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Unformatted text preview: FUL PROPERTIES OF ROTATION
MATRICES:
1.Each column vector of the rotation matrix is
a representation of the rotated axis unit vector
expressed in terms of the axis unit vector of
the reference frame & each row vector is a
representation of the axis unit vector of
reference frame expressed in terms of the
rotated axis unit vectors of the OUVW frame. USEFUL PROPERTIES OF ROTATION MATRICES:
2. Since each row and column frame is a unit vector
representation, the magnitude of each row and
column should be equal to 1. This is a direct property of orthonormal coordinate systems.
The determinant of a rotation matrix is +1 for a right
handed coordinate system and 1 for a left handed
system. USEFUL PROPERTIES OF ROTATION MATRICES: 3. Since each row is a vector representation of
orthonormal vectors, the inner product (dot
product) of each row with each other row
equals zero. Similarly, the inner product of
each column with each other column equals
zero.
4.The inverse of a rotation matrix is the
transpose of the rotation matrix.
•R1 = RT & RRT = I3 A 3*3 rotation matrix does not give any
provision for translation and scaling,
A fourth component is introduced, the matrix
is then known as HOMOGENEOUS matrix
which consists of 4 sub matrices Identify 2 coordinate systems
►the fixed reference coordinate frame OXYZ
►the moving coordinate system OUVW The 4*4 homogeneous
matrix describes spatial
displacement between
these
2
coordinate
systems The 2 coordinate
systems can be
assigned to 2 ends
of a robot link Review
DH transformation matrix for adjacent coordinate
frames, i and i1.
The position and orientation of the ith frame coordinate can be
expressed in the (i1)th frame by the following 4 successive
elementary transformations:
i
Ti −1 = T ( zi −1 , d i ) R ( zi −1 , θ i )T ( xi , ai ) R ( xi , α i ) ⎡Cθ i
⎢ Sθ
=⎢ i
⎢0
⎢
⎣0 − C α i Sθ i Sα i Sθ i Cα i Cθ i
Sα i − Sα i C θ i
Cα i 0 0 ai Cθ i ⎤
ai Sθ i ⎥
⎥
di ⎥
⎥
1⎦
40 Rotation Matrices in 3D –
⎡ cos θ
R z = ⎢...
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 Spring '14
 Mechatronics

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