En robòtica, les matrius de transformació \( \mathbf{T}_i \) per a cada junta es poden expressar en termes de les funcions trigonomètriques \( \sin \) i \( \cos \). Anem a calcular la matriu de transformació final \( \mathbf{T}_6 \) com la multiplicació de les matrius de transformació \( \mathbf{T}_1 \) a \( \mathbf{T}_6 \).
La matriu de transformació per a cada junta \( i \) és:
\[ \mathbf{T}_i = \begin{pmatrix} \cos\theta_i & -\sin\theta_i \cos\alpha_i & \sin\theta_i \sin\alpha_i & a_i \cos\theta_i \\ \sin\theta_i & \cos\theta_i \cos\alpha_i & -\cos\theta_i \sin\alpha_i & a_i \sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
Com ja vam calcular anteriorment, el producte de les matrius \( \mathbf{T}_1 \) i \( \mathbf{T}_2 \) dona:
\[ \mathbf{T}_{12} = \begin{pmatrix} \cos(\theta_1 + \theta_2) & -\sin(\theta_1 + \theta_2) & 0 & a_1 \cos\theta_1 + a_2 \cos(\theta_1 + \theta_2) \\ \sin(\theta_1 + \theta_2) & \cos(\theta_1 + \theta_2) & 0 & a_1 \sin\theta_1 + a_2 \sin(\theta_1 + \theta_2) \\ 0 & 0 & 1 & d_1 + d_2 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
Multiplicant \( \mathbf{T}_{12} \) amb \( \mathbf{T}_3 \) obtenim:
\[ \mathbf{T}_{123} = \mathbf{T}_{12} \times \begin{pmatrix} \cos\theta_3 & -\sin\theta_3 \cos\alpha_3 & \sin\theta_3 \sin\alpha_3 & a_3 \cos\theta_3 \\ \sin\theta_3 & \cos\theta_3 \cos\alpha_3 & -\cos\theta_3 \sin\alpha_3 & a_3 \sin\theta_3 \\ 0 & \sin\alpha_3 & \cos\alpha_3 & d_3 \\ 0 & 0 & 0 & 1 \end{pmatrix} \] \]
Desenvolupant la multiplicació, obtenim:
\[ \mathbf{T}_{123} = \begin{pmatrix} \cos(\theta_{123}) & -\sin(\theta_{123}) \cos\alpha_3 & \sin(\theta_{123}) \sin\alpha_3 & x_{123} \\ \sin(\theta_{123}) & \cos(\theta_{123}) \cos\alpha_3 & -\cos(\theta_{123}) \sin\alpha_3 & y_{123} \\ 0 & \sin\alpha_3 & \cos\alpha_3 & z_{123} \\ 0 & 0 & 0 & 1 \end{pmatrix} \] \]
On:
\[ \theta_{123} = \theta_1 + \theta_2 + \theta_3 \] \]
\[ x_{123} = a_1 \cos\theta_1 + a_2 \cos(\theta_1 + \theta_2) + a_3 \cos(\theta_1 + \theta_2 + \theta_3) \]
\[ y_{123} = a_1 \sin\theta_1 + a_2 \sin(\theta_1 + \theta_2) + a_3 \sin(\theta_1 + \theta_2 + \theta_3) \]
\[ z_{123} = d_1 + d_2 + d_3 \]
Multiplicant \( \mathbf{T}_{123} \) amb \( \mathbf{T}_4 \) obtenim:
\[ \mathbf{T}_{1234} = \mathbf{T}_{123} \times \begin{pmatrix} \cos\theta_4 & -\sin\theta_4 \cos\alpha_4 & \sin\theta_4 \sin\alpha_4 & a_4 \cos\theta_4 \\ \sin\theta_4 & \cos\theta_4 \cos\alpha_4 & -\cos\theta_4 \sin\alpha_4 & a_4 \sin\theta_4 \\ 0 & \sin\alpha_4 & \cos\alpha_4 & d_4 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
Desenvolupant la multiplicació, obtenim:
