Mechanics of bending, torsion, and variable precurvature in multi-tube active cannulas

Active cannulas are a relatively new continuum robot subclass characterized by their use of preshaped tubes that transmit bending moments as they slide within one another and are axially rotated. Previous (experimentally vetted) mechanics-based models of active cannula shape assume piecewise constant precurvature of component tubes, and neglect torsion in curved sections of the device. Recently a general, coordinate-free, energy-based framework for active cannula shape has been formulated that relaxes these requirements and includes all prior models as special cases. However, only the 2-tube, constant-precurvature case has thus far been explored in detail using the framework. In this paper we consider the general case of an arbitrary number of component tubes and precurvatures that vary with arc length, deriving a set of differential equations that capture both bending and torsional effects continuously along the active cannula backbone.We then show how to solve these differential equations numerically to describe active cannula shape.