Constant- and piecewise-constant-curvature models are widely used to simplify the kinematics of continuum robots, although the assumed curvature profiles are not necessarily inherent mechanical properties of the robot and may therefore be violated during operation. Rather than assuming a prescribed curvature profile in the kinematic model and estimating deviations from it, this paper presents a multi-segment scissor-based support structure that mechanically constrains a continuum robot to exhibit prescribed constant or spatially varying curvature profiles. The design uses Taylor approximations to enable an overconstrained multi-joint linkage to predictably distribute global bending among the small-angle movements of multiple flexures. The geometric coupling is realized using compliant flexure joints, enabling the support structure to be manufactured as a single piece. The structure is integrated with a tendon-actuated continuum robot with a flexible backbone. Experiments show that the structure makes the robot resistant to deformations other than the prescribed curvature, significantly enhancing shape precision compared to classical flexible-backbone designs. This constrains the robot from making S-shapes when external loads are applied in the opposite direction of the bending. By mechanically constraining the admissible deformation, the proposed structure reduces the number of independent shape variables that must be estimated.

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