Ivancevic_Applied-Diff-Geom

212 Applied Differential Geometry: A Modern IntroductionHence γ A is an integral curve of the left–invariant vector–field X A . Therefore,the exponential map is given byexp : A ∈ L(R n , R n ) ↦→ exp(A) ≡ e A = γ A (1) =∞∑i=0For each A ∈ Gl(n, R) the corresponding adjoint mapA ii!∈ Gl(n, R).Ad A : L(R n , R n ) → L(R n , R n )is given byAd A B = A · B · A −1 .3.8.4 Application: Lie Groups in Biodynamics3.8.4.1 Lie Groups of Joint RotationsRecall (see [Ivancevic and Ivancevic (2006)]) that local kinematics at eachrotational robot or (synovial) human joint, is defined as a group action ofan nD constrained rotational Lie group SO(n) on the Euclidean space R n .In particular, there is an action of SO(2)−-group in uniaxial human joints(cylindrical, or hinge joints, like knee and elbow) and an action of SO(3)−groupin three–axial human joints (spherical, or ball–and–socket joints, likehip, shoulder, neck, wrist and ankle). In both cases, SO(n) acts, with itsoperators of rotation, on the vector x = {x µ }, (i = 1, 2, 3) of external,Cartesian coordinates of the parent body–segment, depending, at the sametime, on the vector q = {q s }, (s = 1, · · · , n) on n group–parameters, i.e.,joint angles.Each joint rotation R ∈ SO(n) defines a mapR : x µ ↦→ ẋ µ , R(x µ , q s ) = R q sx µ ,where R q s ∈ SO(n) are joint group operators. The vector v = {v s }, (s =1, · · · , n) of n infinitesimal generators of these rotations, i.e., joint angularvelocities, given byv s = −[ ∂R(xµ , q s )∂q s ] q=0∂∂x µ ,constitute an nD Lie algebra so(n) corresponding to the joint rotation groupSO(n). Conversely, each joint group operator R q s, representing a one–parameter subgroup of SO(n), is defined as the exponential map of the