Useful properties of rotation matrices 2 since each

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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. •R-1 = 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 D-H transformation matrix for adjacent coordinate frames, i and i-1. The position and orientation of the i-th frame coordinate can be expressed in the (i-1)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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