Mechanical Computation for Passive Force Control

Force control implemented by a passive mechanical device (perhaps a wrist) has inherent advantages over active implementations. A passive mechanical device can regain some of the versatility of its active counterpart if it incorporates mechanical elements with programmable parameters, e.g. damping coefficients or spring stiffnesses. We wish to characterize the range of accommodation matrices that a passive device may be programmed to possess. Here we review the known theoretical limits on the accommodation (inverse damping) matrices that any linear system of programmable dampers may adopt. Recent results [22, 26] show that such matrices are well suited to force-guided assembly. However, even with fully adjustable damping constants a mechanical device of fixed geometric design can attain only a subset of the all accommodation matrices. In this work we describe the set of attainable accommodation matrices, and show that each such matrix can be composed of a positive linear combinations of a fixed set of basis matrices. We show how the damping coefficients can be chosen to achieve a desired accommodation matrix, i.e. how to program this mechanical computer. We compare the space of attainable matrices to the space of all matrices, and suggest a method of visualizing it in low-dimensional examples.