Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 215corresponding respectively to rotations about x−axis by an angle ϕ, abouty−axis by an angle ψ, and about z−axis by an angle θ.The total three–axial joint rotation A is defined as the product of aboveone–parameter rotations R ϕ , R ψ , R θ , i.e., A = R ϕ · R ψ · R θ is equal⎡⎤cos ψ cos ϕ − cos θ sin ϕ sin ψ cos ψ cos ϕ + cos θ cos ϕ sin ψ sin θ sin ψA = ⎣ − sin ψ cos ϕ − cos θ sin ϕ sin ψ − sin ψ sin ϕ + cos θ cos ϕ cos ψ sin θ cos ψ ⎦ .sin θ sin ϕ − sin θ cos ϕ cos θHowever, the order of these matrix products matters: different order productsgive different results, as the matrix product is noncommutative product.This is the reason why Hamilton’s quaternions 5 are today commonly usedto parameterize the SO(3)−group, especially in the field of 3D computergraphics.The one–parameter rotations R ϕ , R ψ , R θ define curves in SO(3) startingfrom I 3 = 0 1 0 . Their derivatives in ϕ = 0, ψ = 0 and θ = 0 belong( ) 1 0 00 0 1to the associated tangent Lie algebra so(3). That is the correspondinginfinitesimal generators of joint rotations – joint angular velocitiesv ϕ , v ψ , v θ ∈ so(3) – are respectively given byv ϕ =v θ =[ ] 0 0 00 0 −10 1 0[ ] 0 −1 01 1 00 0 0= −y ∂ ∂z + z ∂ ∂y , v ψ == −x ∂ ∂y + y ∂∂x .Moreover, the elements are linearly independent and so⎧⎡⎤ ⎫⎨ 0 −a b⎬so(3) = ⎣ a 0 −γ ⎦ |a, b, γ ∈ R⎩⎭ .−b γ 0[ ] 0 0 10 0 0 = −z ∂−1 0 0 ∂x + x ∂ ∂z ,The Lie algebra so(3) is identified with R 3 by associating to each v =5 Recall that the set of Hamilton’s quaternions H represents an extension of the setof complex numbers C. We can compute a rotation about the unit vector, u by an angleθ. The quaternion q that computes this rotation is„q = cos θ 2 , u sin θ «.2