New metrics for rigid body motion interpolation - helix

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So, right invariance is guaranteed under the condition that BWB T = W , or equivalently, W commutes with all the elements B 2 SO(n), which is equivalent to W = I. Remark 3.2 If right invariance on SO(n) is desired, we can define 2 > W;A = Tr(XY T W )=Tr(Y T WX)=Tr(WXY T ) Metric will be right invariant on SO(n) for W symmetric and positive definite and biinvariant if W = I. The proof is similar to that for metric and is omitted. 3.2 The induced metric on SO(3) Even though the following derivation can be done in the general case of a n(n , 1)=2-dimensional manifold SO(n) in the ambient n 2 -dimensional manifold GL(n; IR), we will restrict to the case n = 3 for practical purposes. Some general results will be presented and proved for the general case though. Let R be an arbitrary element from SO(3). LetX; Y be two vectors from T R SO(3) and R x (t);R y (t) the corresponding local flows so that X = R _ x (0); Y = R _ y (0); R x (0) = R y (0) = R: The metric inherited from GL(3; IR) can be written as: W;R = Tr( _ R T x (0) _ R y (0)W )=Tr( _ R T x (0)RRT _R y (0)W )=Tr(^! T x ^! yW ) where ^! x = R x (0) T _R x (0) and ^! y = R y (0) T _R y (0) are the corresponding twists from the Lie algebra so(3). If we write the above relation using the vector form of the twists, some elementary algebra leads to: W;R = ! x T G! y (11) where G = Tr(W )I 3 , W (12) is the matrix of the metric on SO(3) as defined by (8). A different equivalent way of arriving at the expression of G would be defining the metric in so(3) (i.e. at identity of SO(3)) asbeingthe one inherited from T I GL(3; IR): g ij = Tr(L o i T L o j W ); i; j = 1; 2; 3 (Lo 1 ;Lo 2 ;Lo 3 is the basis in 10
so(3)). Left translating this metric throughout the manifold is equivalent to inheriting the metric at each 3-dimensional tangent space of SO(3) from the corresponding 9-dimensional tangent space of GL(3; IR). Proposition 3.3 G is symmetric if and only if W is symmetric. If W is positive definite, then G is positive definite. If G is positive definite, then W is positive definite if and only the eigenvalues of G are the lengths of the sides of a triangle. Proof: The first part follows immediately from (12). For the second part, note that the eigenvalues of G are given by i (G) = X j (W ) , i (W )= X j j6=i Because W is positive definite, it follows that i (W ) > 0 which implies i (G) > 0, i.e. Gis positive definite. For the third part, we start from the equivalent form of (12) W = 1 Tr(G)I 3 , G 2 j W which implies and the claim is proved. i (W )= 1 2 X j j (G) , i (G) Remark 3.4 In the particular case when W = I; > 0, from (12), we have G =2I, whichis the standard bi-invariant metric on SO(3). This is consistent with the second assertion in Proposition 3.1. For =1, metric (10) will induce the well known Frobenius norm on GL(3; IR). 3.3 The induced metric in SE(3) Similarly, we can get a left invariant metric on SE(n , 1) by inheriting the metric W given ~ by (10) from GL(n; IR). For practical purposes, we will assume the following form for the n n matrix of the metric in (10): ~W = 2 4 W 0 0 w 3 5 (13) 11
Page 1 and 2: New metrics for rigid body motion i
Page 3 and 4: otations SO(3) by representing the
Page 5 and 6: {F} z z’ {M} t x’ 3 {M} z’ y
Page 7 and 8: A vector field generated by Equatio
Page 9: metric (i.e. the kinetic energy) at
Page 13 and 14: Remark 3.6 If W is symmetric and po
Page 15 and 16: Remark 4.5 For the case W = I, it i
Page 17 and 18: e an arbitrary element from SE(n ,
Page 19 and 20: The proof is rather involved and ca
Page 21 and 22: (a) (b) 1 0.75 0.5 0.25 0 1 Displac
Page 23 and 24: 1.6 1.4 1.2 True Projection True Pr
Page 25 and 26: 1.6 1.6 1.6 1.4 1.2 True Projection
Page 27 and 28: By use of the Rodrigues formula aga

New metrics for rigid body motion interpolation - helix

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