Tendon-driven mechanisms used in robotic surgery exhibit strong nonlinearities, particularly a static backlash-like hysteresis, in their motion transmission behavior. In this paper, an extension of a previously developed model is proposed that allows for estimation of angular displacements in multi-DOF tendon-driven devices where special attention is given to the coupling effect between DOFs. The proposed model consists of the conventional coupling matrix and a novel elongation matrix which compensates for the coupled hysteretic effect. The model is applied to the problem of position estimation in three DOFs (one pitch and two grasping DOFs) of a da Vinci^® surgical instrument. As a further extension, a preliminary dynamic model is also suggested to deal with high-frequency inputs. According to the experimental results obtained, the proposed quasi-static model can describe the transmission behavior with goodness-of-fit of 76-92 per cent, and the estimates are improved by 35-72 per cent in terms of the RMSE for the proposed dynamic model as compared to the conventional rigid model.