Testing iterative robotic algorithms by their rate of convergence

Klein, Charles A.; Chu-Jeng, C.; Kittivatcharapong, S.

doi:10.1109/70.97881

Cited by 5 publications

(5 citation statements)

References 3 publications

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“…In [19], the suggestion is made to replace A x with Ax + pCi -p a , where pd -pa is the error between the desired and actual position at the beginning of the time sample. Using this drift correction will reduce the error in position at the end of the trajectory from being on the order of U ( l / k ) to U ( ( l / k ) 2 ) [20]. This drift correction is also easily applied to this method.…”

Section: The Collected System Ismentioning

confidence: 93%

“…First the changes in the null space could now be calculated by (22) and (21). It would appear at first that numerical drift could also be reduced by replacing the right-hand side of (24) with one where residual errors have been subtracted [20]. However, (21) deals strictly with the row space of J but any residual errors in the orthogonality of the null space basis would require a change in the null space.…”

Section: -D Examplesmentioning

confidence: 95%

“…The first step in simplifying (20) is noting that if the null space changes are orthogonal to the null space, then they must be contained in the row space of .I and can be expressed by .…”

Section: A More Numerically Tractable Formulationmentioning

confidence: 99%

“…. Call the square, order ( r + 1)n, matrix in (20) J,,,,. A linear equations routine can now be used to numerically solve (20) greatly improve the numerical efficiency.…”

Section: A More Numerically Tractable Formulationmentioning

confidence: 99%

“…A protocol for testing the correctness of a kinematic algorithm has been presented in [20] using (20) as a sample system for the planar manipulator with 11 = 3.…”

Section: -D Examplesmentioning

confidence: 99%

See 4 more Smart Citations

A new formulation of the extended Jacobian method and its use in mapping algorithmic singularities for kinematically redundant manipulators

Klein

Chu-Jenq²,

Ahmed³

1995

IEEE Trans. Robot. Automat.

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The extended Jacobian method resolves the redundancy of a kinematically redundant manipulator such that a secondary criterion is kept at an optimal value. This paper describes a new formulation of this algorithm that is well suited numerically for systems with multiple degrees of redundancy and for more general choices of the secondary criterion. This formulation is used here for tracing algorithmic singularities, which mark a boundary of operation for any algorithm using the same secondary criterion to resolve the redundancy. Examples of tracing algorithmic singularities are presented for both planar and spatial systems.Index Terms-kinematically redundant robots, extended Jacobian method, algorithmic singularities

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Section: The Collected System Ismentioning

confidence: 93%

Section: -D Examplesmentioning

confidence: 95%

“…The first step in simplifying (20) is noting that if the null space changes are orthogonal to the null space, then they must be contained in the row space of .I and can be expressed by .…”

Section: A More Numerically Tractable Formulationmentioning

confidence: 99%

“…. Call the square, order ( r + 1)n, matrix in (20) J,,,,. A linear equations routine can now be used to numerically solve (20) greatly improve the numerical efficiency.…”

Section: A More Numerically Tractable Formulationmentioning

confidence: 99%

“…A protocol for testing the correctness of a kinematic algorithm has been presented in [20] using (20) as a sample system for the planar manipulator with 11 = 3.…”

Section: -D Examplesmentioning

confidence: 99%

See 3 more Smart Citations

A new formulation of the extended Jacobian method and its use in mapping algorithmic singularities for kinematically redundant manipulators

Klein

Chu-Jenq²,

Ahmed³

1995

IEEE Trans. Robot. Automat.

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Klein

Chu

1997

Journal of Intelligent and Robotic Systems

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The Lie bracket condition as a test of stable, drift-free pseudoinverse trajectories

Klein

1993

IEEE Trans. Robot. Automat.

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In [ 11 K. Kee and I showed with numerical experiments that pseudoinverse control, while generally being nonconservative, in some cases appears to give conservative results. In this paper a numerical procedure was described to find stable, drift-free trajectories.In [2], Shamir and Yomdin discuss using the Lie Bracket Condition (LBC) as a test of the repeatability of redundant manipulator control. In particular, they demonstrated this test on a three-link planar manipulator with absolute joints. An interesting hypothesis is that the stable trajectories of [l], which are done numerically on a threelink, planar manipulator with relative joints, might be predicted by the LBC. If this hypothesis were true, one implication would be that within a closed region, any path should be conservative for a starting point that meets the LBC.A numerical test of this hypothesis has been performed. Reference[2] gives the formula for the LBC for arbitrary joint variables. Based on that formula, the LBC in terms of relative joint angles can be expressed as tan O2 -sin !I3 (1 + cos 83)/( 1 + sin2 8, ) = 0.(1) Fig. 1 is a plot of the left hand side of (1) and the measure pitch, as described in [I], along the closed locus of joint angles which yield the same end effector point. The starting joint angles are 20, 30, and 50 degrees and the 3 link lengths are 200. The square is 65 units on aside. The point where tracing the square has no drift (the pitch being zero) is very close to where the LBC is satisfied, which at first sight appears to verify the hypothesis. However, fine scale examination of the data for this curve does show a small displacement between the zero crossings and this displacement is very significant from a theoretical view. A careful expansion of the data shows that the zero crossings differ by 3.67 degrees which is 0.6% of the horizontal axis of Fig. 1. A natural question is whether this difference could be a numerical or a computational artifact. To test this question the protocol of [3] has been used, and the difference observed does not converge to zero for an increasing number of integration steps. Conversely, when the joint angles are started where the LBC is exactly zero, the pitch is not exactly zero, drift does occur, and this drift also does not converge to zero for increasing number of integration segments.In addition to the numerical tests, there is an analytical reason to explain why the LBC does not describe the stable trajectories observed in this situation. In order for pseudoinverse control to be following the surface determined by the LBC, it would be necessary for joint motion to be orthogonal to the gradient of the LBC function. This can easily be seen not to be the case. Pseudoinverse control for the 3 link planar manipulator problem always moves the joints angles orthogonal to the null space vector. This would imply that the gradient of the LBC and the null space vector would have to be parallel. However, the former has no 81 component while the latter, in general, does. I would conclude that the...

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Testing iterative robotic algorithms by their rate of convergence

Cited by 5 publications

References 3 publications

A new formulation of the extended Jacobian method and its use in mapping algorithmic singularities for kinematically redundant manipulators

A new formulation of the extended Jacobian method and its use in mapping algorithmic singularities for kinematically redundant manipulators

Untitled

The Lie bracket condition as a test of stable, drift-free pseudoinverse trajectories

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