Continuum robots suffer large deflections due to internal and external
forces. Accurate modeling of their passive compliance is necessary for accurate
environmental interaction, especially in scenarios where direct force sensing
is not practical. This paper focuses on deriving analytic formulations for the
compliance of continuum robots that can be modeled as Kirchhoff rods. Compared
to prior works, the approach presented herein is not subject to the
constant-curvature assumptions to derive the configuration space compliance,
and we do not rely on computationally-expensive finite difference
approximations to obtain the task space compliance. Using modal approximations
over curvature space and Lie group integration, we obtain closed-form
expressions for the task and configuration space compliance matrices of
continuum robots, thereby bridging the gap between constant-curvature analytic
formulations of configuration space compliance and variable curvature task
space compliance. We first present an analytic expression for the compliance of
a single Kirchhoff rod. We then extend this formulation for computing both the
task space and configuration space compliance of a tendon-actuated continuum
robot. We then use our formulation to study the tradeoffs between computation
cost and modeling accuracy as well as the loss in accuracy from neglecting the
Jacobian derivative term in the compliance model. Finally, we experimentally
validate the model on a tendon-actuated continuum segment, demonstrating the
model's ability to predict passive deflections with error below 11.5\% percent
of total arc length