Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 585effector can be neglected, this gives a topological solution to the standardinverse kinematics problem.Analogously, all momenta ˆp i have their images as rectified momentaˆp i in the ˆp−-command space – selector I N p . Therefore, the total momentumphase–space manifold T ∗ T N gets its ‘cortical image’ as the ̂(q, p)−commandspace, a trivial 2ND bundle Iq N × Ip N .Now, the simplest way to perform the feedback FC on the cortical̂(q, p)−-command space IqN × Ip N , and also to mimic the cortical–like behavior[1,2], is to use the 2ND fuzzy–logic controller, in pretty muchthe same way as in popular ‘inverted pendulum’ examples [Kosko (1992);Kosko (1996)].We propose the fuzzy feedback–control map Ξ that maps all the rectifiedjoint angles and momenta into the feedback–control 1−formsΞ : (ˆq i (t), ˆp i (t)) ↦→ u i (t, q, p), (4.95)so that their corresponding universes of discourse, ˆM i = (ˆq max i − ˆq min i ),ˆP i = (ˆp maxi − ˆp mini ) and U i = (u maxi − u mini ), respectively, are mapped asΞ :N∏i=1N∏ N∏ˆMM i × ˆP i → U i . (4.96)i=1 i=1The 2N−-D map Ξ (4.95–4.96) represents a fuzzy inference system,defined by (adapted from [Ivancevic et al. (1999b)]):(1) Fuzzification of the crisp rectified and discretized angles, momenta andcontrols using Gaussian–bell membership functionsµ k (χ) = exp[− (χ − m k) 22σ k], (k = 1, 2, . . . , 9),where χ ∈ D is the common symbol for ˆq i , ˆp i and u i (q, p) andD is the common symbol for M i , ˆP i and i ; the mean values m k ofthe seven partitions of each universe of discourse D are defined asm k = λ k D + χ min , with partition coefficients λ k uniformly spanningthe range of D, corresponding to the set of nine linguistic variablesL = {NL, NB, NM, NS, ZE, P S, P M, P B, P L}; standard deviationsare kept constant σ k = D/9. Using the linguistic vector L, the 9 × 9FAM (fuzzy associative memory) matrix (a ‘linguistic phase–plane’), isheuristically defined for each human joint, in a symmetrical weighted