###### Abstract  
It is difficult to install the sensor on the end effector of a minimally invasive surgical robot with a narrow space within which to obtain the position and grasping force; thus, the elongation effect of cable-driven surgical instruments results in low precision control, which may damage the tissues and organs of patients. A novel transmission backlash compensation and grasping force saturation limitation algorithm based on the tension and displacement transmission model of a cable-pulley system is proposed to improve the operation's accuracy and safety. The algorithm considers the force and position transmission characteristics of each stage of the cable-pulley system including the transition stage. Experimental verifications show that the proposed algorithm can significantly improve the position tracking accuracy of surgical instruments and the safety of grasping operations. m _Applied_  
_sciences_  
**Classification: [PERSON]; [PERSON]; [PERSON] Transmission Backlash Compensation and Grasping Force Estimation of Surgical Instruments for the Laparoscopic Minimally Invasive Surgery Robot. tension transmission and first formulated the transmission characteristics with apparent tendon-stiffness and equivalent backlash [10]. [PERSON] et al. presented a viscoelastic mode that describes the tendon-sheath driving system with a lumped parameters model, and also proposed a control strategy to compensate for the nonlinear effects [11; 12]. [PERSON] et al. proposed an asymmetric backlash hysteresis model that characterized the transmission phenomena of the tendon-sheath mechanism during the loading and unloading phases for endoscopic systems, and an efficient parameter identification method was used to estimate the model's parameters [13; 14]. [PERSON] et al. derived a dynamic model of a coupled cable-pulley structure while considering the material tensile characteristics of the cable; this model derived an estimate of the operating force of the end effector as well as the states of all inner pulleys without encoders [15]. [PERSON]. [PERSON] et al. presented a method with which to estimate the end effector parameters using only a force and position sensor at the proximal site [16]. [PERSON] et al. proposed the mathematically rigorous distributed parameter model for cable-conduit actuation with any curvature and initial tension profile across the cable, which is described by a set of partial differential equations in the continuous time-domain, and is also discretized for its effective numerical simulation of cable motion and tension transmission across the cable [17]. [PERSON] et al. proposed a real-time position compensation control method based on the deformation of flexible cables, by establishing a torque transmission model of double tendon-sheaths under arbitrary load conditions, and developed two open-loop control strategies for friction and deformation compensation [18]. [PERSON] et al. proposed a new motion transmission model of a tendon-sheath pulley mechanism to estimate the motion error and force of da Vinci system surgical instruments, and used this model for the position control of the instrument tip to eliminate hysteresis and achieve an accurate trajectories track [19; 20]. For the cable-pulley transmission model, [PERSON] et al. conducted more rigorous analyses to improve the classical capstan equation by including both rod bending rigidity and power-law friction into the formula, and also investigated the effects of important parameters on tension transmission efficiency [21; 22]. [PERSON] et al. proposed analytical methods for predicting the transmission capability of a precise cable drive including the bending rigidity, and also developed the analytical methods required to predict the transverse oscillation frequency of a precise cable drive to enable designers to better assess dynamic performance in the design study phase [23; 24]. [PERSON] et al.