Cooperative multi-input shaping for arbitrary inputs

Input shaping is an effective tool to reduce residual vibration when maneuvering flexible structures. For multiple inputs, a common scheme is to design a shaper with zeros at every vibrational mode (a "single-input shaper") and use it for all inputs. Recently, other techniques have been developed in which the inputs "cooperate" to cancel the modes, resulting in shorter shapers and therefore less delay. Most of these methods require all inputs to be the same or similar, and in some cases require knowledge of what the inputs are beforehand. The paper first presents a cooperative shaping technique for the case where inputs are arbitrary but known, and a procedure for improving robustness is also discussed. A method is then presented for cooperative shaping when the inputs are unknown. Both schemes realize a performance increase over single-input shaping approaches.