\[ \mathbf{T}_{1234} = \begin{pmatrix} \cos(\theta_{1234}) & -\sin(\theta_{1234}) \cos\alpha_4 & \sin(\theta_{1234}) \sin\alpha_4 & x_{1234} \\ \sin(\theta_{1234}) & \cos(\theta_{1234}) \cos\alpha_4 & -\cos(\theta_{1234}) \sin\alpha_4 & y_{1234} \\ 0 & \sin\alpha_4 & \cos\alpha_4 & z_{1234} \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
On:
\[ \theta_{1234} = \theta_1 + \theta_2 + \theta_3 + \theta_4 \]
\[ x_{1234} = x_{123} + a_4 \cos(\theta_{1234}) \]
\[ y_{1234} = y_{123} + a_4 \sin(\theta_{1234}) \]
\[ z_{1234} = z_{123} + d_4 \]
Multiplicant \( \mathbf{T}_{1234} \) amb \( \mathbf{T}_5 \) obtenim:
\[ \mathbf{T}_{12345} = \mathbf{T}_{1234} \times \begin{pmatrix} \cos\theta_5 & -\sin\theta_5 \cos\alpha_5 & \sin\theta_5 \sin\alpha_5 & a_5 \cos\theta_5 \\ \sin\theta_5 & \cos\theta_5 \cos\alpha_5 & -\cos\theta_5 \sin\alpha_5 & a_5 \sin\theta_5 \\ 0 & \sin\alpha_5 & \cos\alpha_5 & d_5 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
Desenvolupant la multiplicació, obtenim:
\[ \mathbf{T}_{12345} = \begin{pmatrix} \cos(\theta_{12345}) & -\sin(\theta_{12345}) \cos\alpha_5 & \sin(\theta_{12345}) \sin\alpha_5 & x_{12345} \\ \sin(\theta_{12345}) & \cos(\theta_{12345}) \cos\alpha_5 & -\cos(\theta_{12345}) \sin\alpha_5 & y_{12345} \\ 0 & \sin\alpha_5 & \cos\alpha_5 & z_{12345} \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
On:
\[ \theta_{12345} = \theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5 \]
\[ x_{12345} = x_{1234} + a_5 \cos(\theta_{12345}) \]
\[ y_{12345} = y_{1234} + a_5 \sin(\theta_{12345}) \]
\[ z_{12345} = z_{1234} + d_5 \]
Finalment, multipliquem \( \mathbf{T}_{12345} \) amb \( \mathbf{T}_6 \) per obtenir la matriu final de transformació \( \mathbf{T}_6 \):
\[ \mathbf{T}_6 = \mathbf{T}_{12345} \times \begin{pmatrix} \cos\theta_6 & -\sin\theta_6 \cos\alpha_6 & \sin\theta_6 \sin\alpha_6 & a_6 \cos\theta_6 \\ \sin\theta_6 & \cos\theta_6 \cos\alpha_6 & -\cos\theta_6 \sin\alpha_6 & a_6 \sin\theta_6 \\ 0 & \sin\alpha_6 & \cos\alpha_6 & d_6 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
El resultat final és:
\[ \mathbf{T}_6 = \begin{pmatrix} \cos(\theta_{123456}) & -\sin(\theta_{123456}) \cos\alpha_6 & \sin(\theta_{123456}) \sin\alpha_6 & x_{123456} \\ \sin(\theta_{123456}) & \cos(\theta_{123456}) \cos\alpha_6 & -\cos(\theta_{123456}) \sin\alpha_6 & y_{123456} \\ 0 & \sin\alpha_6 & \cos\alpha_6 & z_{123456} \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
On:
\[ \theta_{123456} = \theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5 + \theta_6 \]
\[ x_{123456} = x_{12345} + a_6 \cos(\theta_{123456}) \]
\[ y_{123456} = y_{12345} + a_6 \sin(\theta_{123456}) \]
\[ z_{123456} = z_{12345} + d_6 \]