A novel closed-form solutions method based on the product of exponential model for the minimally invasive surgical robot

Purpose This study aims to represent a novel closed-form solutions method based on the product of the exponential model to solve the inverse kinematics of a robotic manipulator. In addition, this method is applied to master–slave control of the minimally invasive surgical (MIS) robot. Design/methodology/approach For MIS robotic inverse kinematics, the closed-form solutions based on the product of the exponential model of manipulators are divided into the RRR and RRT subproblems. Geometric and algebraic constraints are used as preconditions to solve two subproblems. In addition, several important coordinate systems are established on the surgical robot and master–slave mapping strategies are illustrated in detail. Finally, the MIS robot can realize master–slave control by combining closed-form solutions and master–slave mapping strategy. Findings The simulation of the instrument manipulator based on the RRR and RRT subproblems is executed to verify the correctness of the proposed closed-form solutions. The fact that the accuracy of the closed-form solutions is better than that of the compensation method is validated by the contrastive linear trajectory experiment, and the average and the maximum tracking errors are 0.1388 mm and 0.3047 mm, respectively. In the animal experiment, the average and maximum tracking error of the left instrument manipulator are 0.2192 mm and 0.4987 mm, whereas the average and maximum tracking error of the right instrument manipulator are 0.1885 mm and 0.6933 mm. The successful completion of the animal experiment comprehensively demonstrated the feasibility and reliability of the master–slave control strategy based on the novel closed-form solutions. Originality/value The proposed closed-form solutions are error-free in theory. The master–slave control strategy is not affected by calculation error when the closed-form solutions are used in the surgical robot. And the accuracy and reliability of the master–slave control strategy are greatly improved.