Introduction to Mathematical Modeling of Cooperative Systems

Part of the Microprocessor-Based and Intelligent Systems Engineering book series (ISCA, volume 30)

3.5 Summary of the Problem of Mathematical Modeling

Based on the introductory consideration concerning the consistent mathematical procedure for modeling a simple cooperative system it is possible to derive the following general conclusions that could serve as landmarks in the process of modeling complex cooperative systems:
  • The problem of force uncertainty is to be solved by introducing the assumption on elasticity of that part of the cooperative system in which that uncertainty appears.

  • It is convenient to model an elastic system separately in order to ensure an easier and more correct description of its (quasi)statics and dynamics.

  • In modeling an elastic system, it is necessary to first solve the static conditions on the basis of the minimum of potential (deformation) energy (δAd=δU, (13)).

    As a result of this step, we get:
    • the relation F=Ky between the elastic forces F and stiffness characteristics K and displacement of the elastic system with respect to its unloaded state y,

    • the number of state quantities of elastic system ny equal to the dimension of the vector yR n y,

    • singular stiffness matrix K (det K=0, rank K<ny),

    • kinematically unstable (mobile) elastic system,

    • arbitrary choice ny-rank K of displacements of the leader for the given elastic system in space.

  • The relation F=Ky is to be transposed into the dependence of elastic force on the absolute coordinates F=K(Y)Y and deformation energy determined as a function of the absolute coordinates Y, the energy needed to perform the general motion of the elastic system.

  • The kinetic and deformation energies and generalized forces should be determined as a function of absolute coordinates Y and Lagrange formalism is to be applied to generate the equation of motion of the elastic system.

  • A model of the cooperative system dynamics is to be formed by coupling the model of elastic system motion with the models of manipulators and relations describing the contact conditions.


Contact Force Contact Point Internal Force Elastic Force Cooperative Work 
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© Springer 2006

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