A novel geometrical derivation of the Lie product

Cervantes-Sánchez, J. Jesús; Moreno-Báez, Miguel A.; Rico-Martínez, José M.; González-Galván, Emilio J.

doi:10.1016/j.mechmachtheory.2004.05.002

Cited by 7 publications

(14 citation statements)

References 25 publications

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“…1 With a nonredundant motion parameterization supplying each Ti as a six-element column vector, matrix J=[T1…Tn-1] supplies one of the alternative forms of the robot Jacobian . 22,23 This form gives sensitivities of the six elements of the tool link motion parameters TCn to the six robot joint rates C1,…,Cn-1. Giving joint 7 the prescribed rate C7, a path-following method solves equation (1) for the rates of the robot joints.…”

Section: Representations Of Displacement and Motionmentioning

confidence: 99%

“…Â Ã supplies one of the alternative forms of the robot Jacobian. 22,23 This form gives sensitivities of the six elements of the tool link motion parameters T C n to the six robot joint rates C 1 , . .…”

Section: Representations Of Displacement and Motionmentioning

confidence: 99%

“…It also eliminates fixed AE90 corrections to the joint angles i t ð Þ (constant i for a prismatic joint) aligning the local x-axis in relation to the link. Any such AE90 rotations are applied by a cyclic permutation of equations (23) to (27).…”

Section: Updating the Instantaneous Screw Between Link Framesmentioning

confidence: 99%

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Effect of the coordinate frame on high-order expansion of serial-chain displacement

Milenkovic

2019

Proceedings of the Institution of Mechanical Engineers, Part K:

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An algorithmic differentiation technique gives a simpler, faster power series expansion of the finite displacement of a closed-loop linkage. It accomplishes this by using a higher order than what has been implemented by complicated prior formulas for kinematic derivatives. In this expansion, the joint rates and axis lines generate the instantaneous screw of each link. Constraining the terminal link to have a zero instantaneous screw satisfies closure. In order to maintain closure over a finite displacement, it is necessary to track the spatial trajectory of each joint axis line, which in turn is directed by the instantaneous screw of a link to which it is attached. Prior algorithms express these screws in a common ground-referenced coordinate frame. Motivated by the kinematics solver portion of the recursive Newton–Euler algorithm, an alternative formulation uses sparse matrices to update the instantaneous screw between successive link-local frames. The recursive Newton–Euler algorithm, however, conducts the expansion to only second order, where this paper shows local coordinate frames that are only instantaneously aligned with their respective links give identical expressions to those in frames that move with the links. Moving frames, however, require about 40% of the operations of the global-frame formulation in the asymptotic limit. Both incrementally translated (Java) and statically compiled (C++) software implementations offer more modest performance gains; execution profiling shows reasons in order of importance (1) balance of calculation tasks when below the asymptotic limit, (2) Java array bounds checking, and (3) hardware acceleration of loops.

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Section: Representations Of Displacement and Motionmentioning

confidence: 99%

Section: Representations Of Displacement and Motionmentioning

confidence: 99%

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Effect of the coordinate frame on high-order expansion of serial-chain displacement

Milenkovic

2019

Proceedings of the Institution of Mechanical Engineers, Part K:

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“…At this point, it is important to realize that a Lie product, [S j S i ], is also a screw [24,25], whose components depend on the joint variables corresponding to the involved screws. Moreover, since j < i, it follows that the Lie product [S j S i ] may be given as a function of the joint variables q 1 , q 2 , .…”

Section: Second Time Derivative Of a Screwmentioning

confidence: 99%

“…Moreover, because of the global nature of the formulation, they did not pursue obtaining explicit expressions for the time derivatives or the partial derivatives of screws. In this regard, some derivations have appeared in the literature [24][25][26][27]. However, they only compute up to the first time derivative or up to the first partial derivative of screws.…”

Section: Introductionmentioning

confidence: 99%

The differential calculus of screws: Theory, geometrical interpretation, and applications

Cervantes-Sánchez

Rico-Martínez

González-Montiel

et al. 2009

Proceedings of the Institution of Mechanical Engineers, Part C:

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This article presents a novel and original formula for the higher-order time derivatives, and also for the partial derivatives of screws, which are successively computed in terms of Lie products, thus leading to the automation of the differentiation process. Through the process and, due to the pure geometric nature of the derivation approach, an enlightening physical interpretation of several screw derivatives is accomplished. Important applications for the proposed formula include higher-order kinematic analysis of open and closed kinematic chains and also the kinematic synthesis of serial and parallel manipulators. More specifically, the existence of a natural relationship is shown between the differential calculus of screws and the Lie subalgebras associated with the expected finite displacements of the end effector of an open kinematic chain. In this regard, a simple and comprehensible methodology is obtained, which considerably reduces the abstraction level frequently required when one resorts to more abstract concepts, such as Lie groups or Lie subalgebras; thus keeping the required mathematical background to the extent that is strictly necessary for kinematic purposes. Furthermore, by following the approach proposed in this article, the elements of Lie subalgebra arise in a natural way -due to the corresponding changes in screws through time -and they also have the typical shape of the so-called ordered Lie products that characterize those screws that are compatible with the feasible joint displacements of an arbitrary serial manipulator. Finally, several application examples -involving typical, serial manipulators -are presented in order to prove the feasibility and validity of the proposed method.

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