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

doi:10.1007/s11465-022-0681-7

Front. Mech. Eng.

2022, Vol. 17

Issue (2) : 25 https://doi.org/10.1007/s11465-022-0681-7

RESEARCH ARTICLE

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

Tao SONG¹, Bo PAN¹(

), Guojun NIU², Jiawen YAN¹, Yili FU¹

¹. State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China
². School of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China

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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.

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

Corresponding Author(s): Bo PAN

About author:

Tongcan Cui and Yizhe Hou contributed equally to this work.

Just Accepted Date: 22 April 2022 Issue Date: 15 August 2022

Cite this article:

Tao SONG,Bo PAN,Guojun NIU, et al. General closed-form inverse kinematics for arbitrary three-joint subproblems based on the product of exponential model[J]. Front. Mech. Eng., 2022, 17(2): 25.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0681-7
https://academic.hep.com.cn/fme/EN/Y2022/V17/I2/25

Fig.1 General form of the RRR three-joint subproblem.

Fig.2 Form of the unique solution under triaxial parallelism.

Fig.3 General form of the two adjacent axes of the RRR three-joint subproblem.

Fig.4 General form of the RRT three-joint subproblem.

Fig.5 Solution of RRT satisfying

ω 2 ∥ ω 1

and

ω 2 ⊥ v 3

. (a) Unique solution of category 1, (b) infinite number of solutions of category 1.

Fig.6 General form of the RTR three-joint subproblem.

Fig.7 Solution of RTR satisfying

ω 1 ∥ ω 3

and

ω 1 ⊥ v 2

. (a) Unique solution of category 1, (b) infinite number of solutions of category 1.

Fig.8 Solution for the position of point c.

Fig.9 General form of the RTT three-joint subproblem.

Fig.10 Solution of RTT satisfying

ω 1 ⊥ v 2

and

ω 1 ⊥ v 3

Fig.11 General form of the TRT three-joint subproblem.

Fig.12 Solution of TRT satisfying

ω 2 ⊥ v 1

and

ω 2 ⊥ v 3

Fig.13 General form of the TTT three-joint subproblem.

Fig.14 Sample points and joint angles of the inverse solution of RRR subproblem.

Fig.15 Sample points and joint angles of the inverse solution of RRT subproblem.

Fig.16 Sample points and joint angles of the inverse solution of RTR subproblem.

Fig.17 Sample points and joint angles of the inverse solution of RTT subproblem.

Fig.18 Sample points and joint angles of the inverse solution of TRT subproblem.

Fig.19 Sample points and joint angles of the inverse solution of TTT subproblem.

Fig.20 Structure diagram of the instrument manipulator of Huaque III surgical robot.

Fig.21 Sample points and joint angles of the inverse solution for Huaque III surgical robot.

Fig.22 Huaque III minimally invasive surgical robot.

Fig.23 Movement trajectories for two methods: (a) closed-form solution and (b) compensation solution.

Fig.24 End-effector tracking errors for two methods: (a) closed-form solution and (b) compensation solution.

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
$g st (0)$ , $g st (θ)$	Initial transformation matrix and transformation matrix at different joint angles of the tool frame, respectively
l_c2	Distance between points c and p₂
l_ij (i, j = 1, 2, 3)	Distance between points p_i and p_j
m	Difference between the vectors p₃ and p
$p ~ 5$	Homogeneous coordinate of point p₅
$q ~$ , $p ~$	Homogeneous coordinate of points q and p, respectively
$r$	Position vector of the reference point r of the revolute joint axis
r₅	Position vector of point p₅
$v$	Unit directional vector of the translational joint axis
$θ i$	Generalized angle of the ith joint
δ	Distance between two vectors s and t, δ = \|\|s ? t\|\|
$ω$	Unit directional vector of the revolute joint axis
$ω^$	Skew-symmetric matrices of $ω$
$ξ$	Joint twist
$ξ i$ $(i = 1, 2, 3)$	Twist coordinate of the ith joint axis
$ξ^i$ $(i = 1, 2, 3)$	Instantaneous joint twist of the ith joint axis

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