Analysis of Large Amplitude, Short Duration Control Forces for Guiding Underactuated Mechanical Systems into Safe Operating Regions

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In a fully actuated mechanical system, an independent control force is available for each degree of freedom. Systems with fewer independent controls than degrees-of-freedom are called "underactuated." Many important systems are naturally of this class, including certain types of missiles, satellites, underwater vehicles, and bipedal robots. As might be expected from the shortage of control inputs, underactuated systems are challenging to stabilize and to steer. This project improves upon existing approaches by using large-amplitude, short-duration control forces -- called impulsive forces -- to expand the region of attraction of the system. The region of attraction is the set of initial configurations and velocities from which the system will be able to reach its desired operating point. If the system ever leaves its region of attraction, it will depart from its desired behavior, corresponding to, for example, the loss of control of a missile or spacecraft. Expanding the region of attraction means safer operation, since the system can tolerate upsets due to large disturbance forces or unexpected initial conditions. A major new contribution of this project will be to bridge the gap between purely theoretical studies and practical applications, through careful experimental validation on widely accepted benchmark test bed systems. Dynamical systems are continuously subjected to disturbances and their ability to reject them largely depend on the stability property of their equilibrium configurations. This research will enlarge the region of stable operation of underactuated dynamical systems and reduce the incidence of unstable behavior, behavior that typically has negative consequences. The region of stable operation will be enlarged by application of impulsive forces, which will be included in the set of admissible control inputs. In this project, impulsive control will be implemented using high-gain feedback over short intervals of time and mathematical tools such as singular perturbation methods will be used to analyze and design the high-gain feedback systems. The implementation of high-gain feedback will require the use of very fast observers to estimate the unmeasured velocities. To this end, high-gain observers will be designed and analyzed via multi-time-scale singular perturbation methods. There are many challenging technical issues that need to be solved to make such designs feasible. This includes the development of efficient algorithms that can transfer the system configuration from outside the region of stable operation to inside the region, using a single application or multiple applications of impulsive inputs. The developed algorithms will be tested experimentally on simple underactuated systems; but the methods will be applicable to more complex problems such as orbital transfer of satellites, rapid maneuvering of missiles and underwater vehicles, and active prosthetic devices.