Ivancevic_Applied-Diff-Geom

222 Applied Differential Geometry: A Modern Introduction3.8.4.3 Group Structure of Biodynamical ManifoldPurely Rotational Biodynamical ManifoldKinematics of an n−-segment human–body chain (like arm, leg orspine) is usually defined as a map between external coordinates (usually,end–effector coordinates) x r (r = 1, . . . , n) and internal joint coordinatesq i (i = 1, . . . , N) (see [Ivancevic and Snoswell (2001); Ivancevic (2002);Ivancevic and Pearce (2001b); Ivancevic and Pearce (2001b); Ivancevic(2005)]). The forward kinematics are defined as a nonlinear map x r =x r (q i ) with a corresponding linear vector functions dx r = ∂x r /∂q i dq i ofdifferentials: and ẋ r = ∂x r /∂q i ˙q i of velocities. When the rank of theconfiguration–dependent Jacobian matrix J ≡ ∂x r /∂q i is less than n thekinematic singularities occur; the onset of this condition could be detectedby the manipulability measure. The inverse kinematics are defined converselyby a nonlinear map q i = q i (x r ) with a corresponding linear vectorfunctions dq i = ∂q i /∂x r dx r of differentials and ˙q i = ∂q i /∂x r ẋ r of velocities.Again, in the case of redundancy (n < N), the inverse kinematicproblem admits infinite solutions; often the pseudo–inverse configuration–control is used instead: ˙q i = J ∗ ẋ r , where J ∗ = J T (J J T ) −1 denotes theMoore–Penrose pseudo–inverse of the Jacobian matrix J.Humanoid joints, that is, internal coordinates q i (i = 1, . . . , N), constitutea smooth configuration manifold M, described as follows. Uniaxial,‘hinge’ joints represent constrained, rotational Lie groups SO(2) i cnstr,parameterized by constrained angles qcnstr i ≡ q i ∈ [qmin i , qi max]. Three–axial, ‘ball–and–socket’ joints represent constrained rotational Lie groupsSO(3) i cnstr, parameterized by constrained Euler angles q i = q φ icnstr (in thefollowing text, the subscript ‘cnstr’ will be omitted, for the sake of simplicity,and always assumed in relation to internal coordinates q i ).All SO(n)−-joints are Hausdorff C ∞ −-manifolds with atlases (U α , u α );in other words, they are paracompact and metrizable smooth manifolds,admitting Riemannian metric.Let A and B be two smooth manifolds described by smooth atlases(U α , u α ) and (V β , v β ), respectively. Then the family (U α × V β , u α × v β :U α × V β → R m × R n ) ( α, β) ∈ A × B is a smooth atlas for the directproduct A × B. Now, if A and B are two Lie groups (say, SO(n)), thentheir direct product G = A × B is at the same time their direct productas smooth manifolds and their direct product as algebraic groups, with the