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By Mark W. Spong
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It is coincident with the frame o0 x0 y0 z0 . After the rotation by π, the block’s coordinate frame, which is rigidly attached to the block, is also rotated by π. If we denote this rotated frame by o1 x1 y1 z1 , we obtain −1 0 0 R10 = Rz,π = 0 −1 0 0 0 1 In the local coordinate frame o1 x1 y1 z1 , the point pb has the coordinate representation p1b . 9), giving p0b = Rz,π p1b The key thing to notice is that the local coordinates, p1b , of the corner of the block do not change as the block rotates, since they are defined in terms of the block’s own coordinate frame.
54) to mean that two axis/angle representations may be combined using standard rules of vector algebra as doing so would imply that rotations commute which, as we have seen, in not true in general. 6 RIGID MOTIONS We have seen how to represent both positions and orientations. We combine these two concepts in this section to define a rigid motion and, in the next section, we derive an efficient matrix representation for rigid motions using the notion of homogeneous transformation. 1 A rigid motion is an ordered pair (d, R) where d ∈ R3 and R ∈ SO(3).
Of course, in order to assign coordinates it is necessary to specify a coordinate frame. 1. We could specify the coordinates of the point p with respect to either frame o0 x0 y0 or frame o1 x1 y1 . 2)T . So that the reference frame will always be clear, we will adopt a notation in which a superscript is used to denote the reference frame. 2 Geometrically, a point corresponds to a specific location in space. We stress here that p is a geometric entity, a point in space, while both p0 and p1 are coordinate vectors that represent the location of this point in space with respect to coordinate frames o0 x0 y0 and o1 x1 y1 , respectively.