Estimate of the two smallest singular values of the Jacobian Matrix: Application to damped least-squares inverse kinematics

In this paper, a novel design of a multi-sectioned, remotely-actuated, continuum type manipulator is presented. Spatially Hyper-Redundant Robot (SHRR) is based on a continuous backbone model which is divided into four sections. In the area of hyper redundant robotics, kinematic redundant systems result in non square Jacobian matrix which requires a pseudo inverse method to inverse the matrix. A methodology has been devised to solve the Inverse Kinematics (IK) problem of SHRR by predicting the curvature values of each of the section. Redundant IK techniques like Pseudo-Inverse Method (PIM), Jacobian Transpose Method (JTM), Damped Least Squares Method (DLS) and Selectively Damped Least Squares Method (SDLS) are tested on the formulated kinematic model of SHRR using MATLAB and a comparative study has been made.

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An Experimental Study of Resolved Acceleration Control of Robots at Singularities: Damped Least-Squares Approach

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2801224 ◽

1997 ◽

Vol 119 (1) ◽

pp. 97-101 ◽

Cited By ~ 10

Author(s):

M. Kirc´anski ◽

N. Kirc´anski ◽

D. Lekovic´ ◽

M. Vukobratovic´

Keyword(s):

Least Squares ◽

Degrees Of Freedom ◽

Jacobian Matrix ◽

Control Level ◽

Singular Values ◽

Control Scheme ◽

Resolved Acceleration Control ◽

Robot Task ◽

Damped Least Squares ◽

Value Decomposition

Most of the robot task space control methods based on inverse Jacobian matrix suffer from instability in singular regions of workspace. Methods based on damped least-squares algorithm (DLS) for matrix inversion have been developed but not experimentally confirmed. The application of DLS method at the kinematic control level has been reported in (Chiaverini et al., 1994). In this article, a modified DLS method combined with the resolved-acceleration control scheme, is experimentally verified on two degrees of freedom of a PUMA-560 robot. In order to decrease the position error introduced by the damping, only small singular values are damped, in contrast to the conventional damping method were all the singular values are damped. The symbolic expressions of the singular value decomposition of the Jacobian matrix were used, to decrease the computational burden.

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Improved Damped Least Squares Inverse Kinematics with Joint limits for 7-DOF “T-FLoW” Humanoid Robot Manipulator

10.1109/ies53407.2021.9593970 ◽

2021 ◽

Author(s):

Muhammad Ramadhan Hadi Setyawan ◽

Sanggar Dewanto ◽

Bayu Sandi Marta ◽

Dadet Pramadihanto

Keyword(s):

Least Squares ◽

Inverse Kinematics ◽

Humanoid Robot ◽

Robot Manipulator ◽

Damped Least Squares ◽

Joint Limits

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Regularization of the Spatial Inverse Positioning Problem of Revolute Joint Manipulators

Periodica Polytechnica Electrical Engineering and Computer Science ◽

10.3311/ppee.8607 ◽

2017 ◽

Vol 61 (3) ◽

pp. 279

Author(s):

Dániel András Drexler

Keyword(s):

Inverse Kinematics ◽

Closed Loop ◽

Singular Values ◽

Singular Configurations ◽

Revolute Joints ◽

Singular Direction ◽

Inverse Kinematics Problem ◽

Singular Configuration ◽

Damped Least Squares ◽

Kinematic Singularities

Inverse kinematics is a central problem in robotics, and its solution is burdened with kinematic singularities, i.e. the task Jacobian of the problem is singular. A subproblem of the general inverse kinematics problem, the inverse positioning problem is considered for spatial manipulators consisting of revolute joints, and a regularization method is proposed that results in a regular task Jacobian in singular configurations as well, provided that the manipulator’s geometry makes movement in singular directions possible. The conditions of regularizability are investigated, and bounds on the singular values of the regularized task Jacobian are given that can be used to create stable closed-loop inverse kinematics algorithms. The proposed method is demonstrated on the inverse positioning problem of an elbow manipulator and compared to the Damped Least Squares and the Levenberg-Marquardt methods, and it is shown that only the proposed method can leave the singular configuration in the singular direction.

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