978-1-7281-8878-2/20/$31.00 ©2020 IEEE 13 Multivariable Control System of the Stewart Platform Oleg Gasparyan 1 , Azatuhi Ulikyan 2 National Polytechnic University of Armenia Yerevan, Armenia 1 ogasparyan@gmail.com, 2 azatuhi.ulikyan@gmail.com Abstract—The paper is devoted to the problem of designing matrix regulators for a special class of multivariable feedback control systems that are described by the so-called simple symmetrical transfer matrices. The typical example of such systems is the control system of a Stewart platform, also known as hexapod. The exposition is based on the characteristic transfer function method, which allows reducing the investigation of an N -dimensional (i.e. having N inputs and N outputs) cross-connected control system to investigation of N isolated fictitious systems with one input and one output. It is shown that the design of matrix regulators for simple symmetrical systems can be accomplished, irrespective of the system dimension N , by the design of two scalar regulators based on the conventional methods of classical feedback control. Keywords—Stewart platform, hexapod, multivariable control system, simple symmetrical system, characteristic transfer function, matrix regulator I. INTRODUCTION Demand on high precision motion systems has been increasing in recent years. Especially this concerns the parallel manipulators that possess high stiffness, fast motion and accurate positioning capability. Stewart platforms (also called hexapods) belong to the most popular types of parallel manipulators that are widely used in such applications as flight simulators, positioning of large telescopes’ mirrors, underwater research, orthopedic surgery, and many others [1- 4]. The control systems of hexapods belong to the class of multiple-input multiple-output (MIMO) control systems, the design of which is one of the central issues in multivariable feedback control [5, 6]. The paper is devoted to the task of designing matrix regulators for hexapods’ control systems. The proposed approach is based on the characteristic transfer functions (CTF) method [5, 7], which allows reducing the design of an N -dimensional square (i.e. having N inputs and N outputs) MIMO control system to the design of N one- dimensional independent systems on the basis of conventional methods of classical feedback control [8]. II. CONTROL SYSTEMS OF HEXAPODS Physically, a hexapod consists of a movable payload platform connected to the fixed base by six variable-length struts (or legs). The length of each strut can be controlled independently by six linear actuators and sensors, to achieve independent translational and rotational motions along the X, Y, and Z axes. The simplified kinematic scheme of the hexapod is shown in Fig. 1. Fig. 1. The kinematic scheme of the hexapod Generally, the control systems of hexapods belong to the class of MIMO (or multivariable) feedback control systems, One of the most effective methods of anaysis and design of such systems is the CTF method [5, 7]. In accordance to that method, the open-loop and closed-loop transfer matrices () Ws and () s Φ of the MIMO system in Fig. 1 can be represented, using the similarity transformation, in the following canonical form: { } 1 () () () () i Ws C s diag q s C s - = (1) 1 1 () () [ ( )] () () () 1 () i i q s s I Ws Ws C s diag C s q s - -   Φ = + =   +   , (2) where I denotes an identity matrix, ()( 1, 2,..., ) i q s i N = are the characteristic transfer functions (“eigenvalues”) of the open-loop system (we assume for simplicity that all () i q s are distinct), and the modal matrix () Cs is composed of the linearly-independent vectors () i c s forming the canonical basis of the system [5]. The stability of the MIMO system in Fig. 2 is determined by the roots of the characteristic equation [ ] [ ] 1 det () 1 () 0 N i i I Ws q s = + = + = ∏ , (3) which splits into the following N characteristic equations of the so-called one-dimensional characteristic systems 1 () 0, 1, 2,..., . i q s i N + = = (4) Authorized licensed use limited to: University of Gothenburg. Downloaded on December 19,2020 at 20:54:15 UTC from IEEE Xplore. Restrictions apply.