A Kinematic Tree

Many kinematic structures consist of a ``tree'' of rigid bodies, as opposed to a chain, as shown in Figure 2.13.a. The human body, with its joints and limbs attached to the torso, provides another example that can be modeled as a tree of rigid links. The position and orientation of bodies in a tree can be handled in the same way as those in a chain.

A single link can serve as the root of the tree. The position and orientation of a link, ${\cal A}_i$ is determined by applying the product of homogeneous matrices (2.12) or (2.14) along the sequence of links from ${\cal A}_i$ to the root link. All other branches in the three should be ignored when assigning the DH parameters.

Suppose that for the case of a 2D tree of bodies, all child links of ${\cal A}_i$ are attached the same joint. In this case the a_i parameter may be assigned in the usual way. If the child links are attached in at different joints, then the situation becomes more complicated. For each path in the tree, the chain of links can be handled in the usual way by assigning a distinct a parameter for each joint that connects a child. The required local frame for defining ${\cal A}_i$ will be different for each joint because the X axis must pass through the joint that connects the child link. One way to fix this problem is to abandon (2.12), and define a special homogeneous transformation matrix for each case. The situation becomes even more complicated for a 3D tree of bodies.

 

Figure 2.13:   General linkages: a) Instead of a chain of rigid bodies, a ``tree'' of rigid bodies can be considered; b) if there are loops, the problem becomes much more difficult because configurations must be assigned in a consistent way.