Ivancevic_Applied-Diff-Geom

224 Applied Differential Geometry: A Modern IntroductionReduction of the Rotational Biodynamical ManifoldThe biodynamical configuration manifold M (Figure 3.6) can be (forthe sake of the brain–like motor control) reduced to N−-torus T N , in threesteps, as follows.First, a single three–axial SO(3)−joint can be reduced to the directproduct of three uniaxial SO(2)−joints, in the sense that three hinge jointscan produce any orientation in space, just as a ball–joint can. Algebraically,this means reduction (using symbol ‘’) of each of the three SO(3) rotationmatrices to the corresponding SO(2) rotation matrices⎛⎞1 0 0 ( )⎝ 0 cos φ − sin φ ⎠ cos φ − sin φsin φ cos φ0 sin φ cos φ⎛⎞cos ψ 0 sin ψ (⎝ 0 1 0 ⎠cos ψ− sin ψ cos ψ− sin ψ 0 cos ψ)sin ψ⎛⎞cos θ − sin θ 0( )⎝ sin θ cos θ 0 ⎠ cos θ − sin θsin θ cos θ0 0 1In this way we can set the reduction equivalence relation SO(3) SO(2) ✄ SO(2) ✄ SO(2), where ‘✄’ denotes the noncommutative semidirectproduct (see (3.8.4.2) above).Second, we have a homeomorphism: SO(2) ∼ S 1 , where S 1 denotes theconstrained unit circle in the complex plane, which is an Abelian Lie group.Third, let I N be the unit cube [0, 1] N in R N and ‘∼’ an equivalencerelation on R N get by ‘gluing’ together the opposite sides of I N , preservingtheir orientation. The manifold of human–body configurations (Figure 3.6)can be represented as the quotient space of R N by the space of the integrallattice points in R N , that is a constrained ND torus T N (3.221),R N /Z N = I N / ∼ ∼ =N∏Si 1 ≡ {(q i , i = 1, . . . , N) : mod 2π} = T N . (3.58)i=1Since S 1 is an Abelian Lie group, its N−-fold tensor product T N is also anAbelian Lie group, the toral group, of all nondegenerate diagonal N × Nmatrices. As a Lie group, the biodynamical configuration space M ≡ T N