<p>In minimally invasive surgery(MIS), the motion of surgical robots is constrained by access-port geometry and nonlinear actuation coupling, making remote center of motion (RCM) essential for safe operation. To address the complexity and limited generality of conventional mechanical RCM mechanisms, this paper proposes an RCM control framework compatible with multiple types of surgical instruments, aiming to enhance robotic motion performance and adaptability under nonlinear constraints. First, a quick-change flexible surgical actuator is developed, which integrates a ball-and-socket continuum manipulator(BSCM) and a universal driving mechanism, enabling dexterous end-effector motion under RCM constraints. To characterize the spatial configuration evolution of the BSCM under actuation, a curvature-parameterized modeling approach is adopted. By combining static equilibrium analysis with numerical approximation and accounting for the effects of friction and external load, nonlinear coupled forward and inverse kinematic models are established. On this basis, RCM constraints are incorporated into the nonlinear kinematic framework using geometric constraints and nonlinear optimization methods, leading to the development of RCM algorithms applicable to both rigid and flexible instruments, which significantly enhance the robot’s motion flexibility in multiple surgical scenarios. Experimental results demonstrate that the proposed flexible surgical robot achieves a spatial motion error of 0.72 mm (2.38%) and an RCM trajectory tracking error of 1.79 mm (5.9%), validating the effectiveness of the nonlinear kinematic model and the RCM motion algorithms. This study provides a feasible modeling and motion-planning framework for achieving safe and highly dexterous operation of minimally invasive surgical robots under complex constraints.</p>