(shown below) - in which the sum of the virtual power produced by the external forces, at each instant, is equal to zero [16]:
𝑃∗ = ∑
𝑛𝑐
𝑖=1
𝑓𝑖𝑞̇𝑖∗ ≡ 𝑞̇𝑖∗𝑇𝑓 = 0, (13)
where the vector of all the forces that produce virtual power f expressed by:
𝑓 = 𝑀𝑞̈ − 𝑔. (14)
Knowing that 𝑀𝑞̈ representing the vector of inertial forces, M is the global mass matrix, 𝑞̈ is the vector of generalized accelerations, 𝑔 is the vector of generalized applied forces. The internal forces do not produce virtual power since the internal forces are associated in action-reaction pairs, and thus are not represented in equation (13). However, it is possible to calculate them through the method of Lagrange's mul-tiplier:
𝑔Φ = ΦqT𝜆, (15)
where 𝑔Φ is the generalized forces vector, and 𝜆 is the column vector of Lagrange multipliers. The magnitude of the internal constraint forces is obtained through these multipliers, while the direction of these forces is given by the lines of the Jacobian matrix. The equation of the virtual power is obtained from the combination of equa-tions 13 to 15:
𝑃∗ = 𝑞̇∗𝑇(𝑀𝑞̈ − 𝑔 + Φ𝑞𝑇𝜆) = 0. (16) The complete equation of motion is shown below and is solved so as to obtain the unknowns of the system:
{𝑀𝑞̈ + Φ𝑞𝑇𝜆 = 𝑔
Φ𝑞𝑞̈ = 𝛾 . (17)
This methodology assembles and solves the equations of motion (17) of the bio-mechanical system. Through the different ways of getting the solutions of equations (17), one will then have Forward Dynamic Analysis (Simulation) or Inverse Dynamic Analysis (Analysis). This problem (inverse dynamic analysis) can then be used to determine the external forces (that produce the observed motion), and the external and internal forces previously unknown are the results that could be obtained [23].
meth-prospects.Orthotists and engineers may benefit from the information obtained from this review article. The information obtained may be applied to understand the bio-mechanics mathematically behind the ankle-foot orthoses models, and moreover to develop new models.
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JÁRATTERVEZÉSI FELADATOK ÉS IRÁNYZATOK AZ INTELLIGENS ELOSZTÁSI LOGISZTIKA MEGVALÓSÍTÁSÁRA
Szabó Adél Anett1, Illés Béla2, Bányainé Tóth Ágota3
1PhD hallgató, 2PhD, habil, egyetemi tanár, 3PhD, egyetemi docens
1,2,3Logisztikai Intézet, Gépészmérnöki és Informatikai Kar
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