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 to . Assume that the point in the body frame of is permanently fixed to a point in the body frame of . This implies that the translation of is completely determined once and are given. Note that and depend on , , and . This implies that and have a total of four degrees of freedom when attached. The independent parameters are , , , and . The task in the remainder of this section is to determine exactly how the models of , , , are transformed when they are attached in a chain, and to give the expressions in terms of the independent parameters.
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, . 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 and for each . For convenience that will soon become evident, the -axis for the body frame of is defined as the line through the two joints that lie in , as shown in Figure 3.10. For the last link, , the -axis can be placed arbitrarily, assuming that the origin is placed at the joint that connects to . The body frame for the first link, , can be placed using the same considerations as for a single rigid body.
Steven M LaValle 2012-04-20