A mathematical basis for the design and design optimization of adaptive trusses in precision control

A mathematical basis for the optimal design of adaptive trusses to be used in supporting precision equipment is provided. The general theory of adaptive structures is introduced, and the global optimization problem of placing a limited number, q, of actuators, so as to maximally achieve precision control and provide prestress, is stated. Two serialized optimization problems, namely, optimal actuator placement for prestress and optimal actuator placement for precision control, are addressed. In the case of prestressing, the computation of a 'desired' prestress is discussed, the interaction between actuators and redundants in conveying the prestress is shown in its mathematical form, and a methodology for arriving at the optimal placement of actuators and additional redundants is discussed. With regard to precision control, an optimal placement scheme (for q actuators) for maximum 'authority' over the precision points is suggested. The results of the two serialized optimization problems are combined to give a suboptimal solution to the global optimization problem. A method for improving this suboptimal actuator placement scheme by iteration is presented.