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| <?xml version="1.0" encoding="UTF-8"?>
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<titleStmt>
<title level="a" type="main">Accounting for Member Deformations in the Determination of the Dynamic Behavior of Robot-Type Structures</title>
</titleStmt>
<publicationStmt>
<publisher/>
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<licence/>
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<sourceDesc>
<biblStruct>
<analytic>
<author>
<persName>
<forename type="first">A</forename>
<surname>Barraco</surname>
</persName>
<affiliation>
<orgName type="institution">Ecole Nationale Sup~rieure d'Arts et Mbtiers</orgName>
<address>
<addrLine>151, Bd de l'HSpital</addrLine>
<postCode>75640</postCode>
<settlement>Paris Cedex 13</settlement>
<country key="FR">France</country>
</address>
</affiliation>
</author>
<author>
<persName>
<forename type="first">B</forename>
<surname>Cuny</surname>
</persName>
<affiliation>
<orgName type="institution">Ecole Nationale Sup~rieure d'Arts et Mbtiers</orgName>
<address>
<addrLine>151, Bd de l'HSpital</addrLine>
<postCode>75640</postCode>
<settlement>Paris Cedex 13</settlement>
<country key="FR">France</country>
</address>
</affiliation>
</author>
<author>
<persName>
<forename type="first">G</forename>
<surname>Ishiomin</surname>
</persName>
<affiliation>
<orgName type="institution">Ecole Nationale Sup~rieure d'Arts et Mbtiers</orgName>
<address>
<addrLine>151, Bd de l'HSpital</addrLine>
<postCode>75640</postCode>
<settlement>Paris Cedex 13</settlement>
<country key="FR">France</country>
</address>
</affiliation>
</author>
<title level="a" type="main">Accounting for Member Deformations in the Determination of the Dynamic Behavior of Robot-Type Structures</title>
</analytic>
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<imprint>
<date/>
</imprint>
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<note>283</note>
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<profileDesc>
<textClass>
<keywords>
<term>Dynamic equations</term>
<term>Mechanisms</term>
<term>Structural deformations</term>
<term>Finite elements</term>
<term>Robotics</term>
</keywords>
</textClass>
<abstract>
<p>Two themes are being investigated: development of conception , simulation and visualization tools for rigid body mechanisms , accounting for member deformations in the dynamic behavior of mechanical systems. The equations for the dynamic behavior of systems are established from the application of an updated Lagrangian method to finite element discretiza-tions.</p>
</abstract>
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<text xml:lang="en">
<body>
<div>
<head n="1">Introduction</head>
<p>The purpose of this research is to integrate member deformation effects into the mechanical models used to analyse the dynamic behavior of robot-type structures. This approach is indeed justified by the utilization conditions of such systems which look forward to have a maximal weight moved in a minimal time by an as light as possible structure, while keeping high precision and repeatability in the movement. From a mechanical standpoint, we can distinguish two types of problems which account for deformations in two different ways: Problem Nr. 1: in most multiple-connected systems , the major loss of precision originates in the connexions themselves. They always show some flexibility, free play or friction which induces a poor output. Problem Nr. 2: in some multiple-connected systems , in addition to the above connexion located deformations, structural members also show significant deformations, so that the mechanical systems can be called fully flexible. From these two problems therefore originate two different topics of research: Topic Nr. 1: different mechanical models do exist for such systems. The research is then aimed at selecting the "best" model while elaborating user-oriented procedures that allow a simple usage of the chosen model. We have used Computer</p>
<figure>
<trash>Professor Guy Ishiomin graduated as engineer from the "E cole Nationale Sup&ieure d'Arts & M&iers" in 1970. He has been a Computer Science Prolessor at E.N.S.A.M, since 1971 and specializes in Computer-Aided De- sign.</trash>
</figure>
</div>
<div>
<head>Computerv m Industc~'</head>
