The homogeneous transformation matrix for 3D bodies

As in the 2D case, a homogeneous transformation matrix can be defined. For the 3D case, a $4 \times 4$ matrix is obtained that performs the rotation given by $R(\alpha,\beta,\gamma)$, followed by a translation given by $x_t,y_t,z_t$. The result is

$$T = {\small \begin{pmatrix}\cos\alpha \cos\beta & \cos\alpha \sin... ...ta \sin\gamma & \cos\beta \cos\gamma & z_t 0 & 0 & 0 & 1 \end{pmatrix} } .$$ (3.50)

Once again, the order of operations is critical. The matrix $T$ in (3.50) represents the following sequence of transformations:

1. Roll by $\gamma$		3. Yaw by $\alpha$
2. Pitch by $\beta$		4. Translate by $(x_t,y_t,z_t)$.

The robot primitives can be transformed to yield ${\cal A}(x_t,y_t,z_t,\alpha,\beta,\gamma)$. A 3D rigid body that is capable of translation and rotation therefore has six degrees of freedom.