Homogeneous transformation matrices for 2D chains

We are now prepared to determine the location of each link. The location in ${\cal W}$ of a point in $(x,y) \in {\cal A}_1$ is determined by applying the 2D homogeneous transformation matrix (3.35),

$$T_1 = \begin{pmatrix}\cos\theta_1 & -\sin\theta_1 & x_t \sin\theta_1 & \cos\theta_1 & y_t 0 & 0 & 1 \end{pmatrix}.$$ (3.51)

As shown in Figure 3.10, let $a_{i-1}$ be the distance between the joints in ${\cal A}_{i-1}$. The orientation difference between ${\cal A}_i$ and ${\cal A}_{i-1}$ is denoted by the angle $\theta _i$. Let $T_i$ represent a $3 \times 3$ homogeneous transformation matrix (3.35), specialized for link ${\cal A}_i$ for $1 < i \leq m$,

$$T_i = \begin{pmatrix}\cos\theta_i & -\sin\theta_i & a_{i-1} \sin\theta_i & \cos\theta_i & 0 0 & 0 & 1 \end{pmatrix} .$$ (3.52)

This generates the following sequence of transformations:

1. Rotate counterclockwise by $\theta _i$.
2. Translate by $a_{i-1}$ along the $x$-axis.

The transformation $T_i$ expresses the difference between the body frame of ${\cal A}_i$ and the body frame of ${\cal A}_{i-1}$. The application of $T_i$ moves ${\cal A}_i$ from its body frame to the body frame of ${\cal A}_{i-1}$. The application of $T_{i-1} T_i$ moves both ${\cal A}_i$ and ${\cal A}_{i-1}$ to the body frame of ${\cal A}_{i-2}$. By following this procedure, the location in ${\cal W}$ of any point $(x,y) \in {\cal A}_m$ is determined by multiplying the transformation matrices to obtain

$$T_1 T_2 \cdots T_m \begin{pmatrix}x y 1 \end{pmatrix} .$$ (3.53)

Example 3..3 (A 2D Chain of Three Links)   To gain an intuitive understanding of these transformations, consider determining the configuration for link ${\cal A}_3$, as shown in Figure 3.11. Figure 3.11a shows a three-link chain in which ${\cal A}_1$ is at its initial configuration and the other links are each offset by $\pi/4$ from the previous link. Figure 3.11b shows the frame in which the model for ${\cal A}_3$ is initially defined. The application of $T_3$ causes a rotation of $\theta_3$ and a translation by $a_2$. As shown in Figure 3.11c, this places ${\cal A}_3$ in its appropriate configuration. Note that ${\cal A}_2$ can be placed in its initial configuration, and it will be attached correctly to ${\cal A}_3$. The application of $T_2$ to the previous result places both ${\cal A}_3$ and ${\cal A}_2$ in their proper configurations, and ${\cal A}_1$ can be placed in its initial configuration. $\blacksquare$

Figure 3.11: Applying the transformation $T_2 T_3$ to the model of ${\cal A}_3$. If $T_1$ is the identity matrix, then this yields the location in ${\cal W}$ of points in ${\cal A}_3$.

For revolute joints, the $a_i$ parameters are constants, and the $\theta _i$ parameters are variables. The transformed $m$th link is represented as ${\cal A}_m(x_t,y_t,\theta_1,\ldots,\theta_m)$. In some cases, the first link might have a fixed location in the world. In this case, the revolute joints account for all degrees of freedom, yielding ${\cal A}_m(\theta_1,\ldots,\theta_m)$. For prismatic joints, the $a_i$ parameters are variables, instead of the $\theta _i$ parameters. It is straightforward to include both types of joints in the same kinematic chain.