This paper proposes a planar tendon-driven joint with a soft cylinder contacting a rigid plane and its modelbased controller. In previous studies of planar tendon-driven joints with soft materials, those angles were only controlled despite the position displacement due to material deformation. Concurrently, position and angle displacements are addressed in the field of the robotic hand. Although some of the studies designed controllers using soft rolling contact models, their models cannot be applied in practical cases simply due to the effect of the property of the soft material. Even if it can be modeled, their feedback controllers also cannot be applied to the joint if the model is different from theirs. Although a controller design based on the obtained model is required, this realization is difficult because a constraint due to the soft rolling contact becomes a non-holonomic one. To resolve those problems, we set the soft rolling contact as a gray-box model and design the controller based on this modeled constraint. In the controller, the modeled constraint is used to generate a reference path. A soft rolling contact model is constructed through experiments, and then we validate the proposed controller by simulation using the modeled constraint.