The collaborative transportation of objects between humans and robots is a fundamental task in physical human-robot interaction. Most of the literature considers the rigid co-grasping of non-deformable items in which both the human and the robot directly hold the transported object with their hands. In this letter, we implement a control strategy for the collaborative manipulation of a cable-suspended platform. The latter is an articulated and partially deformable object that can be used as a base where to place the transported object. In this way, the human and the robot are not rigidly coupled, ensuring a greater flexibility in the partners' motions and a safer interaction. However, the uncertain dynamics of the platform introduces a greater possibility of unintended collisions with external objects, which must be distinguished from contacts arising when a load is placed on or removed from the platform. This letter proposes a contact detection and distinction strategy to address this challenge. The proposed cable-suspended manipulation framework is based only on force sensing at the robot end-effector, and was tested with ten users.