Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 213corresponding joint group generator v sR q s = exp(q s v s ). (3.53)The exponential map (3.53) represents a solution of the joint operator differentialequation in the joint group–parameter space {q s }dR q sdq s = v s R q s.Uniaxial Group of Joint RotationsThe uniaxial joint rotation in a single Cartesian plane around a perpendicularaxis, e.g., xy−plane about the z axis, by an internal joint angle θ,leads to the following transformation of the joint coordinatesẋ = x cos θ − y sin θ, ẏ = x sin θ + y cos θ.In this way, the joint SO(2)−group, given by{ ( ) }cos θ − sin θSO(2) = R θ =|θ ∈ [0, 2π] ,sin θ cos θacts in a canonical way on the Euclidean plane R 2 by(( ) ( ))cos θ − sin θ xSO(2) =, ↦−→sin θ cos θ yIts associated Lie algebra so(2) is given by{( ) }0 −tso(2) = |t ∈ R ,t 0( x cos θ −y sin θx sin θ y cos θsince the curve γ θ ∈ SO(2) given by( )cos tθ − sin tθγ θ : t ∈ R ↦−→ γ θ (t) =∈ SO(2),sin tθ cos tθpasses through the identity I 2 =ddt∣ γ θ (t) =t=0( ) 1 0and then0 1( ) 0 −θ,θ 0so that I 2 is a basis of so(2), since dim (SO(2)) = 1.).