Refined Wave-Based Control Applied to Nonlinear, Bending, and Slewing Flexible Systems

The problem of controlling the position and vibration of open-chain flexible structures undergoing fast maneuvers is of wide interest. In this work, the general flexible structure is actuated by a single actuator at one end, which, depending on the case of interest, is capable of rotating, translating, or simultaneously translating and rotating the root of the flexible system. The goal is to control the motion of the entire flexible system from rest to rest. This needs a simultaneous synthesis of position control and active vibration damping. A new strategy is presented based on further developments of wave-based control. As before it views the actuator motion as simultaneously launching and absorbing mechanical waves into and out of the system. But a new simple method of resolving the actuator motion into two waves is presented. By measuring the elastic forces exchanged at the interface between the actuator and the rest of the system, a returning displacement wave can be resolved. This is then added to a set, launch wave to determine the actuator motion. Typically the launch wave is set to reach half the target displacement, and the addition of the return wave absorbs the vibration while simultaneously moving the system the second half of the target displacement, neatly achieving the two goals in one controlled motion. To date wave-based control has been applied to lumped, second-order, longitudinally vibrating systems. The refined method avoids a difficulty that previously arose in some contexts, thereby making wave-based control even more generic. It can easily control nonlinear elastic systems, laterally bending systems (in the sense of Euler–Bernoulli beams), and slewing systems where lateral translation and system rotation are strongly coupled. Numerical simulation results are presented for controlled, rest-to-rest maneuvers of representative flexible structures, all controlled using the same (linear) algorithm. The first case is control of a string of rigid bodies interconnected by nonlinear springs. The second problem is the rotational control of a very flexible one-link planar manipulator. Finally, in an extension of the previous system, the actuator both translates and rotates, slewing the flexible system to a target lateral displacement and a target rotation angle simultaneously. The strategy is found to be remarkably effective with many advantages. It seamlessly integrates position and vibration control. It is rapid, robust, energy efficient, and computationally light. It requires little sensing, little knowledge of the flexible system dynamics, and copes well with nonideal actuator behavior. It is generic and easily handles a wide variety of flexible systems. It can get the entire system to stop dead exactly at target with little vibration in transit.