The Singular Value Decomposition: Computation and Applications to Robotics

Maciejewski, Anthony A.; Klein, Charles A.

doi:10.1177/027836498900800605

Cited by 176 publications

(97 citation statements)

References 40 publications

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“…There exist other numerical methods of matrix inversion based on a factorization of the matrix (for example, singular value decomposition (Maciejewski and Klein, 1989) or LU factorization (Golub and Van Loan, 1996)). As a rule they provide an easy way to get matrix inversion, if only factorization has been performed.…”

Section: Jacobian-based Methods Of Inverse Kinematics and Their Evalumentioning

confidence: 99%

“…α 3 is the average of α 1 and α 2 , while α 4 is a dumped version of α 3 with the dumping factor equal to 2. Here α 5 corresponds to the ideal case when the maximal singular value is known (in fact, it was computed numerically for a given manipulator and a given point in the configuration space using the SVD algorithm (Maciejewski and Klein, 1989)). Although α 5 does not have got any practical significance (exact λ max computation is too costly) it may certainly be useful in a comparative study with other values (α 1−4 ).…”

Section: Simulationsmentioning

confidence: 99%

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A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators

Dulźba¹,

Opałka²

2013

International Journal of Applied Mathematics and Computer Science

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The objective of this paper is to present and make a comparative study of several inverse kinematics methods for serial manipulators, based on the Jacobian matrix. Besides the well-known Jacobian transpose and Jacobian pseudo-inverse methods, three others, borrowed from numerical analysis, are presented. Among them, two approximation methods avoid the explicit manipulability matrix inversion, while the third one is a slightly modified version of the Levenberg-Marquardt method (mLM). Their comparison is based on the evaluation of a short distance approaching the goal point and on their computational complexity. As the reference method, the Jacobian pseudo-inverse is utilized. Simulation results reveal that the modified Levenberg-Marquardt method is promising, while the first order approximation method is reliable and requires mild computational costs. Some hints are formulated concerning the application of Jacobian-based methods in practice.

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Section: Jacobian-based Methods Of Inverse Kinematics and Their Evalumentioning

confidence: 99%

Section: Simulationsmentioning

confidence: 99%

A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators

Dulźba¹,

Opałka²

2013

International Journal of Applied Mathematics and Computer Science

View full text Add to dashboard Cite

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“…The modelling and control for a redundant mobile modular manipulator is more challenging since self-motions have to be taken into consideration, (Li & Liu, 2006b;Maciejewski & Klein, 1989). The proposed algorithms are just verified by simulations, there is still a lot of work to do to verify the algorithms by real experiments.…”

Section: Future Research Workmentioning

confidence: 99%

Dynamics and Control for Nonholonomic Mobile Modular Manipulators

Li¹,

Liu²

2006

Mobile Robotics, Moving Intelligence

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“…1 was 5.13 ms. By comparison, the Golub-Reinsch algorithm in the IMSL package required an average of 32.3 ms. Details of the algorithm and its implementation can be found in [7].…”

Section: Dealing With Singularitiesmentioning

confidence: 99%

Real-time SVD for the control of redundant robotic manipulators

Maciejewski

1989

IEEE International Conference on Systems Engineering

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By taking advantage of the well-behaved nature of singular values and vectors, the computational expense of determining the SVD of the Jacobian matrix can be reduced to such an extent that on-line calculation becomes feasible. This permits more intelligent use of the extra degrees of freedom present with redundant manipulators, particularly with respect to the optimization of various secondary criteria, including obstacle avoidance, under the constraint of a specified end effector trajectory.However, the control of robotic systems is not based on the solution of arbitrary matrix equations but quite frequently involves the solution of equations based on the Jacobian matrix. The current Jacobian for a system can be regarded as a perturbation of a previously known matrix for which perturbation bounds on the singular values and singular vectors can be established. Knowledge of the previous state can be exploited during the current calculation of the SVD in order to reduce the overall computational burden. This results in a computational scheme capable of calculating the SVD of the Jacobian for use in real-time control of manipulators. u,v; (4) ;=1 II. Perturbation Bounds on the SVD,=k+lThe perturbation bounds on the rotation of subspaces defined by singular vectors are not as widely known but are also well-behaved [11]. In particular, consider a partitioning of the Jacobian into two singular subspaces of the form All SVD algorithms possess an iterative component designed to orthogonalize the columns (or rows) of the matrix being decomposed. Clearly, the more orthogonal these columns are to begin with, the fewer the number of calculations are required to reach convergence. Thus, if one considers the current manipulator Jacobian, denoted by J(t + At), to be a perturbation of the previous Jacobian (2)(1) J(t) = U(t)D(t)VT(t)the SVD of which is known and given by then the matrix J(t + At)V(t) will have nearly orthogonal columns provided the perturbation AJ(t) is small relative to J(t). The foundation of the above lies in the fundamentally well-behaved nature of the SVD of a matrix. The perturbation bounds on singular values, denoted by ai, are very well-known and it is easy to show that lai(J(t + At)) -a,(J(t))1~IILlJ(t)ll.(3)

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The Singular Value Decomposition: Computation and Applications to Robotics

Abstract: Abstract

Cited by 176 publications

References 40 publications

A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators

A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators

Dynamics and Control for Nonholonomic Mobile Modular Manipulators

Real-time SVD for the control of redundant robotic manipulators

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