Analytic Geometry and Singularities of Mechanisms

Lerbet, Jean

doi:10.1002/(sici)1521-4001(199810)78:10<687::aid-zamm687>3.0.co;2-t

Cited by 37 publications

(27 citation statements)

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“…In other words the series expansion does not require analytic derivatives but only involves vector cross products. This has been first reported in (Karger 1996) and was later advanced in (Lerbet 1998(Lerbet , 1999(Lerbet , 2003Muller and Rico 2008;Muller 2012;Rico et al 1995).…”

Section: Higher-order Analysismentioning

confidence: 85%

Manipulator Velocities and Static Forces

Zhang¹,

Zhang²,

Müller³

2014

Handbook of Manufacturing Engineering and Technology

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Robotic manipulators are mechanisms that are used to transmit motions and forces. Their kinematic and static properties are thus basic characteristics that must be analyzed when controlling them but also within the design phase. Such kinematic properties are the transmission of joint rates to the end-effector velocities. The dual property is the transformation of end-effector forces to joint forces. This requires adequate modeling of the manipulators, where serial and parallel manipulators must be distinguished. Special situations where these transformation properties change drastically are so-called singularities.In this chapter, the kinematic modeling is reviewed making use of recursive frame transformation. The velocity and force transformation relation is derived and explained for several manipulators, serial as well as parallel manipulators. The phenomenon of singularities is discussed, and the conditions for the existence of singularities are presented.The motion planning and control requires the solution of the inverse kinematic problem. This is also discussed in this paper. In particular, kinematic redundancy and redundant actuation are introduced, and the inverse problem discussed.where V is the platform velocity and _ q are the actuator joint rates. This kinematic model involves two Jacobians, each of which can become singular. This has been first addressed in (Gosselin and Angeles 1990). Configurations where B becomes singular are called type I singularities, and when Handbook of Manufacturing Engineering and Technology

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Section: Higher-order Analysismentioning

confidence: 85%

Manipulator Velocities and Static Forces

Zhang¹,

Zhang²,

Müller³

2014

Handbook of Manufacturing Engineering and Technology

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“…The so‐called first and higher order mobilities introduced by Kuznetsov had then been directly and explicitly connected to appropriate combinations of Lie brackets of vectors of each Lie algebra attached to the links of the mechanisms. Following these investigations, a full classification of singularities of mechanisms has been proposed in a founding paper in 1998 [] which extended the solution of the problem of the infinitesimal motions to singular situations by use of analytic geometry and the computation of the tangent cone at a singularity. It is interesting to note that the transversality concept is also the crux concept to tackle the singular mechanisms even tough it it then used in another complete different way than in the present paper.…”

Section: Examplementioning

confidence: 99%

Coordinate free nonlinear incremental discrete mechanics

et al. 2018

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In this paper we extend the Kinematic Structural Stability (KISS) issues obtained in the punctual linear case to a full non linear framework. Kinematic Structural Stability (KISS) refers to the stability of conservative or non-conservative elastic systems under kinematic constraints. We revisite and sometimes introduce some new mechanical and geometrically-related concepts, using the systematic language of vector bundles: transversality-stability, tangent stiffness tensor, loading paths and KISS. We bring an intrinsic geometrical meaning to these concepts and we extend the main results to a general nonlinear framework. The lack of connection on the configuration manifold in order to make derivatives of sections is bypassed thanks to the key rule of the nil section of the cotangent bundle. Two mechanical examples illustrate the interest of the new concepts, namely a discrete nonconservative elastic system under nonconservative loading (Ziegler's column) and, in more details, a four-sphere granular system with nonconservative interactions called diamond pattern. K E Y W O R D Scoordinate free approach, fiber bundle, kinematic structural stability, transversality

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“…Lerbet first applied the concept of tangent cone to the kinematic analysis of linkages [Lerbet (1999)]. The cone is introduced noticing its interpretation as set of tangents to curves.…”

Section: Local Analysismentioning

confidence: 99%

Kinematic Singularities of Mechanisms Revisited

Müller¹,

Zlatanov²

2015

Proceedings of the IMA Conference on Mathematics of Robotics

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The paper revisits the definition and the identification of the singularities of kinematic chains and mechanisms. The degeneracy of the kinematics of an articulated system of rigid bodies cannot always be identified with the singularities of the configuration space. Local analysis can help identify kinematic chain singularities and better understand the way the motion characteristics change at such configurations. An example is shown that exhibits a kinematic singularity although its configuration space is a smooth manifold.

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Analytic Geometry and Singularities of Mechanisms

Cited by 37 publications

References 0 publications

Manipulator Velocities and Static Forces

Manipulator Velocities and Static Forces

Coordinate free nonlinear incremental discrete mechanics

Kinematic Singularities of Mechanisms Revisited

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