Closed Kinematic Chains

If there are closed loops of rigid bodies, as shown in Figure 2.13.b, the problem becomes much more difficult because a set of constraints must be maintained. For example, suppose we take a 2D chain of rigid bodies and attach ${\cal A}_m$ to ${\cal A}_1$ with a revolute joint. We are no longer free to choose $\theta_1$, $\theta_2$,$\ldots$, $\theta_m$ independently. Many choices will cause ${\cal A}_m$ to become detached ${\cal A}_1$ to become detached. The solution to this problem can be considered as a special case of the inverse kinematics problem in robotics.