Attaching bodies

When bodies are attached in a kinematic chain, degrees of freedom are removed. Figure 3.9 shows two different ways in which a pair of 2D links can be attached. The place at which the links are attached is called a joint. For a revolute joint, one link is capable only of rotation with respect to the other. For a prismatic joint is shown, one link slides along the other. Each type of joint removes two degrees of freedom from the pair of bodies. For example, consider a revolute joint that connects ${\cal A}_1$ to ${\cal A}_2$. Assume that the point $(0,0)$ in the body frame of ${\cal A}_2$ is permanently fixed to a point $(x_a,y_a)$ in the body frame of ${\cal A}_1$. This implies that the translation of ${\cal A}_2$ is completely determined once $x_a$ and $y_a$ are given. Note that $x_a$ and $y_a$ depend on $x_1$, $y_1$, and $\theta_1$. This implies that ${\cal A}_1$ and ${\cal A}_2$ have a total of four degrees of freedom when attached. The independent parameters are $x_1$, $y_1$, $\theta_1$, and $\theta_2$. The task in the remainder of this section is to determine exactly how the models of ${\cal A}_1$, ${\cal A}_2$, $\ldots$, ${\cal A}_m$ are transformed when they are attached in a chain, and to give the expressions in terms of the independent parameters.

Figure 3.9: Two types of 2D joints: a revolute joint allows one link to rotate with respect to the other, and a prismatic joint allows one link to translate with respect to the other.

Consider the case of a kinematic chain in which each pair of links is attached by a revolute joint. The first task is to specify the geometric model for each link, ${\cal A}_i$. Recall that for a single rigid body, the origin of the body frame determines the axis of rotation. When defining the model for a link in a kinematic chain, excessive complications can be avoided by carefully placing the body frame. Since rotation occurs about a revolute joint, a natural choice for the origin is the joint between ${\cal A}_i$ and ${\cal A}_{i-1}$ for each $i > 1$. For convenience that will soon become evident, the $x_i$-axis for the body frame of ${\cal A}_i$ is defined as the line through the two joints that lie in ${\cal A}_i$, as shown in Figure 3.10. For the last link, ${\cal A}_m$, the $x_m$-axis can be placed arbitrarily, assuming that the origin is placed at the joint that connects ${\cal A}_m$ to ${\cal A}_{m-1}$. The body frame for the first link, ${\cal A}_1$, can be placed using the same considerations as for a single rigid body.

Figure 3.10: The body frame of each ${\cal A}_i$, for $1 < i < m$, is based on the joints that connect ${\cal A}_i$ to ${\cal A}_{i-1}$ and ${\cal A}_{i+1}$.