This paper presents the first kinematic modeling of tendon-driven rolling-contact joint mechanisms with general contact surfaces subject to external loads. We derived the kinematics as a set of recursive equations and developed efficient iterative algorithms to solve for both tendon force actuation and tendon displacement actuation. The configuration predictions of the kinematics were experimentally validated using a prototype mechanism. Our MATLAB implementation of the proposed kinematic is available at https://github.com/hjhdog1/RollingJoint.