  Soft robot are celebrated for their propensity to enable compliant and
complex robot-environment interactions. Soft robotic manipulators, or slender
continuum structure robots have the potential to exploit these interactions to
enable new exploration and manipulation capabilities and safe human-robot
interactions. However, the interactions, or perturbations by external forces
cause the soft structure to deform in an infinite degree of freedom (DOF)
space. To control such system, reduced order models are needed; typically
models consider piecewise sections of constant curvature although external
forces often deform the structure out of the constant curvature hypothesis. In
this work we perform an analysis of the trade-off between computational
treatability and modelling accuracy. We then propose a new kinematic model, the
Piecewise Affine Curvature (PAC) which we validate theoretically and
experimentally showing that this higher-order model better captures the
configuration of a soft continuum body robot when perturbed by the external
forces. In comparison to the current state of the art Piecewise Constant
Curvature (PCC) model we demonstrate up to 30\% reduction in error for the end
position of a soft continuum body robot.
