Ivancevic_Applied-Diff-Geom

586 Applied Differential Geometry: A Modern Introductionformµ kl = ϖ kl exp{−50[λ k + u(q, p)] 2 }, (k, l = 1, 2, . . . , 9)with weights ϖ kl ∈ {0.6, 0.6, 0.7, 0.7, 0.8, 0.8, 0.9, 0.9, 1.0}.(2) Mamdani inference is used on each FAM–matrix µ kl for all humanjoints:(i) µ(ˆq i ) and µ(ˆp i ) are combined inside the fuzzy IF–THEN rules usingAND (Intersection, or Minimum) operator,µ k [ū i (q, p)] = minl{µ kl (ˆq i ), µ kl (ˆp i )}.(ii) the output sets from different IF–THEN rules are then combined usingOR (Union, or Maximum) operator, to get the final output, fuzzy–covariant torques,µ[u i (q, p)] = max {µ k[ū i (q, p)]}.k(3) Defuzzification of the fuzzy controls µ[u i (q, p)] with the ‘center of gravity’methodu i (q, p) =∫µ[ui (q, p)] du i∫dui,to update the crisp feedback–control 1−forms u i = u i (t, q, p). Theserepresent the cortical FC–level corrections to the covariant torques F i =F i (t, q, p).Operationally, the construction of the cortical ̂(q, p)−-command spaceI N q × I N p and the 2ND feedback map Ξ (4.95–4.96), mimic the regulationof locomotor conditioned reflexes by the motor cortex [Houk et al. (1996)],giving the cortical correction to the covariant driving torques F i . Togetherthey form the nonlinear loop functor N L = Ξ[L] : ST ⇒ ST ∗ .A sample output from the leading human–motion simulator, HumanBiodynamics Engine (developed by the authors in Defence Science & TechnologyOrganisation, Australia), is given in Figure 4.9, giving the sophisticated264 DOF analysis of adult male running with the speed of 5 m/s.