WAVE-BASED MODELLING AND CONTROL OF LUMPED, MULTIBODY FLEXIBLE SYSTEMS

The motivation for this work is the control of flexible mechanical systems, such as long, light robot arms, gantry cranes, and large space structures, modelled as a series of rigid bodies or lumped masses interconnected by translational or rotational elastic elements, with an actuator at one end and a "free" boundary at the other end. The actuator is attempt- ing to position a load at the far end through an intermediate system that is flexible, while con- trolling the vibrations due to the flexibility. Powerful, generic control strategies have been developed based on interpreting the actuator motion as launching and absorbing mechanical waves into and out of the flexible system. This paper explores the validity and nature of such wave concepts in lumped systems. A new wave-based model of uniform mass-spring systems is first proposed and verified. A standard case is n lumped masses and springs in series. The wave-based model then has a string of n transfer functions going out, n coming back, one at the free boundary and a summing junction at the actuator. For the uniform case, all transfer functions are identical, and of low order, no matter how long or short the system. For the non-uniform case they vary, but are still of low order. Then a wave model for non-uniform systems is presented and analyzed. Useful simplifications and approximations are also pre- sented. The conclusion is that, unexpectedly, wave-based models for all such dynamic systems are possible. There are some ambiguities, but the superposed motions are exact. The models prove decisively useful for control. By measuring and then responding to the "returning wave", a controller can achieve superb end-point control, getting a flexible system to stop dead at the target position. The returning wave reveals the system dynamics and tells the con- troller exactly how to bring the flexible system to rest by affecting a time-reversed image of the start-up motion. These control systems are robust to actuator limitations and even to large unknown system changes. They are generic, require very little system modelling, need only local sensing, are computationally light, and are easy to implement. Beyond the control ap- plication, the wave approach provides a new analysis tool of interest for a range of flexible mechanical systems, and for dynamics systems in general.