# Solving the Time-Varying Inverse Kinematics Problem for the Da Vinci Surgical Robot
## 1 Introduction  
In recent years, medical robots have increasingly been used as fundamental surgical instrument/equipment [1], among which the Da Vinci system developed by Intuitive Surgical Company is the most successful [2]. The Da Vinci robotic system consists of three components: the surgeon console (master manipulators), the patient trolley (slave manipulators), and the imaging system [3]. The surgeon manipulates two master handles at the master remote console, which can acquire high-resolution, binocular, three-dimensional, magnified views of the operative field as compared with open surgery [4]. However, this kind of master-slave operation mode poses a great challenge to the real-time and accurate performance of the master-slave mapping control, among which the inverse kinematics problem is the first obstacle. The Da Vinci system and similar surgical robot systems satisfy the following two features:  
1. contain mechanical structures to generate Remote Center of Motion [5] (RCM, a fixed point on the active arm, there are two methods to generate RCM based motion: (a) virtual RCM, (b) physically constrained RCM [6]), and the structure can be simplified;
2. do not satisfy the Pieper principle. At present, the main methods of robotic inverse kinematics include the analytic method and the numerical method. The analytic approach uses vector, spinor or Lie algebra to obtain results [7; 8; 9], which can only obtain satisfied solutions in limited robotic configurations that should satisfy the Pieper principle [10; 11], etc. However, to imitate the surgeon operation, the manipulators of most thoracic and abdominal surgery robots, including the Da Vinci system, contain rotational/translational kinematic pairs, tilt angle joints, and offset joints at the tip (artificial wrist part) that are designed for additional flexibility, which is beyond the solution capability of the analytic approach. One feasible strategy is to simplify the kinematic model to acquire analytical results. For example, [PERSON] et al. and [PERSON] et al. [12; 13] obtained an approximate solution by separating the position and orientation of the slave manipulators, the solution deviation of which needs to be manually eliminated by experienced surgeons via visual inspection. [PERSON] et al. [14] solved the inverse kinematics of a 4-DOF surgical robot using the elimination method, and [PERSON] et al. [15] used unit dual quaternions to model the kinematics, however, outcomes of their studies benefited strongly from the simplicity of the arm configuration. In addition, [PERSON] et al. [16; 17] obtained an approximate solution based on the differential transformation that increases error feedbacks to compensate the cumulative error, the final precision of which can reach 0.18 mm. Another way to solve the inverse kinematics problem of the surgical robot that does not satisfy the Pieper principle is the Jacobian matrix-based numerical method [18]. However, this method involves a Jacobian matrix, which may lose versatility due to its singularity problem as well as the low convergence speed of the iterative solutions. To improve the numerical solution performance, a numerical method based on the position increment of robotic joints was proposed [18]. Specifically, to make this method suitable for the inverse kinematics analysis of a robotic manipulator with an offset wrist, [PERSON] et al. [19] used the method of decoupling DOFs based on a cut-off point that is limited by the manipulator configuration. In addition, modern methods such as the neural network method [20], the genetic algorithm [21], the electromagnetism-like mechanism [22; 23], the hybrid electromagnetism-like mechanism [24], etc., have been gradually explored for possible solutions of the inverse kinematics of surgical robots. However, these methods either require a large amount of sample data or suffer from poor convergence speed; therefore, these methods are not suitable for the rapid development of a master-slave manipulator in surgical robots.