Wave-based control of planar motion of beam-like mass–spring arrays

Wave-based control (WBC) is a simple and relatively new technique for motion control of under-actuated flexible systems. To date it has been mainly applied to rectilinear lumped flexible systems. The current work focuses on a development of WBC to control two-dimensional beam-like structures in which an actuator, attached to one end, acts to translate and rotate the structure through an arbitrary path in the plane. In this work, first a lumped model of a beam is developed using mass–spring arrays. The lumped beam model is of interest here as a benchmark control challenge. It can also be considered as a model of various lumped or distributed mass structures. To check the latter, the mode shapes and frequencies are first compared with those of classical beam theory. This involved a new technique to find mode shapes and frequencies for arrays. The control strategy is then presented and tested for a range of manoeuvres. As a system to be controlled, the mass–spring array presents many challenges. It has many degrees of freedom, many undamped vibration modes, is highly under-actuated, and sensing of system states is difficult. Despite these challenges, WBC performs well, combining a fairly rapid response with active vibration damping and zero steady-state error. The controller is simple to implement and of low order. It does not need or use any system model and is very robust to system changes.