This paper investigates the problem of closed-loop control of a small number of parameters by allocating actuation in a system with many binary degrees of freedom, using an actual large-scale air-jet table as an example. In this system, the desired force and torque are produced by a large number of spatially distributed binary air jets directing individual forces on an object. Various algorithms for solving the force allocation problem-determining the appropriate valve states in this hyper-redundant system-are investigated. The algorithms range from discrete optimal search to continuous constrained optimization to a hybrid hierarchical approach that can be distributed. The latter consists of using the continuous optimal solutions to recursively break the large optimization problems into smaller problems that can be solved using optimal search methods or precomputed lookup tables. A tradeoff between computation time and allocation error was found. The optimal algorithms yield low errors but the time is exponential in the number of actuators, while the continuous solutions execute quickly but yield larger errors. The hybrid hierarchical optimal algorithms give the best compromise between these conflicting goals, and their applicability spans the full range of degrees of freedom from a few to many thousands. These hierarchical algorithms are useful in many such highly redundant systems.
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