By Samuel Eilenberg
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Additional resources for Automata, languages, and machines
If so, explain how this can occur. 2 RIGID MOTIONS AND HOMOGENEOUS TRANSFORMATIONS A large part of robot kinematics is concerned with the establishment of various coordinate systems to represent the positions and orientations of rigid objects, and with transformations among these coordinate systems. Indeed, the geometry of three-dimensional space and of rigid motions plays a central role in all aspects of robotic manipulation. 1 Homogeneous transformations combine the operations of rotation and translation into a single matrix multiplication, and are used in Chapter 3 to derive the so-called forward kinematic equations of rigid manipulators.
Note that the rotations themselves are not performed in reverse order. Rather they are performed about the fixed frame instead of about the current frame. To see why this is so, suppose we have two frames o0 x0 y0 z0 and o1 x1 y1 z1 related by the rotational transformation R10 . 1 that the representation for R in the current frame o1 x1 y1 z1 is given by (R10 )−1 RR10 . 20) COMPOSITION OF ROTATIONS z1 z0 z0 φ + x0 x1 z1 θ y0 Fig. 10 x1 y1 y0 z0 z2 = x0 45 x0 φ θ x1 x 2 y2 y0 , y1 Composition of rotations about fixed axes.
1, is not defined since the frames o0 x0 y0 and o1 x1 y1 are not parallel. Thus, we see a clear need, not only for a representation system that allows points to be expressed with respect to various coordinate systems, but also for a mechanism that allows us to transform the coordinates of points that are expressed in one coordinate system into the appropriate coordinates with 32 RIGID MOTIONS AND HOMOGENEOUS TRANSFORMATIONS y0 y1 x1 sin θ θ x0 o0 , o1 cos θ Fig. 2 Coordinate frame o1 x1 y1 is oriented at an angle θ with respect to o0 x0 y0 .
Automata, languages, and machines by Samuel Eilenberg