Explain the importance of homogeneous coordinates in modeling of robotic manipulators.
Answers
Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as translation, rotation, scaling and perspective projection to be represented as a matrix by which the vector is multiplied.
Hope it helps you. Plz mark this as brainliest bcoz I need only two brainliest answers to gain my next rank. Plz mark it plz plz plz plz.
The homogeneous transformation matrix
The transformation $ T_i$, for each $ i$ such that $ 1 < i \leq m$, is
$\displaystyle T_i = Q_{i-1} R_i = \begin{pmatrix}\cos\theta_i & -\sin\theta_i &...
...{i-1} & \cos\alpha_{i-1} & \cos\alpha_{i-1}d_i 0 & 0 & 0 & 1 \end{pmatrix} .$ (3.56)
This can be considered as the 3D counterpart to the 2D transformation matrix, (3.52). The following four operations are performed in succession:
Translate by $ d_i$ along the $ z_i$-axis.
Rotate counterclockwise by $ \theta _i$ about the $ z_i$-axis.
Translate by $ a_{i-1}$ along the $ x_{i-1}$-axis.
Rotate counterclockwise by $ \alpha _{i-1}$ about the $ x_{i-1}$-axis.
As in the 2D case, the first matrix, $ T_1$, is special. To represent any position and orientation of $ {\cal A}_1$, it could be defined as a general rigid-body homogeneous transformation matrix, (3.50). If the first body is only capable of rotation via a revolute joint, then a simple convention is usually followed. Let the $ a_0,\alpha_0$ parameters of $ T_1$ be assigned as $ a_0 = \alpha_0 = 0$ (there is no $ z_0$-axis). This implies that $ Q_0$ from (3.55) is the identity matrix, which makes $ T_1 =
R_1$.
The transformation $ T_i$ for $ i
> 1$ gives the relationship between the body frame of $ {\cal A}_i$ and the body frame of $ {\cal A}_{i-1}$. The position of a point $ (x,y,z)$ on $ {\cal A}_m$ is given by
$\displaystyle T_1 T_2 \cdots T_m \begin{pmatrix}x y z 1 \end{pmatrix} .$
HOPE THIS HELPS.
GOOD LUCK.
MARK AS BRAINLIEST.