapunov theorems, the stability of the presented algorithm is confirmed in the presence of uncertainties. Simulation results show the proposed control is easy and effective to realize path tracking, and it exhibits excellent performance and high efficiency without a precision dynamics of plant. Furthermore, the presented algorithm is well suitable for most industrial applications. Keywords: Parallel manipulator, hydraulic system, approximate feedback linearization, path tracking. 1. INTRODUCTION Parallel manipulator has been extensively investigated due to the advantages of high force-to-weight ratio, high stiffness and accuracy, and its widespread applications in various fields such as machine tools, high fidelity simulators, and so on [1, 2]. Additionally, spatial hydraulic parallel manipulator has the characteristics of rapid responses and large output force, etc. Spatial 6-DOF hydraulic parallel manipulators have been applied as the motion system of flight simulators, vehicle simulators, ship simulators, and large spacecraft-mounted system. However, the high nonlinearity and strong coupling, resulted from the complex dynamic properties of spatial hydraulic parallel manipulator, always exists in such a system. Hence, it is very important to develop a simple and effective controller, with the merits of both classical PID control, simple and better real time that other controllers, and traditional feedback linearization approach, excellent static and dynamic control performances, for spatial 6-DOF hydraulic parallel manipulator to develop its merits. Many approaches have been employed for parallel manipulator [3-8], the strategies may be divided into two schemes, joint-space control [9-12], and workspace control [13-15]. The current joint space controllers are developed and implemented to treat the system as independent SISO system in joint space. A classical proportional plus integral plus derivative control in joint space has been applied in industry, but it can not always guarantee high control perfor
 *Address correspondence to this author at the Building 2F, Room 325, Science Park of Harbin Institute of Technology, No.2 Yi-kuang Street, Nangang District, Harbin 150001, China; Tel: +86-451-86402642-325 (Office), +86-15045107623 (Mobile); Fax: +86-451-86412558; E-mail: [email protected] mances for spatial parallel manipulator [3]. Su presented a robust auto-disturbance rejection controller in joint space for a 6-DOF parallel manipulator [16]. The adaptive control scheme [17, 18], artificial neural networks and fuzzy control [19, 20] are also suggested to spatial parallel manipulator. However, the characteristics of hydraulically driven system are not taken into account in the above controllers. Kim proposed a robust nonlinear control scheme in joint space for a hydraulic parallel manipulator based on Lyapunov redesign method, and the dynamics of hydraulic parallel manipulator is involved in the controller [21]. Yet, the dynamic coupling is not considered, which only be treated as disturbance. On the other words, the workspace control scheme seems to have the potential to provide a superior DOF control only after system states information is acquired via costly direct measurements or cumbersome state estimation. Davliakos presented a model-based control for a 6-DOF electro-hydraulic Stewart-Gough platform without considering uncertainties [22]. Chen studied the feedback linearization control of a two-link robot using a multi-crossover genetic algorithm [23]. Unfortunately, the method is not suitable for high realtime control system, especially for spatial 6-DOF hydraulic parallel manipulator. In this paper, the theoretical and simulation studies are performed to develop an approximate feedback linearization control scheme for spatial 6-DOF hydraulic parallel manipulator. The parallel manipulator is described as thirteen rigid bodies, and the hydraulic system is given with hydromechanics principle. The novel contribution is that a non-modelbased approximate feedback linearization control is presented in this paper. The presented controller can avoid the drawbacks of classical PID approach and traditional feedback linearization method, which is not based on a priori knowledge of dynamics but exploits a novel feedback linearization under PID control scheme. The data information of 118 The Open Mechanical Engineering Journal, 2011, Volume 5 Yang et al. position and pressure in previous step of each leg is employed in the control scheme, as the feedback of spatial 6DOF hydraulic parallel manipulator. The aim of the developed approximate feedback linearization control is to improve the control performance of path tracking of hydraulic parallel manipulator, through implementing the approximate linearization of nonlinear and strong coupling spatial hydraulic parallel manipulator, without aggravating the computational burden. 2. SYSTEM MODEL The spatial 6-DOF hydraulic parallel manipulator are configured with a moving platform and a fixed base connected by six identical hydraulic legs in parallel, illustrated in Fig. (1). Fig. (1). Spatial 6-DOF parallel manipulator. The OU-XUYUZU is body coordinate system fixed to the moving platform, and the origin OU is the center of mass of the moving platform with payload. The OL-XLYLZL is the inertial coordinate system. OL and OU are the same point in the initial pose of the spatial hydraulic parallel manipulator. Only one leg of the spatial 6-DOF parallel manipulator should be taken into account for deriving the dynamic equations, since all the legs of parallel manipulator are identical. The inertial forces of leg resulted from linear motions are described as, Jai,q T Jaci,ai T ma ! vaci + Jbci,ai T mb ! vbci ( ) = Flvi (1) where m a ,m b are the mass of rod and cylinder, respectively. v aci , v bci are 3x1 velocity vector of mass center of rod and cylinder, Jai,q !" 3x6 , J aci,ai , J bci,ai !" 3x3 are Jacobian matrices. vaci = Jaci,ai Jai,q ! q , vbci = Jbci,ai Jai,q ! q (2) In the inertial coordinate system, the inertial force of leg stemmed from rotation is formulated as, Jai,q T Jwi,ai T (Iai + Ibi ) ! ! li +! li " (Iai + Ibi )! li ( ) = Flai (3) where J wi,ai !" 3x3 is a Jacobian matrix, ! li "# 3 is angu-lar velocity vector of leg in inertial frame, I ai , I bi are the inertia matrices of rod and cylinder in inertial frame. Iai = Rl Ia a Rl T = Iax a lnilni T + Iay a (I ! lnilni T ) (4) Ibi = Rl Ib b Rl T = Ibx b lnilni T + Iby b (I ! lnilni T ) (5) ! li = Jwi,ai vai = Jwi,ai Jai,q ! q (6) where I a a , I b b !" 3x3 are the center inertia matrices of rod and cylinder of leg, l ni !" 3 is unit leg vector, computed by, l ni = (Ra i + c ! b i ) || Ra i + c ! b i || (7) where a i ,b i are the upper joint and lower joint coordinate, respectively, c is the position vector of mass center of the moving platform in inertial frame, R is a 3x3 rotation matrix, referred in [2]. The inertial forces and moments of the moving platform can be given as,