General closed-form inverse kinematics for arbitrary three-joint subproblems based on the product of exponential model

Abstract

The inverse kinematics problems of robots are usually decomposed into several Paden–Kahan subproblems based on the product of exponential model. However, the simple combination of subproblems cannot solve all the inverse kinematics problems, and there is no common approach to solve arbitrary three-joint subproblems in an arbitrary postural relationship. The novel algebraic geometric (NAG) methods that obtain the general closed-form inverse kinematics for all types of three-joint subproblems are presented in this paper. The geometric and algebraic constraints are used as the conditions precedent to solve the inverse kinematics of three-joint subproblems. The NAG methods can be applied in the inverse kinematics of three-joint subproblems in an arbitrary postural relationship. The inverse kinematics simulations of all three-joint subproblems are implemented, and simulation results indicating that the inverse solutions are consistent with the given joint angles validate the general closed-form inverse kinematics. Huaque III minimally invasive surgical robot is used as the experimental platform for the simulation, and a master–slave tracking experiment is conducted to verify the NAG methods. The simulation result shows the inverse solutions and six sets given joint angles are consistent. Additionally, the mean and maximum of the master–slave tracking experiment for the closed-form solution are 0.1486 and 0.4777 mm, respectively, while the mean and maximum of the master–slave tracking experiment for the compensation method are 0.3188 and 0.6394 mm, respectively. The experiments results demonstrate that the closed-form solution is superior to the compensation method. The results verify the proposed general closed-form inverse kinematics based on the NAG methods.

Graphical abstract

Keywords

inverse kinematics / Paden–Kahan subproblems / three-joint subproblems / product of exponential / closed-form solution

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Nomenclature

Abbreviations
D–H	Denavit–Harbenterg
DOF	Degree-of-freedom
NAG	Novel algebraic geometric
POE	Product of exponential
R, T	revolute joint and translational joint, respectively
RCM	Remote center of motion
Variables
c, d, p, p1, p2, p3, and q	Position vectors of points c, d, p, p1, p2, p3, and q, respectively
{\mathit{g}}_{\text{st}}(\mathbf{0}), {\mathit{g}}_{\text{st}}(\mathit{\theta })	Initial transformation matrix and transformation matrix at different joint angles of the tool frame, respectively
lc2	Distance between points c and p2
lij (i, j = 1, 2, 3)	Distance between points pi and pj
m	Difference between the vectors p3 and p
{\stackrel{\mathbf{~}}{\mathit{p}}}_5	Homogeneous coordinate of point p5
\stackrel{\mathbf{~}}{\mathit{q}}, \stackrel{\mathbf{~}}{\mathit{p}}	Homogeneous coordinate of points q and p, respectively
\mathit{r}	Position vector of the reference point r of the revolute joint axis
r5	Position vector of point p5
\mathit{v}	Unit directional vector of the translational joint axis
{\theta }_i	Generalized angle of the ith joint
δ	Distance between two vectors s and t, δ = ||s − t||
\text{ω}	Unit directional vector of the revolute joint axis
\widehat{\text{ω}}	Skew-symmetric matrices of \text{ω}
\mathit{\xi }	Joint twist
{\mathit{\xi }}_i \left(i=1,2,3\right)	Twist coordinate of the ith joint axis
{\hat{\mathit{\xi }}}_i \left(i=1,2,3\right)	Instantaneous joint twist of the ith joint axis

Acknowledgements

This study was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51521003), the National Natural Science Foundation of China (Grant No. 61803341), and the Self-planned Task of State Key Laboratory of Robotics and System (Harbin Institute of Technology) (Grant No. SKLRS202009B). No conflicts of interest exist in this paper.

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