Koordinat Sistemlerinin Birbirine Göre İfadesi

\(\begin{bmatrix} n_x & s_x & a_x & P_x \\ n_y & s_y & a_y & P_y \\ n_z & s_z & a_z & P_z \\ 0 & 0 & 0 & 1 \end{bmatrix}\)

n, s, a sonraki/hedef koordinat sistemi 'nin x', y', z' eksenleridir.

x, y, z ise önceki/referans koordinat sistemi 'nin x, y, z eksenleridir.

P ise sonraki/hedef koordinat sisteminin önceki/referans koordinat sistemine göre konumudur.

Döndürme Matrisleri

\( T_{x,\theta} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & cos \theta & -sin \theta & 0 \\ 0 & sin \theta & cos \theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \)

\( T_{y,\theta} = \begin{bmatrix} cos \theta & 0 & sin \theta & 0 \\ 0 & 1 & 0 & 0 \\ -sin \theta & 0 & cos \theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \)

\( T_{z,\theta} = \begin{bmatrix} cos \theta & -sin \theta & 0 & 0 \\ sin \theta & cos \theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \)

Öteleme Matrisi

\( T_{x,y,z} = \begin{bmatrix} 0 & 0 & 0 & x \\ 0 & 0 & 0 & y \\ 0 & 0 & 0 & z \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \)

Denavit - Hartenberg Notasyonuna Göre Dönüşüm Matrisi

\(^{i-1}T_i = T_{z,d} T_{z,\theta} T_{x,a} T_{x,\alpha}\)

\( = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & d_i \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} cos \theta_i & -sin \theta_i & 0 & 0 \\ sin \theta_i & cos \theta_i & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & a_i \\ 0 & 1& 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & cos \theta_i & -sin \theta_i & 0 \\ 0 & sin \theta_i & cos \theta_i & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \)

\( = \begin{bmatrix} cos \theta_i & -cos \alpha_i sin \theta_i & sin \alpha_i sin \theta_i & a_i cos \theta_i \\ sin \theta_i & cos \alpha_i cos \theta_i & -sin \alpha_i cos \theta_i & a_i sin \theta_i \\ 0 & sin \alpha_i & cos \alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix} \)


denavit hartenberghomojen transformasyon matrisirobotikrotationtranslation

Previous PostNext Post