This paper proposes a joint control method for tendon-driven mechanisms (TDMs) with branching tendons, in which multiple tendons are connected at a point, and which are often found in musculoskeletal systems. TDMs usually require the same number of tendons as actuators, which are one of the heaviest components in a robotic system. The utilization of branching tendons is useful for reducing the number of actuators needed when making lightweight robotic mechanisms, such as prosthetic hands. However, the under-actuation of branching tendons makes it difficult to accurately control the joint motion of TDMs. Therefore, TDMs with branching tendons have been used only for simple adaptive grasping mechanisms. In this paper, we derive the tendon kinematics of TDMs with branching tendons and design a joint PD controller for the mechanisms. We demonstrate the stability of the control system using Lyapunov's direct method. We show that the bias force setting, which does not appear in conventional TDMs, is important in the realization of accurate control in TDMs with branching tendons. Simulations were performed to evaluate the proposed method.