Control systems and methods for a medical instrument use measurements to determine and control the tensions that actuators apply through instrument transmission systems. The use of tension and feedback allows control of a medical instrument having transmission systems that provide non-negligible compliance between joints and actuators even when the positions of joints cannot be directly related to actuator positions. One embodiment determines joint torques and tensions from differences between desired and measured joint positions. Another embodiment determines joint torques and tensions from differences between desired and measured positions of a tip of the instrument. Determination of tensions from joint torques can be performed using sequential evaluation of joints in an order from a distal end of the instrument toward a proximal end of the instrument.