<p>Aided Design facilities to develop both a modelisation of actuators and an automatic procedure to compute member inertia characteristics. Topic Nr. 2: Since no mechanical model exists for such fully flexible systems, the research work resides in defining the theoretical principles and the way they must be applied. After computer implementation , these models must be checked against the known models applicable to rigid body mechanisms , so that utilization domains may be identified for any specific method. 2. The C.A.D. Developments</p>
</div>
<div>
<head n="2.1">Purpose</head>
<p>Using the graphics code EUCLID developed by MATRA DATAVISION, we have created a set of utilitary subroutines enabling an engineer to define and analyse a robot in an interactive fashion (e.g. to determine from the dynamic analysis, relative to a predefined tool trajectory, forces and torques generated in members or at member con- nexions).</p>
</div>
<div>
<head n="2.2">Developments</head>
<p>Since different programs were involved (EUCLID for the graphical considerations, PAMTIN for the dynamic analysis of systems built of rigid bodies, and GENTRA for the geometry and kinematics problems of trajectory generation ), part of the work was concerned with interfacing these different programs so that the resulting software chain appeared easy to handle to the user. GENTRA is elaborated in a conversational manner and enables the user to define any tool trajectory composed of a succession in space of straight lines and circle portions to be run over with a specific velocity pattern
<ref type="bibr" target="#b0" coords="2,178,32,546,72,20,96,8,86">[1]</ref>
<ref type="bibr" target="#b1" coords="2,178,32,546,72,20,96,8,86">[2]</ref>
</p>
<figure>
<figDesc>.</figDesc>
</figure>
</div>
<div>
<head n="3">The Different Mechanical Models of the Dynamic Behavior of Mechanisms Composed of Non-Rigid Elements 3.1. Problem Nr. 1: Deformation Is Located at Member Connexions</head>
<p>As was indicated in the Introduction, different mechanical models are available (Lagrange's equations , Newton-Euler formalism, "D'Alembert's principle). Althrough their easy application mostly depends on which problem is being considered, they all lead to the following type of relationships:</p>
<figure>
<trash>A(c~t)'~t+B(~t, (~t)+B*((~t, ~I-)t)=~; (1) in which c,</trash>
<figDesc>:column vector of relative position variable parameters, :column vector of known efforts at member connexions, :matrix of relative acceleration quanti- ties, :column vector accounting for Coriolis acceleration. The B(~ t, i~t) column vector accounts for structural deformations. It depends only on ~t when member connexions are purely elastic, but depends on both ff~t and ~t for viscous damping dissipation at connexions. In this formulation, the major difficulty arises from not knowing any real characteristics of actuators: mass, inertia and damping values.</figDesc>
</figure>
</div>
<div>
<head n="3.2">Problem Nr. 2." Deformation also Induced in Structural Members</head>
<p>The deformed shape of structural members is then known by adding to the previous rigid body degrees of freedom ~t complementary quantities accounting for member deformations. Let Ut, V t and F t be the column vectors of this complementary d.o.f.'s and of their first and second time derivatives, velocity and acceleration. The mechanical model equations then come up to be:</p>
<p>
<formula>Ka(~,, i~t, ~b,). Ut+D(¢~,, ~,). Vt + M(~,)./', +h(tI~t)'~.lt+B(trfft, (IJt)+B*(tlJt, ~t) = Ct (2)</formula>
</p>
<p>Equation (2) includes all quantities present in equation (1) for the first problem. Moreover there appears a coupling between the rigid body system movements, parameterized by ~t and the part of the movement due to member deformations and parametrized by U, V t and Ft. Let us state here that displacements U t must have small amplitudes in order to use beam theory or shell theory equa- tions.</p>
<p>In order to get a simplified formulation of these equations, the last known deformed position at time (t-dt) is used as reference position instead of</p>
</div>
<div>
<head>Computers in Industry A. Barraco et al. / Dynamic Behavior of Robot-Type Structures</head>
<p>
<formula>285 lJ ~ • :/ 0 /" t/l ~1~# Q Fig. 1</formula>
</p>
