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Fundamentals_of_Robotic_Mechanical_Systems Part 3

Tham khảo tài liệu 'fundamentals_of_robotic_mechanical_systems part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 46 2. Mathematical Background Moreover the Euler-Rodrigues parameters should not be confused with the Euler angles which are not invariant and hence admit multiple definitions. The foregoing means that no single set of Euler angles exists for a given rotation matrix the said angles depending on how the rotation is decomposed into three simpler rotations. For this reason Euler angles will not be stressed here. The reader is referred to Exercise 18 for a short discussion of Euler angles Synge 1960 includes a classical treatment while Kane Likins and Levinson provide an extensive discussion of the same. Example 2.3.9 Find the Euler-Rodrigues parameters of the proper orthogonal matrix Q given as .-122 Q 2 -1 2 đ 2 2 -1 Solution Since the given matrix is symmetric its angle of rotation is 7T and its vector linear invariant vanishes which prevents US from finding the direction of the axis of rotation from the linear invariants moreover expressions 2.77 do not apply. However we can use eq. 2.49 to find the unit vector e parallel to the axis of rotation i.e. eeT 1 1 Q or in component form e. e.e. e. e. 1 1 1 1 e. e. e . e. e. 3 1 1 r e. e. e. e. e . Ill A simple inspection of the components of the two sides of the above equation reveals that all three components of e are identical and moreover of the same sign but we cannot tell which sign this is. Therefore e 1 1 1 Moreover from the symmetry of Q we know that ộ 7T and hence r e sin 1 2 3 1 r. cos 4 2 2.4 Composition of Reflections and Rotations 47 2.4 Composition of Reflections and Rotations As pointed out in Section 2.2 reflections occur often accompanied by rotations. The effect of this combination is that the rotation destroys the two properties of pure reflections symmetry and self-inversion as defined in Section 2.2. Indeed let R be a pure reflection taking on the form appearing in eq. 2.5 and Q an arbitrary rotation taking on the form of eq. 2.48 . The product of these two transformations QR denoted by T is .