Approximation and control of curvilinear shapes via deployable mechanisms with two degrees of freedom

This paper proposes a multi-unit mechanism, which can be used to approximate, with two independent degrees of freedom, the shape of the geometric outline of an arbitrarily large area. The new device is a variant of a recently introduced planar deployable mechanism with two uncoupled degrees of freedom, built from identical units, each combining Sarrus and scissor linkages. Similar units, but with varying sizes, are used in the new device, which is able to change its elliptical physical boundary by varying independently the two parameters in the standard ellipse equation. The size and placement of the deployable units and their links are analyzed and selected for getting the expected geometric shape. The relationships between the number of dividing lines and the approximating accuracy and the degree of overconstraint, respectively, are calculated. A similar deployable mechanism controlling a hyperbola is also outlined. Kinematic analysis and simulated models show that the mechanisms can be used to approximate geometric curves, as desired.