<p>. Closed kinematic chain definition. ~2 = 600 tr/mn 11 = 36 in; 12 =12 in; 13 = 36 in; 14 = 30 in. Cross section I x 1 in 2. E = 107 psi p = 0.101 lb/in 3. the rigid body position at time t, and then equations for time (t-dt) and t are subtracted. While introducing the displacement, velocity and acceleration increment vectors u, v and 7, we obtain the following incremental form.</p>
<figure>
<trash>#,_,,,, +,_,,,).,, + • v + M(Ot_a, ) "7 = AC (3)</trash>
</figure>
<p>The solution of this equation was turned into a computer program based on the finite element technique. Spatial mechanism cases show high</p>
<figure>
<trash>' /)4/02 --@--04, s Fig. 3</trash>
</figure>
<p>. Output velocity b 4 versus input velocity ~2 diagram. complexity because of the great number of quantities involved. Plane mechanism cases are much more simple, for which we show calculation resuits in the following paragraph.</p>
</div>
<div>
<head n="3.3">Examples</head>
<p>The plane problems dealt with in this part were chosen so as to offer direct comparisons with results obtained by other approaches and reported in the literature. The mechanical system under study is built with three deformable members</p>
<figure>
<trash>e en P2 x 10 "4 (Psi) 1.5 0 4 04, s 0.5 1 2 3 0 2 I / I I I I~ 7r / 2 /r Fig. 2</trash>
</figure>
<p>. Output angle 0 4 versus input angle 0 2 diagram.</p>
<p>
<formula>2 1 -1 -2 --3 1</formula>
</p>
<p>A.,/2 Fig. 4. Bending stress at the middle point P2 of member 2 (solution in [4]).</p>
</div>
<div>
<head>Industrial Robotics in Discrete Manufacturing</head>
<p>
<formula>Computers m lndu,~trv 4 g. 3 0 × 1 09 .~--2 m 3 -4 P2 <</formula>
</p>
<note>%.~'~ /t - Mechanism 1 ',;, 7220 -/ ..... 1-2--1 --.--1 3--1 Fig. 5</note>
<p>. Effect of division on coupler stress.</p>
<figure>
<trash>1 Damping C = 302 M, 2 Damping C = 2 02 M, 3 Damping C = 02 M, 4 Damping C = 0.</trash>
</figure>
<p>nected at their ends by hinges. This structure is a closed kinematic chain.</p>
<p>For this mechanism shown in Fig. 1, resulting values are compared: -on one hand, between rigid and deformable member structures. Of particular interest are the relationships between the input angle and angular velocity and their output counterparts [3] and Figs. 2 and 3. -on the other hand, between the different formulations available that account for member deformations [4] and Figs. 4 and 5.</p>
</div>
<div>
<head>From Fig. 2</head>
<p>that relates the 94, a output angle variations for deformable elements to its 64. ,</p>
<figure>
<trash>en P4 x 10 4 (Psi) 5 4 3 2 1 1 2 3 Fig. 6</trash>
<figDesc>. Bending stress at the middle point P4 of member 3. terpart for rigid members, we can check that: ..... the time diagram for 04, , well matches the classical solution proposed in [3]</figDesc>
</figure>
<p>; -the 04, J values show oscillations about 04,, solution , the amplitude of these variations depending on both the flexibility of mechanical elements and the importance of the inertia wheel installed on the output axis. Fig. 3 shows how the output angular velocities/94, a and 04,s behave when members are deformable or rigid. Once again the 04,d values show oscillations about the 04,, variation diagram, while the amplitude of these variations is larger than that of Fig. 2</p>
<figure>
<figDesc>.</figDesc>
</figure>
</div>
<div>
<head n="4">Conclusions</head>
<p>We have shown that the requirement for CAD tools is to integrate them in a continuous software chain for conception and simulation of mecha- nisms. Concerning the different mechanical models of dynamic behavior under consideration, examples for plane mechanisms have been tested and checked against other authors' analyses. The problem is now to properly limit domains in which different types of simplifications are acceptable and thus lead to knowing what specific formulation is applicable to each particular case.</p>
</div>
</body>
<back>
<div type="acknowledgement">
<head>Acknowledgement</head>
<p>This work is supported by the "Automatisme Robotique Avanc6e" research project, sponsored by the Centre National de La Recherche Scien- tific.</p>
</div>
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<title level="j">D~partement STRUCTURES-ENSAM</title>
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<biblStruct xml:id="b2">
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</div>
</back>
</text>
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