Abstract
A dialytic-elimination and Newton-iteration based quasi-analytic inverse kinematics approach is proposed for the 6 degree of freedom (DOF) active slave manipulator in the Da Vinci surgical robot and other similar systems. First, the transformation matrix-based inverse kinematics model is derived; then, its high-dimensional nonlinear equations are transformed to a high-order nonlinear equation with only one unknown variable by using the dialytic elimination with a unitary matrix. Finally, the quasi-analytic solution is eventually obtained by the Newton iteration method. Simulations are conducted, and the result show that the proposed quasi-analytic approach has advantages in terms of accuracy (error < 0.00004 degree (or mm)), solution speed (<20 ms) and is barely affected by the singularity during intermediate calculations, which proves that the approach meets the real-time and high-accuracy requirements of master–slave mapping control for the Da Vinci surgical robots and other similar systems. In addition, the proposed approach can also serve as a design reference for other types of robotic arms that do not satisfy the Pieper principle.
Highlights
In recent years, medical robots have increasingly been used as fundamental surgical instrument/equipment [1], among which the Da Vinci system developed by Intuitive SurgicalCompany is the most successful [2]
A finite number of values are obtained by extracting values within these available ranges at a certain angle interval to solve the forward kinematics, the results of which are stored as the initial value database for the inverse kinematics solution
Vinci surgical robot established by using a coordinate transformation matrix method according to
Summary
Medical robots have increasingly been used as fundamental surgical instrument/equipment [1], among which the Da Vinci system developed by Intuitive Surgical. Fu 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]. The dialytic elimination method [25] is used to transform the high-dimensional equations into a nonlinear equation containing only one unknown variable, which is solved by the Newton iteration method In applying this method to the inverse kinematics solution of slave manipulator, we show dual superiorities: (a) barely affected by the singularity problem and (b) maintain rapidity. Da Vinci system; Section 3 solves the inverse kinematics by the Newton iteration method; Section 4
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