Experimental results in the vibration control of a highly flexible manipulator

The aim of this study is to investigate the motion of a two-axis single-link flexible manipulator. In particular the motion in the vertical plane is modelled, and a mixed sensitivity H controller is designed. This controller is found to give a better tracking response than a classical controller, and can be designed to be robust to payload variation. However, careful choice of weighting functions is required to ensure good performance. INTRODUCTION A robot manipulator which has been designed to reduce weight and improve speed, or to have a particularly long reach, is liable to exhibit significant structural flexibility in one or more links of the arm. If neglected this will cause vibration and static deflection. Thus the flexibility must be modelled and controlled if the manipulator design is to be viable. Many researchers have addressed the problem of controlling flexible manipulators over the last twenty years. Initial studies concentrated on the motion of a single link flexible arm driven in the horizontal plane. Latterly multi-link manipulators have been investigated, still largely in the horizontal plane; this presents a complex non-linear vibration control problem. In the present study a two-axis single link manipulator is considered, driven about a revolute hub in both vertical and horizontal planes. The objectives are: • to model the manipulator, including any interactions between horizontal and vertical motions • to design a robust H-infinity controller • to tackle the particular difficulties encountered in controlling the manipulator in the vertical plane under the influence of gravity. This paper focusses on motion in the vertical plane, and follows on from simulation results contained in [1]. A number of H controller design approaches have been proposed for flexible manipulators. Lenz et al [2] developed a mixed sensitivity H controller that showed considerable promise. The majority of the work that has been conducted has concentrated on single payload loading conditions. However, Cashmore [3] used an uncertainty model to take into account some variation in payload conditions so utilising the robustness of the H controller. Recent work on H controllers for flexible manipulators has been published by Matsuno & Tanaka [4], Estiko et al [5], Landau et al [6], and Jovik & Lennartson [7]. All of these authors highlight the difficulties associated with choosing the weighting functions. EXPERIMENTAL FLEXIBLE MANIPULATOR An experimental single link two degree of freedom flexible manipulator has been constructed (Figures 1 and 2). The two degrees of freedom allow the end effector to be moved in the horizontal and vertical planes. Table 1 contains component specifications. FIGURE 1: Experimental flexible manipulator FIGURE 2: Flexible manipulator schematic TABLE 1: Experimental manipulator specifications Flexible link Length (L) 1.00 m Mass (Mb) 0.34 kg Offset from motor axes(R) 0.06 m Flexural rigidity (EI) 72.2 Nm Payloads Mass (Mp ) variable from 0.075 kg to 0.753 kg Hub Horizontal inertia (Jhy) 0.468 kgm 2 Vertical inertia (Jhz) 0.249 kgm 2 Torque motors Maximum torque 33 Nm Rotor inertia 0.013 kgm Torque constant 1.2 Nm/A Maximum control signal ± 8.5V Encoders Line count 3600 (incremental) Resolution (x4 counting) 4.4×10 rad Potentiometers Linearity ±0.2% Linearity wrt link tip position ±3×10 rad Interface A/D (0-10V differential inputs) 14bit D/A ( ±10V outputs) 16 bit Encoder interface card 24 bit The flexible link is an homogeneous, cylindrical, aluminium rod with constant properties along its length. The motion of the link is not artificially constrained in any plane. The link is symmetrical about its longitudinal axis, and this axis intersects both horizontal and vertical drive axes at the hub. Each of the interchangeable payloads is symmetrical about the longitudinal axis of the link, with centre of gravity positioned coincident with the link tip. The vertical and horizontal displacements of the tip of the link are measured. Each measurement is composed of two parts as shown in Figure 3. Firstly, the position of the link tip in relation to the hub axis is measured using a precision potentiometer (θflex). This is combined with the angle of the hub relative to earth axes, measured by an incremental optical encoder (θhub). These signals are interfaced to a PC which is used to implement control algorithms. R L tot θflex θhub Hub Flexible link FIGURE 3: The components of link tip position MODELLING Link element model The Newton-Euler method for modelling transverse vibrations in beams will be used. Only motion in the vertical plane will be considered. The approach broadly follows that of Meirovitch [8], with extensions to accommodate gravitational effects. A three dimensional free body diagram of a small element, length Δx, of the flexible link is shown in Figure 4. In the figure, shear forces are denoted by Vz , and bending moments by Mz . Neglecting angular inertia and shear deformation of the element, and neglecting second order effects, the following link element equation for vertical motion can be derived: − − = mg EI Z x t x m Z x t t n ∂ ∂ ∂ ∂ 4